diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzznzpl" "b/data_all_eng_slimpj/shuffled/split2/finalzznzpl" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzznzpl" @@ -0,0 +1,5 @@ +{"text":"\\section{INTRODUCTION}\n\n\nThe density of states (DoS): $d(E) = \\sum_i \\delta(E-E_i)$ of a quantum\nsystem -in other words, the density of eigenstates $E_i$ at a given energy $E$- plays a\nkey role in the field of ``quantum chaos''. Gutzwiller \\cit{Gut90} found\na semiclassical approximation for the oscillatory part of the DoS, in\nterms of the periodic orbits of the (chaotic) classical system. However\nthe DoS, generally, is not probed directly in experiments, as they\nmeasure an observable $I(E)$. Often the latter can be related to the DoS\nby a sort of Fermi golden rule:\n \\begin{equation} I(E) = \\sum_i \\langle \\psi_i |\n\\hat{\\cal A} | \\psi_i \\rangle \\delta(E-E_i) \\com{.} \n \\label{mat} \n \\end{equation} \n In other words, the measured quantity $I(E)$ is the DoS weighted by the\nexpectation value of an observable $\\hat{\\cal A}$ over the eigenstate $|\n\\psi_i \\rangle$ with eigenenergy $E_i$. \n\nExperiments probing such a weighted DoS include ,among others,\nspectroscopic studied of atoms in static fields \\cit{Hol88}, molecular\n(Franck-Condon) transitions \\cit{SchEn90} and electronic transport in\nmicrocavities \\cit{BJS93}. While the Periodic Orbits (POs) have offered a\npowerful tool for understanding these experiments, a quantitative\nanalysis requires one to go beyond the Gutzwiller formula. Semiclassical\nexpressions for the ``matrix elements'' may also be investigated\nusing the stationary phase approximation (SPA).\n\nThe first semiclassical approximation for matrix elements \\cit{EFMW92}\ninvolved POs, and in essence reduced to the Gutzwiller trace formula\n(GTF) of the density of states, weighted by the Wigner transform $A_W$ of\nthe observable $\\hat{\\cal A}$, evaluated along each PO.\nThis result was found by assuming that $A_W$ is smooth in phase space,\nand by neglecting it in the SPA.\nOn the other hand, the photoabsorption rate of the hydrogen atom was\nexpressed in terms of closed orbits (COs) passing through the nucleus\n\\cit{DuDe88}, because the observable $\\hat{\\cal A}$ is very localized.\nAnother semiclassical formula, for the conductance fluctuations in\nmicrostructures, involved ``angle orbits''\ndefined by an angle related to the width of the leads \\cit{BJS93}.\n\nHence the the type of contributing trajectories depends\nstrongly on the relative smoothness of $\\hat{\\cal A}$ and the\nsemiclassical Green's function used to represent the DoS. Further, the\ntype of semiclassical expressions also depended on the level of\napproximation used in the SPA integrations. For the semiclassical matrix\nelements \\cit{EFMW92}, the variations of $A_W$ in both the SPA condition\nand integrations were neglected. A more refined version developed for\nmolecular transitions \\cit{ZoAl93} also neglected them in the SPA\ncondition (which yielded periodic orbits), but included them in the\nintegrations. Finally, ``angle orbits'' were obtained by considering\n{\\em both} the observable and the semiclassical Green's function when\napplying the SPA.\n\nA broad range of situations involve Gaussian matrix elements, given\nby an observable which contains a projector on a Gaussian:\n molecular excitation from a low vibrational state \\cit{ZoAl93}, in the conductance of microcavities with\nparabolic leads \\cit{Nari98} as well as the tunnelling diode experiments\nwhich form the subject of the present study. \nThe Gaussian matrix elements, conveniently, yield fairly simple integrals, \nand also have a very clear localization length scale.\n\nThe resonant tunneling diode in tilted fields (RTD) ---which is a\nmesoscopic realization of quantum wells with tunneling barriers--- was\nintroduced recently as a new probe of quantum chaos \\cit{From94,Mul95}.\nThe oscillations of the measured current (as a function of the applied\nvoltage) were linked to unstable \\cit{From94} and stable \\cit{Mul95}\nperiodic oscillations, following a heuristic application of the\nGutzwiller trace formula \\cit{Gut90} and taking into account the\naccessibility of the periodic orbits (POs) to the tunneling electrons.\n\nThe RTD experiments generated considerable interest and prompted the\ndevelopment of a series of semiclassical theories. One study proposed\ntwo separate formulae (obtained by two different level of approximation\nin the integrations) using normal orbits starting and finishing\nperpendicular to the barrier \\cit{BR98}. Another study proposed a\nformula using periodic orbits \\cit{NSB98}, similar to the approach of\nZobay and Alber \\cit{ZoAl93}. However, it was shown \\cit{SMR98} that the\nnormal and periodic orbits could not give satisfactory results in all\nexperimental regimes, and that a new type of complex and non-periodic\ntrajectories (the saddle orbits) \\cit{SM98b} provided an accurate\nsemiclassical description. Subsequently, Closed orbits, as well as\norbits having a minimum transverse momentum were proposed \\cit{NS99} in\norder to achieve similar results using only real trajectories.\nHere we will review and also extend our study of RTD problem, in order\nto clarify previous work and to prescribe the best\nsemiclassical description for this type of problem. \n\nIn\nsection \\ref{sc} we derive a general semiclassical expression for\nGaussian matrix elements, expressed in terms of an {\\em arbitrary} type\nof orbits [eq. \\rf{gammagen}]. We outline the RTD experiments \nand show that they are a realization of Gaussian matrix elements.\n\n\nIn section \\ref{discu} we explain\ndifferent reasonable assumptions one can make regarding the localization of the\nGaussian observable in position or phase space. We show that different\nassumptions yield various types of contributing trajectories \n(POs, COs, normal orbits, etc.), and correspondingly\ndifferent expressions for the current formula. We approach the problem\nfrom three levels of approximation: $(i)$ no approximation, yielding\n{\\em saddle orbits} (SOs); $(ii)$ the intermediate formulae, where we\nneglect one term in the SPA condition; $(iii)$ the ``hard limit'' level,\nwhere we neglect one term in both the SPA condition and the SPA\nintegrations (for POs, this corresponds to the result of Eckhardt {\\em et al.}\n\\cit{EFMW92}).Our general formula enable us to reproduce easily any of\nthe five semiclassical theories of the current in the RTD that have been\nproposed in the literature \\cit{BR98,NSB98,NS99,SM98b}. We also propose\na new type of trajectories, we term {\\em minimal orbits},selected by\nrequiring that the gradient of the argument of the exponential is minimal\n(instead of zero as in the standard SPA). \n\nIn section \\ref{comp} we test the different formulae against quantum\nmechanical calculations and experimental results (obtained from M\\\"uller \\al\n\\cit{Mul95} and analyzed in \\cit{SM98}). We focus on the most\ninteresting regimes, beyond the scope of standard PO theory: regimes\nwhere there is no real contributing PO (``ghost'' regions), or where\nnon-isolated POs give non-separable contributions; in these cases the\nsaddle orbits succeed while the PO formula fails. Here we find that the\nclosed orbit formula \\cit{NS99} represents an improvement over the PO\nformulae, but requires a complicated strategy where one switches between\ndifferent types of real trajectories in different regimes. On the other\nhand, the minimal orbits achieve the goal of reproducing the accuracy of\nthe complex SOs ---across the whole transition from regularity to\nchaos--- by using only real dynamics. We also investigate\nbifurcation-type phenomena for saddle orbits, where their contribution\npeaks sharply, somewhat reminiscent of the effect of bifurcations on the\nGTF. More detail on this work can be found in \\cit{Sara99}.\n\n\n\\section{THEORY: Derivation of general semiclassical formula\n and the RTD as an example of Gaussian matrix elements}\n\\label{sc}\n\n\\subsection{Semiclassical Gaussian matrix elements}\n\nWe wish to find a semiclassical approximation to the quantity\n\\begin{equation}\nI(E) = \\sum_i \\langle \\psi_i | \\hat{\\cal A} | \\psi_i \\rangle\n\\delta(E-E_i)\n=-\\frac{1}{\\pi} {\\rm I} \\hspace{-1.0mm} {\\rm Im} \\: {\\rm Trace} \\:[ \\hat{\\cal A} \\hat{G} ]\n=-\\frac{1}{\\pi} {\\rm I} \\hspace{-1.0mm} {\\rm Im} \\: \\int d\\bfm{q} \\int d\\bfm{q'} {\\cal A}(\\bfm{q},\\bfm{q'})\nG(\\bfm{q}',\\bfm{q}) \n\\com{,}\n\\label{semimat}\n\\end{equation}\nwhere $| \\psi_i \\rangle$ is an eigenstate of the system with\neigenenergy $E_i$.\nWe also introduced the position matrix elements of the observable,\n${\\cal A}(\\bfm{q},\\bfm{q'})= \n\\langle \\bfm{q}| \\hat{\\cal A} | \\bfm{q}' \\rangle$, and of the energy\nGreen's function \n$G(\\bfm{q'},\\bfm{q})= \\langle \\bfm{q'}| \\hat{G} | \\bfm{q} \\rangle =\n\\lim_{\\eta\\to 0_+} \\langle \\bfm{q'}| (E+i\\eta-\\hat{H})^{-1} | \\bfm{q} \\rangle$.\nWe consider here a closed system with two degrees of freedom \n$\\bfm{q} =(x,z)$, and described by a Hamiltonian $\\hat{H}$.\nWe consider here the special case of {\\em Gaussian} matrix elements, i.e., \nprojectors on a Gaussian: \n\\begin{equation}\n\\hat{\\cal A}=| \\Phi \\rangle \\langle \\Phi | \\com{,}\n\\Phi(x,z)=\\phi(z) \\chi(x) \\com{,}\n \\phi(z) =\\left( \\frac{\\beta}{\\pi\n\\hbar} \\right)^{1\/4} e^{\\textstyle -\\frac{\\beta}{2 \\hbar}z^2}\n\\com{.}\n\t\\label{init}\n\\end{equation} \n$\\chi(x)$ can also be a Gaussian (in Franck-Condon transitions) or a\ncut $\\delta(x)$ (in microcavities or the RTD).\nTo derive the semiclassical approximation, one proceeds in two steps.\nFirst, one uses the semiclassical expression of the Green's function\ninvolving all classically allowed trajectories going from $\\bfm{q}$ to\n$\\bfm{q'}$ with energy $E$ \\cit{Gut90}:\n\\begin{equation}\nG(\\bfm{q'},\\bfm{q}) \\stackrel{\\hbar \\to 0}{\\simeq} \n\\frac{2 \\pi}{(2 \\pi i \\hbar)^{3\/2}} \\sum_{\\bfm{q} \\to \\bfm{q'}}\n\\frac{m}{\\sqrt{p_x p'_x m_{12}}} \ne^{i S(\\bfm{q},\\bfm{q'})\/\\hbar } \\com{,}\n\t\\label{semigreen}\n\\end{equation}\nwhere $S$ is the action of the trajectories, $p_x$ the\nmomentum in $x$, and $m_{12}=\\pder{z'}{p_z}$.\nNote that we define our monodromy matrix \n$M=\\partial (z',p_z')\/(\\partial z, \\partial p_z)$ with respect to $z$\nand not to \nthe coordinate perpendicular to the trajectory as usual.\nWe did not include the phase $(-i\\mu \\pi\/2)$ arising from the number $\\mu$\nof conjugate (or focal) points along the trajectory.\n\nThe second step requires the use of the stationary phase approximation \n(SPA) in eq. \\rf{semimat}.\nThis states that an exponential integral can be approximated by\na quadratic expansion of the argument of the exponential around the\npoint where the argument is stationary.\nDepending on the relative smoothness of $\\hat{\\cal A}$ and $\\hat{G}$,\ndifferent {\\em further} approximations can be made.\n\n\\subsection{Description of the RTD}\n\nA resonant tunneling diode (RTD) can be constructed by adding different\nlayers of semiconductors and applying a voltage $V$ between the\nemitter (where the electrons \naccumulate before entering the well) and the collector.\nIn effect, this will create a wide quantum well between two tunneling\nbarriers (see Fig. \\ref{rtdexpt}). \nOne also applies a magnetic field $\\bfm{B}$ at tilt angle $\\theta$ in the\n$X-Z$ plane, which creates instability in the classical dynamics in the \nwell.\n\n\n\\begin{figure}[!]\n\\vspace{-0cm}\n\\centerline{\\psfig{figure=fig1.eps,angle=0.,height=7.cm,width=12.cm}}\n\\vskip 1.cm \n\\caption{Schematic diagram of the RTD.\nWe show the experimental setup (not to scale), with the conduction\nband profile (effective voltage).\nBelow is the 3-$D$ coordinates axis; the magnetic field $\\bfm{B}$ is at\ntilt angle $\\theta$ with the electric field $-\\bfm{F}$ in the $X-Z$ plane.\nWe also show a representation of the distribution of the electrons in\nthe emitter setback: a Gaussian\ndistribution $\\phi$ in $Z$ due to the magnetic field $B \\cos\\theta$, and an\nAiry function $\\chi$ in $X$ due to the triangular well.\nThe width of the well is $L=120 \\: {\\rm nm} =2267 \\:{\\rm a.u.} $.\n}\n\\label{rtdexpt}\n\\end{figure}\n\n\nThe Hamiltonian describing the motion of the electrons in the well can \nbe reduced to two dimensions \\cit{NS98b,BR98} and reads\n\\begin{equation}\nH(\\bfm{p},\\bfm{q}) = \\frac{1}{2 m} (p_x^2 + p_z^2) - F x + \n\\frac{B^2}{2 m} (x \\sin\\theta-z \\cos\\theta)^2 \\com{,}\n\t\\label{2dham1}\n\\end{equation}\nwhere we used atomic units ($e=m_e=\\hbar=1$), $F=V\/L$ and the\neffective mass of the electron is $m=0.067$.\nThe length of the well is $L=120 \\: {\\rm nm} = 2267 \\:{\\rm a.u.} $.\nWe consider the barriers to be of infinite height; the classical\nelectrons will undergo specular bounces ($p_x \\to -p_x$) at the\nbarriers ($x=0,L$), while the quantum wave function $\\psi_i$\nof the (isolated) well has vanishing boundary conditions \n[$\\psi_i(x=0,z)=\\psi_i(x=L,z)=0$].\n\n\t\\subsection{Bardeen expression for the RTD current}\n\nThe current $I(E)$ can be calculated using the assumption of weak\ntunneling across the emitter barrier, the collector playing no \nimportant role as all the sites are free for the outgoing electrons\n(which are accelerated by the voltage drop).\nIn that case one can use the\nBardeen \\cit{Bar61} formalism, which is a sort of first-order\nperturbation theory for a high barrier \\cit{BR98}:\n\\begin{eqnarray}\nJ &=& \\frac{2 \\pi}{\\hbar} \n\\sum_i |{\\cal M}_i|^2 \\delta(E-E_i) \n\\com{,}\n{\\cal M}_i = \\frac{\\hbar^2}{2 m} \n\\int d\\bfm{q} \\left\\{ \\Phi^*(\\bfm{q}) \\pder{\\psi_i}{x}(\\bfm{q}) -\n\\pder{\\Phi^*}{x}(\\bfm{q}) \\psi_i(\\bfm{q}) \\right\\} \\delta(x) \n\\com{,} \\bfm{q}=(x,z) \\com{.}\n\t\t\\label{bard1}\n\\end{eqnarray}\nIn essence, this is an overlap between $\\psi_i$ and the ``initial\nstate'' $\\Phi$, which is the state of the electron in the emitter\nregion prior to tunneling.\nThe overlap is made on a cut taken on the emitter barrier at $x=0$.\nIt was shown \\cit{MDFB97}\nthat the use of finite or infinite barriers does not change\nsignificantly the important features of the current.\nNote that the Bardeen expression is formally a matrix element\nlike eq. \\rf{semimat}, if one writes\n${\\cal M}_i =i \\langle \\Theta | \\psi_i \\rangle$ with\n$ | \\Theta \\rangle = \\hbar\/(2 m) | \\big ( \\hat{p}_x\n\\hat{\\delta}_x + \\hat{\\delta}_x \\hat{p}_x \\big ) \\otimes 1 \\hspace{-1.2mm} 1 _z |\n\\Phi \\rangle , \\quad \\hat{\\delta}_x =|x=0 \\rangle \\langle x= 0|.$\n\nIn the RTD, one can assume \\cit{BR98} a separable form for $\\Phi$:\nan Airy function $\\chi(x)$ induced by the triangular well, and a Landau state\n$\\phi(z)$ induced by the effective magnetic field $\\beta:=B \\cos\\theta$.\nIn the experiments under consideration and for small $\\theta$, one\ncan further assume that only the lowest Landau level [eq. \\rf{init}] is occupied\n \\cit{From94}.\n\nIt is well known that the imaginary part of the Green's function can\nreproduce the sum of the delta functions of the energies, as \nwritten in eq. \\rf{semimat}.\nThe current is therefore given by\n\\begin{eqnarray}\nJ &=& -\\frac{\\hbar^3}{2 m^2} {\\rm I} \\hspace{-1.0mm} {\\rm Im} \\: \\int dz \\int dz' \\left\\{\n\\Phi(\\bfm{\\bar{q}}) \\Phi^*(\\bfm{\\bar{q}'}) \n\\partial^2_{x x'} G(\\bfm{\\bar{q}'},\\bfm{\\bar{q}}) -\n\\partial_x \\Phi(\\bfm{\\bar{q}}) \\Phi^*(\\bfm{\\bar{q}'}) \n\\partial_{x'} G(\\bfm{\\bar{q}'},\\bfm{\\bar{q}}) \n\\right. \n\\nonumber \\\\ && \\left. \n-\\Phi(\\bfm{\\bar{q}}) \\partial_{x'} \\Phi^*(\\bfm{\\bar{q}'}) \n\\partial_x G(\\bfm{\\bar{q}'},\\bfm{\\bar{q}}) +\n\\partial_x \\Phi(\\bfm{\\bar{q}}) \\partial_{x'} \\Phi^*(\\bfm{\\bar{q}'}) \nG(\\bfm{\\bar{q}'},\\bfm{\\bar{q}})\n\\right\\} \\com{.}\n\t\t\\label{bard2}\n\\end{eqnarray}\nThen one can use the semiclassical expression \\rf{semigreen} for the\nGreen's function.\nBecause of the Bardeen cut, only trajectories going from and to the\nleft barrier $\\bfm{\\bar{q}}=(x=0,z)$ can contribute to the current.\nNote that the derivatives of $\\psi_i$ in eq. \\rf{bard1} will be\ntransfered to derivatives of the Green's function, yielding factors\n$-p_x=\\partial S\/\\partial x$ and $p_x'=\\partial S\/\\partial x'$.\nIt has been shown \\cit{NS99,BR98} that the Green's function, as well\nas its first derivatives, vanish at the hard barrier (this can be\nunderstood by the Dirichlet conditions for $\\psi_i$).\nHence only the first term in eq. \\rf{bard2} contributes, and one is\nleft with:\n\\begin{equation}\nJ = -\\frac{2 |\\chi(0)|^2}{m \\sqrt{2 \\pi \\hbar}} {\\rm I} \\hspace{-1.0mm} {\\rm Im} \\: i^{-3\/2} \n{\\cal I} \\com{,}\n{\\cal I}= \\int dz \\int dz'\n\\sum_{(x=0,z) \\to (x'=0,z')} \\sqrt{\\frac{p_x p'_x}{-m_{12}}}\n\\phi(z) \\phi^*(z') e^{i S(x=0,z;x'=0,z')\/\\hbar} \t\\com{.}\n\t\t\\label{bard3}\n\\end{equation}\n\n\\subsection{Derivation of the general semiclassical formula}\n\nFirst we rewrite eq. \\rf{bard3} using the Gaussian form of the initial \nstate:\n\\begin{eqnarray}\n{\\cal I} & =& \\sqrt{\\frac{\\beta}{\\pi \\hbar}} \\sum_{\\ell}\n\\int \\! \\!\\ \\! \\! \\int_{\\Omega_{\\ell}} \\! \\! dz \\: dz'\n\\sqrt{\\frac{p_x p'_x}{-m_{12}}} e^{\\varphi(z,z')\/\\hbar} \n\\com{with} \n\\varphi(z,z') := i S(z,z') - \\frac{\\beta}{2}(z^2 +z'^2) \\com{.}\n\t\t\\label{varphi}\n\\end{eqnarray}\nWe have also taken the sum over trajectories $(x=0,z) \\to (x'=0,z')$\noutside the\nintegrals where it becomes a sum over all the different ``families''\n$\\{ \\ell \\}$ of trajectories existing in a domain $\\{ (z,z') \\in\n\\Omega_{\\ell} \\}$. \n\nThe tool used for this kind of integration is the {\\em stationary phase \napproximation} (SPA).\nBriefly, it states that only trajectories $z_0 \\to z_0'$\nhaving a stationary phase\n[$\\partial \\varphi(z_0,z_0')\/\\partial z=\n\\partial \\varphi(z_0,z_0')\/\\partial z'=0$] \ncontribute to the integrals, all the others being either damped\nby the Gaussian or destroyed by the random cancellations of the\noscillations due to the action.\nThen one has to expand the phase quadratically around the contributing \ntype of orbit and integrate, pushing the limit of the integration \n$\\Omega \\to {\\rm I} \\hspace{-0.9mm} {\\rm R} ^2$.\n\nWe saw in the introduction that a variety of contributing orbits have\nbeen proposed in the case of the RTD.\nTherefore, we shall not specify the type of orbits yet, but\nrather develop a general formula valid for {\\em any} type, and discuss \nthe different choices in section \\ref{discu}.\n\nOne can relate the second derivatives of the action to the monodromy\nmatrix $M$ \\cit{Gut90}, and the quadratic expansion of the action reads:\n\\begin{eqnarray}\nS(z,z') & \\simeq & S(z_0,z'_0) + \n\\delta z \\pder{S}{z}(z_0,z'_0) + \\delta z' \\pder{S}{z'}(z_0,z'_0)\n\\nonumber \\\\\n&& + \\frac{1}{2} \\left[ \n\\delta z^2 \\frac{\\partial^2 S}{\\partial z^2}(z_0,z'_0) +\n2 \\delta z \\delta z' \\frac{\\partial^2 S}{\\partial z \\partial z'}(z_0,z'_0)\n+\\delta z'^2 \\frac{\\partial^2 S}{\\partial z'^2}(z_0,z'_0)\n\\right] \\\\\n & = & S_0 - p^0_z \\delta z + p^0_{z'} \\delta z' + \n\\frac{1}{2 m_{12}} \\left [ \n\\delta z^2 m_{11} - 2 \\delta z \\delta z' + \\delta z'^2 m_{22} \\right] \\\\\n& =: & {\\cal S}_2(\\delta z, \\delta z';z_0,z'_0)\n\\com{,}\t\t\\label{quadaction}\n\\end{eqnarray}\nwith $\\delta z=z-z_0$ and $\\delta z' = z'-z_0'$.\nThe ``phase'' of the initial state is already quadratic.\nWe follow the techniques of Bogomolny and Rouben \\cit{BR98}, and\ncomplete the square: \n\\begin{eqnarray}\n& {\\cal I} & \\stackrel{\\rm SPA}{\\simeq} \n\\sqrt{\\frac{\\beta}{\\pi \\hbar}} \\sum_{\\ell_0} \n\\int_{ {\\rm I} \\hspace{-0.9mm} {\\rm R} ^2} d\\bfm{\\gamma} \\sqrt{\\frac{p_x p'_x}{-m_{12}}}\n e^{\\varphi_2(\\bfm{\\gamma})\/\\hbar} \n\t\t\\label{integral2} \\\\\n& \\varphi_2(\\bfm{\\gamma}) & := \ni {\\cal S}_2(\\bfm{\\gamma};z_0,z'_0) - \\frac{\\beta}{2}(z^2 +z'^2) \\\\\n&& \\stackrel{\\rf{quadaction}}{=} i S_0 - \\frac{\\beta}{2}(z_0^2 +z_0'^2) + \n\\bfm{\\xi}^{\\rm T} \\bfm{\\gamma} + \n\\frac{1}{2}\\bfm{\\gamma}^{\\rm T} {\\cal H} \\bfm{\\gamma} \n\t\t\\label{lint}\t\\\\\t\t\n&& = i S_0 - \\frac{\\beta}{2}(z_0^2 +z_0'^2) - \n\\frac{1}{2} \\bfm{\\xi}^{\\rm T} {\\cal H}^{-1} \\bfm{\\xi} + \n\\frac{1}{2} (\\bfm{\\gamma} + \\bfm{\\gamma}_1)^{\\rm T} {\\cal H} \n(\\bfm{\\gamma} + \\bfm{\\gamma}_1)\n\\label{varphi2}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n&& \\bfm{\\gamma}=(\\delta z, \\delta z') = (z-z_0,z'-z'_0) \\com{,} \\\\\n&& \\bfm{\\xi} = (-\\beta z_0 - i p_z^0, -\\beta z'_0 + i p_{z'}^0) \\com{,}\n\\bfm{\\gamma}_1= {\\cal H}^{-1} \\bfm{\\xi} \\com{,} \\\\\n&& {\\cal H} = \\left ( \\begin{array}{cc}\n\\displaystyle - \\beta + i \\frac{m_{11}}{m_{12}} &\n\\displaystyle -i \\frac{1}{m_{12}} \\\\ \\\\\n\\displaystyle -i \\frac{1}{m_{12}} &\n\\displaystyle - \\beta + i \\frac{m_{22}}{m_{12}} \n\\end{array} \\right) \\com{.}\n\\end{eqnarray} \n$\\ell_0$ denotes the different contributing trajectories.\nWith the change of variables $\\bfm{\\gamma'} = \\bfm{\\gamma} +\n\\bfm{\\gamma}_1$, the last term in eq. \\rf{varphi2} gives a pure two-dimensional\nGaussian, equal to $2 \\pi \\hbar\/\\sqrt{\\det {\\cal H}}$.\nThe final result is\n\\begin{equation}\n{\\cal I} = \\sum_{\\ell_0 \\atop (z_0 \\to z'_0)} \n2 \\sqrt{\\frac{\\beta \\pi \\hbar p_x p_x'}{- \\cal D}} \ne^{\\left[ i S_0 + \\Gamma(z_0,z'_0)\n\\right] \/\\hbar } \\com{,}\n\t\t\\label{generalformula}\n\\end{equation}\nwith \n\\begin{eqnarray}\n{\\cal D} &=& -m_{21} - i \\beta {\\rm Tr} M + \\beta^2 m_{12}\n\\label{cald}\t\t\\\\\n\\Gamma (z,z') \n&=& \n -\\frac{\\beta}{2 {\\cal D}} \\times \\Bigg\\{\nz^2 \\left[ -m_{21} - i \\beta m_{11} \\right] +\nz'^2 \\left[ -m_{21} - i \\beta m_{22} \\right] +\n2 i \\beta z z' \t \\nonumber \\\\ \n& + & \n \\frac{1}{\\beta^2}p_z^2 \\left[ - i \\beta m_{22} + \\beta^2 m_{12} \\right] +\n\\frac{1}{\\beta^2}p_{z'}^2\\left[ - i \\beta m_{11} + \\beta^2 m_{12} \\right] +\n2 i \\frac{1}{\\beta} p_z p_{z'} \\nonumber \\\\ \n& + & \n 2 \\frac{i}{\\beta} z p_z \\left[ i \\beta m_{22} - \\beta^2 m_{12} \\right] +\n2 \\frac{i}{\\beta} z' p_{z'} \\left[ -i \\beta m_{11} + \\beta^2 m_{12} \\right] +\n2 z p_{z'} - 2 z' p_z \n\\Bigg\\} \\: \\com{.}\n\t\t\t\t\t\\label{gammagen}\n\\end{eqnarray}\n\n\nThe above formula describes the {\\em oscillatory} part of the \ncurrent.\nWe do not consider here the ``smooth'' part, obtained by considering\nzero-length trajectories \\cit{BR98};\nthis part varies slowly with the energy, and corresponds to the Weyl\nterm in Gutzwiller's theory of the density of states \\cit{Gut90}.\n\nThe loss of coherence due to phonon scattering is {\\em not} considered \nin the formalism presented here.\nIt can be modeled by adding an exponential factor $\\exp(-T\/\\tau)$ in\nthe sum, where $T$ is the real part of the total time of each trajectory\nand $\\tau$ is the damping (decoherence) time.\nWe shall proceed the other way round, canceling the effects of the \ndamping in the experiments.\n\nThe formula is valid only for {\\em isolated} expansion orbits. \nAlso, we did not write explicitly the phase arising from \nthe number $\\mu$ of conjugate points along the\ntrajectory, as we shall primarily consider individual contributions\nfrom isolated trajectories. \n\n\\section{THEORY: Semiclassical formulae for specific types of trajectories}\n\t\t\t\t\t\\label{discu}\n\nAt this stage one should go back to the SPA applied to \neq. \\rf{varphi}, and examine which choices of expansion orbits have\nbeen or can be made. \nFirst we mention one \nremarkable feature of the RTD, which is that for {\\em any starting} $z$, \nthere exists, generically, a starting momentum $\\check{p}_z$ for which the \ntrajectory is a time-reversed duplicate of itself and therefore closed. \nWe call such trajectories {\\bf time-symmetric (TS)}.\nThey are defined by\n\\begin{equation}\n\\bom{TS} (z,\\check{p}_z) \\to (z',p_z') = (z,-\\check{p}_z) \n\t\t\\label{condsrco}\n\\end{equation}\nand satisfy the important relation $m_{11} = m_{22}$. \nThe existence of TS orbits is a consequence of the fact that for\nany starting $z$, one can find a starting $p_z$\nso that there is either a perpendicular bounce on a wall\n[$p_z(x=0 \\:{\\rm or} \\:L)=0$], or a turning point on the energy surface\nat a point where $\\bfm{p}=0$.\nNote that some self-retracing trajectories with $p_z^0 \\neq 0$ or\n$z' \\neq z$ are not TS.\n\nOne actually finds that each choice of expansion orbit shown below\ncontains a subset which \nis time-symmetric (TS), and that {\\em in almost all cases only the TS\nsubset contribute to the current}. \nThis is the reason why we first write \\rf{gammagen} for \ntime-symmetric (TS) orbits. \nUsing \\rf{condsrco} and $m_{11}=m_{22}$, one finds\n\\begin{equation}\n\\Gamma_{\\rm TS}(z_0)= - \\frac{\\beta}{1-\\delta} \\Bigg\\{\nz_0^2 - \\left[ \\frac{1}{\\beta^2} (p_z^0)^2 - \n2 \\frac{i}{\\beta}z_0 p_z^0 \\right] \\delta \\Bigg\\} \n\t\t\\label{gammasrco}\n\\end{equation}\nwhere we have defined\n\\begin{equation}\n\\quad \\delta=-i \\beta \\frac{m_{12}}{m_{11}-1} \\com{.}\n\t\t\\label{delta}\n\\end{equation}\nWe now consider different possible choices for the expansion points.\n\t\\subsection{Saddle orbits (SOs)}\n\nThe first expansion orbits we consider here are given by the strict\napplication of the stationary phase condition on \\rf{varphi}:\n\\begin{equation} \\bom{SO}\n\\left\\{ \\begin{array}{ll}\n& p_z^0=i \\beta z_0 \\\\\n& p_{z'}^0= -i \\beta z'_0 \\com{.} \\end{array} \\right. \n\t\t\\label{condso}\n\\end{equation}\nWe call such trajectories {\\bf saddle orbits (SOs)} \\cit{SM98b}.\nInserting eq. \\rf{condso} in eq. \\rf{gammagen}, one finds \n$\\Gamma_{\\rm SO}(z_0,z'_0) = -\\frac{\\beta}{2} \\left( z_0^2 + z_0^{'2}\n\\right)$, i.e.\n\\begin{equation}\n{\\cal I}_{\\rm SO} = \\sum_{{\\rm SOs} \\atop (z_0 \\to z'_0)} \n2 \\pi \\hbar \\sqrt{\\frac{ p_x p_x'}{- \\cal D}} \ne^{i S_0\/\\hbar} \\phi(z_0) \\phi(z_0')\n \\com{.}\n\t\t\\label{gammaso}\n\\end{equation}\nIn the case where SOs are TS {\\bf (TSSOs)} one has\n$\\Gamma_{\\rm TSSO}(z_0) = -\\beta z_0^2 $.\nWe shall show in section \\ref{comp} that the SOs are the most\nsuccessful types of orbits for a semiclassical description of the\nquantum current. \nOne difficulty with SOs is the fact that they are {\\em complex}. \nThis is the reason why Bogomolny and Rouben \\cit{BR98} decided to\navoid them (and considered real trajectories),\nalthough they were well aware of the fact that the stationary phase \napproximation yields SOs.\n\nThe SOs are {\\em non-periodic}; this means that one cannot \nlook at repetitions of a ``primitive'' SO, as it would not satisfy the \nSO condition.\nInstead one must search for an SO with a higher period.\nContrary to complex POs, the complex conjugate of an SO is {\\em not} an SO.\n\n\t\\subsection{Normal orbits (NOs)}\n\t\\label{normorb}\n\nTo obtain real trajectories, one has to make a further approximation\nand neglect one term in the SPA condition \\rf{condso}.\nBogomolny and Rouben \\cit{BR98} considered that the dynamics in the\nwell are very \nchaotic; in that case one should \nexpect the oscillations due to the action to dominate the Gaussian damping\nof the initial state. \nFormally, this corresponds to taking the limit $\\beta \\to 0$ in\n\\rf{condso}, and \nyields {\\bf normal orbits (NOs)}:\n\\begin{equation} \\bom{NO}\n\\left\\{ \\begin{array}{ll}\n& p_z^0=0 \\\\\n& p_{z'}^0=0 \\com{.} \\end{array} \\right. \n\t\t\\label{condno}\n\\end{equation}\nEssentially, this states that the contributing trajectories are determined\nsolely by the oscillations of the Green's function: they must cancel\nthe variations of the action, however small their accessibility to the\ninitial state is.\nMoreover, TS normal orbits {\\bf (TSNOs)} have $z_0'=z_0$. \nThis implies that TSNOs are actually a special subset of periodic orbits,\nthat are time-symmetric (TSPO) and start with $p_z$=0.\nIn the non-TS case ($z_0'\\neq z_0$), the {\\em second} repetition of the NO\nis actually a TSPO; \nthe first repetition of a non-TS NO is in a sense a ``half PO''. \nOne finds\n\\begin{equation}\n\\Gamma_{\\rm NO}(z_0,z'_0) = -\\frac{\\beta}{2 {\\cal D}} \\Bigg\\{ \nz_0^2 \\left[ -m_{21}- i \\beta m_{11} \\right] + \n(z_0')^2 \\left[ -m_{21} - i \\beta m_{22} \\right] + 2 i \\beta z_0 z'_0 \\Bigg\\} \n\\com{,} \n\t\t\\label{gammano}\n\\end{equation}\nwhich is equation (109) of Bogomolny and Rouben (1999) \\cit{BR98}.\nIn the TS case, one has\n$\\Gamma_{\\rm TSNO}(z_0) = -\\frac{\\beta}{1-\\delta}z_0^2$, and \n the current can be written\n\\begin{equation}\n{\\cal I}_{\\rm TSNO}=\\sum_{{\\rm NOs} \\atop (z_0 \\to z'_0)} \n2 \\sqrt{\\frac{ \\beta \\pi \\hbar p_x p_x'}{- |{ \\cal D}|} } \ne^{-\\beta z_0^2 \\left( 1-\\gamma \\right)\/\\hbar }\ne^{ i (S+ \\Delta S -\\arg{\\cal D}\/2 )\/\\hbar } \\com{,}\n\\gamma = \\frac{|\\delta|^2}{1+|\\delta|^2} \\com{,}\n\\Delta S = \\beta z_0^2 \\frac{|\\delta|}{1+|\\delta|^2}\n\\com{.} \\label{gammasrno}\n\\end{equation}\t\t\nWe shall call this expression the {\\em PO\/NO formula}.\nNote the shift $\\Delta S$ of the frequency of the oscillations\nfrom the action $S$ of the PO.\nAlso, the term $\\gamma$ can reduce the Gaussian damping due to the\ninitial state, for trajectories with large $z_0$.\nIn fact, this formula takes into\naccount torus quantization effects {\\em \\`a la} Miller \\cit{Mil75}\noccurring in large islands of stability surrounding stable POs.\nIt has been shown \\cit{BR98} that it is\nanalytically equivalent to a model \\cit{SM98} building the current as\nan overlap between the initial Gaussian and the wavefunction in the well\napproximated by the harmonic oscillator state corresponding to Miller tori.\n\n\n\nIt is also interesting to consider the case when $\\beta$ is {\\em very} small.\nThe first terms of an expansion of $\\Gamma$ and ${\\cal D}$ in\npowers of $\\beta$ yield\n\\begin{equation}\n{\\cal D}_{\\rm HLNO} \\to -m_{21} \\com{,} \n\\Gamma_{\\rm HLNO} \\to -\\frac{\\beta}{2} \\left[z_0^2 + (z_0')^2 \\right]\n\\quad \\Rightarrow \\quad\n{\\cal I}_{\\rm HLNO}=\\sum_{{\\rm NOs} \\atop (z_0 \\to z'_0)} \n2 \\pi \\hbar \\sqrt{\\frac{ p_x p_x'}{m_{21}}} \ne^{i S\/ \\hbar} \\phi(z_0) \\phi(z_0')\n\t\t\\label{hlno}\n\\end{equation}\nWe refer to this kind of expansion as {\\bf ``hard limit''\n(HL)}.\nThis was the first formula proposed by Bogomolny and Rouben \\cit{BR98}.\nIt is justified in the case of extremely chaotic dynamics, where the\noscillations of the Green's function are\nsupposed to be much stronger than the Gaussian decay of the\ninitial state.\nIn the case of a TS orbit, the precise condition for the validity of\nthis theory is \\cit{BR98} \n\\begin{equation}\n\\frac{ \\left| \\frac{\\partial^2 S}{\\partial z^2} \\right| }{\\beta} =\n\\left| \\frac{ m_{11}-1}{\\beta m_{12}} \\right| =\n\\frac{1}{\\left| \\delta \\right|} \\gg 1\n\\com{.}\t\\label{stabfact}\n\\end{equation}\n\nThe HL result corresponds to the SPA method applied to the Green's\nfunction only.\nAs $\\beta$ is supposed to be {\\em very} small, the initial state\nfunction is {\\em neglected} in the integrations \\rf{integral2}, and is \ntaken out of them; \nit is evaluated at the NO and gives the simple Gaussian factor\n$\\Gamma_{\\rm HL}$. \nThe integral is carried out only over the Green's function, \nbringing in a prefactor $m_{21}$ and the exponential of the pure action.\nNote that this is the usual way of proceeding with the SPA method,\nwhile considering the variation of both functions in the integral\n\\rf{integral2} is not standard.\n\n\t\\subsection{Central closed orbits (CCOs)}\n\nNarimanov and Stone \\cit{NS99} proposed a semiclassical approach\\footnote{This was a\ncomplement to their periodic orbit formula presented in Narimanov \\al \\cit{NSB98}\nand discussed below in subsection \\ref{po}.} which \neffectively amounts to consideration of the other extreme case, where \nthe Gaussian damping dominates the action oscillations; \nthis assumption can be justified for fairly regular dynamics.\nThis corresponds to taking the limit $\\beta \\to \\infty$ in \\rf{condso}, \nand yields {\\bf central closed orbits (CCOs)}:\n\\begin{equation} \\bom{CCO}\n\\left\\{ \\begin{array}{ll}\n& z_0=0 \\\\\n& z'_0=0 \\com{.} \\end{array} \\right. \n\t\t\\label{condcco}\n\\end{equation}\nHere the contributing trajectories give maximal accessibility to\nthe initial state, while they do not cancel the variations of the action.\nIn this case one finds\n\\begin{equation}\n\\Gamma_{\\rm CCO}(p_z^0,p_{z'}^0) = -\\frac{1}{2 \\beta {\\cal D}} \\Bigg\\{ \n(p_z^0)^2 \\left[ -i \\beta m_{22} + \\beta^2 m_{12} \\right] + \n(p_{z'}^0)^{2} \\left[ - i \\beta m_{11} + \\beta^2 m_{12} \\right] \n+ 2 i \\beta p_z^0 p_{z'}^0 \\Bigg\\} \n\\com{.} \n\t\t\\label{gammacco}\n\\end{equation}\nThis formula is equivalent to equation (14) of Narimanov and Stone\n\\cit{NS99}. (They derived a general formula for any number of excited \nLandau levels in the initial state.)\nTS central closed orbits {\\bf (TSCCOs)} have $p_{z'}^0=-p_z^0$, and\ngive\n$\\Gamma_{\\rm TSCCO}(z_0) = \\frac{1}{\\beta} \\frac{\\delta}{1-\\delta}(p_z^0)^2$.\n\n\n\nOne can also consider the hard limit, i.e. the first order expansion\nin ${\\cal O}(1\/\\beta)$:\n\\begin{equation}\n{\\cal D}_{\\rm HLCCO} \\to \\beta^2 m_{12} \\com{,}\n\\Gamma_{\\rm HLCCO} \\to -\\frac{1}{2 \\beta} \\left[ (p_z^0)^2 + (p_{z'}^0)^2 \\right]\n\\quad \\Rightarrow \\quad\n{\\cal I}_{\\rm HLCCO}=\\sum_{{\\rm CCOs} \\atop (z_0 \\to z'_0)} \n2 \\pi \\hbar \\sqrt{\\frac{ p_x p_x'}{-m_{12}}} \ne^{i S\/ \\hbar} \\tilde{\\phi}(p_z^0) \\tilde{\\phi}(p_{z'}^0)\n\\com{.} \t\\label{hlcco}\n\\end{equation}\nHere we introduced the initial state (i.e., the observable) in\nmomentum representation: $\\tilde{\\phi}(p_z)=(\\beta \\pi \\hbar)^{-1\/4}\n\\exp[-p_z^2\/(2 \\beta \\hbar) ]$.\nThe hard limit is equivalent to neglecting the quadratic \nterm of the action in the\nintegral \\rf{integral2}.\nThe integration of the linear term with the\ninitial state is in effect a Fourier transform, and brings in the\nvalue of the\ninitial state in momentum representation at the CCO.\nAlternatively, one can express the Green's function and the initial\nstate in momentum representation, and argue that the latter is smooth and\ncan be taken out of the integral by stationary phase approximation.\nA similar expression in terms of ``closed orbits at the nucleus'' and\ninvolving a weighting by $m_{12}^{-1\/2}$ was found in the\nsemiclassical theory of the photoabsorption spectra of a hydrogen atom\nin external fields \\cit{DuDe88}.\nThe similarity is somewhat limited, as the\nexpression for the photoabsorption spectra is much more complicated\nthan mere momentum wave functions (it involves matching the\nsemiclassical Green's function to a quantum one in the vicinity of the \nnucleus). \n\n\t\t\\subsection{Periodic orbits (POs)}\n\t\t\\label{po}\n\t\t\nPeriodic orbits are a natural choice, as it follows the expansion around POs\nfound in the derivation of the Gutziller trace formula for\nthe density of states. \nA discussion of this choice is more adequately made using a\nphase space formalism, described in appendix \\ref{pssc}.\nAlternatively, a more direct route is to \ndefine ``average-difference'' coordinates\n\\begin{equation} \\left\\{ \\begin{array}{ll}\n& \\bar{z}=\\frac{1}{2}(z'+z) \\\\\n& \\Delta z=z'-z \\end{array} \\right. \\com{,}\n\\left\\{ \\begin{array}{ll}\n& \\bar{p_z}=\\frac{1}{2}(p_z'+p_z) \\\\\n& \\Delta p_z=p_z'-p_z \\end{array} \\right. \n\\com{,}\t\t\\label{variable}\n\\end{equation}\nwhich one uses to write the $z$-observable $\\hat{A}=| \\phi \\rangle \\langle \n\\phi |$ in position space as:\n\\begin{equation}\n\\bar{A}(\\bar{z},\\Delta z) = \n \\phi(\\bar{z}-\\frac{1}{2} \\Delta z) \\phi^*(\\bar{z}+\\frac{1}{2}\\Delta z)=\n \\sqrt{\\frac{\\beta_{\\bar{z}}^{1\/2}\\beta_{\\Delta z}^{1\/2}}{\\pi \\hbar}}\n e^{\\textstyle -\\frac{\\beta_{\\bar{z}}}{\\hbar} \\bar{z}^2 -\n\\frac{\\beta_{\\Delta z}}{4 \\hbar} \\Delta z^2 } \n\\com{,}\t\t\t\\label{initdiff}\n\\end{equation}\nwhile the Wigner transform is defined by\n\\begin{equation}\nW(\\bar{z},\\bar{p}_z)=\\frac{1}{2 \\pi \\hbar} \\int d \\Delta z \\:\ne^{i \\bar{p}_z \\Delta z \/\\hbar} \\bar{A}(\\bar{z},\\Delta z) =\n\\frac{1}{\\pi \\hbar} \ne^{\\textstyle -\\frac{\\beta_{\\bar{z}}}{\\hbar} \\bar{z}^2 - \n\\frac{1}{\\beta_{\\Delta z} \\hbar} \\bar{p}^2} \n\t\\label{wign} \\com{.}\n\\end{equation}\nHere we have written two different Gaussian widths $\\beta_{\\bar{z}}$\nand $\\beta_{\\Delta z}$ for respectively $\\bar{z}$ and $\\Delta z$.\nOf course in reality we have $\\beta_{\\bar{z}} = \\beta_{\\Delta z} = \\beta$,\nbut retaining the distinction clarifies the following discussion.\nThe action is\n$\\bar{S}(\\bar{z},\\Delta z)= S(\\bar{z}-\\Delta z\/2,\\bar{z}+\\Delta\/2)$, \nand its quadratic expansion around a point $(\\bar{z}_0,\\Delta\nz_0)$ reads:\n\\begin{eqnarray}\n\\bar{{\\cal S}}_2(\\bar{z},\\Delta z) &=&\n\\bar{S}_0 + \\Delta p_z^0 \\delta \\bar{z} + \\bar{p}_z^0 \\delta \\Delta z\n \\nonumber \\\\ &+&\n\\frac{1}{2 m_{12}} \\left [ \n\\delta \\bar{z}^2 ({\\rm Tr} M-2) + \\delta \\bar{z} \\delta \\Delta z (m_{22}-m_{11})\n + \\frac{1}{4} \\delta \\Delta z^2 ({\\rm Tr} M+2) \\right] \\com{} \\label{s2phase}\n\\end{eqnarray}\nwith $\\delta \\bar{z}= \\bar{z}-\\bar{z}_0$ and \n$\\delta \\Delta z=\\Delta z -\\Delta z_0$.\n\nThe idea is to apply the SPA method to the integral\n$ {\\cal I} \\propto \\int \\Delta z \\int \\bar{z} \\:\n\\bar{A}(\\bar{z},\\Delta z) \n\\exp [i \\bar{{\\cal S}}_2(\\bar{z},\\Delta z)\/\\hbar ]$, i.e., \nwith respect to the variables \\rf{variable}.\nThe SPA condition reads:\n\\begin{equation} \\left\\{ \\begin{array}{ll}\n& \\bar{p}_z^0 =- \\frac{1}{2} i \\beta_{\\Delta z} \\Delta z_0 \\\\\n& \\Delta p_z^0= -2 i \\beta_{\\bar{z}} \\bar{z}_0 \\com{.} \\end{array} \\right.\n\t\t\\label{condsovar}\n\\end{equation}\n\nIn Eckhardt {\\em et al.} \\cit{EFMW92}, one assumes the Wigner transform to be\nsmooth as a function of $(\\bar{z},\\bar{p}_z)$.\nThis corresponds to the case $\\beta_{\\bar{z}} \\to 0$ and \n$\\beta_{\\Delta z} \\to \\infty $, which gives for \\rf{condsovar}:\n\\begin{equation} \\bom{POs} \\left\\{ \\begin{array}{ll}\n& \\Delta z_0 =0 \\\\\n& \\Delta p_z^0= 0 \\com{,} \\end{array} \\right.\n\t\t\\label{condpo2}\n\\end{equation}\nthat is, {\\bf periodic orbits (POs)}.\nHence POs arise naturally when one consider a smooth Wigner\ntransform; as $\\bar{A}(\\bar{z}, \\Delta z)$ \n is its Fourier transform [see eq. \\rf{awign}], \nit is smooth in $\\bar{z}$, but {\\em localized} in $\\Delta z$.\nThis corresponds to a ``local'' operator, in the sense that\n$\\bar{A}(\\bar{z},\\Delta z) \\sim \\bar{a}(\\bar{z}) \\delta(\\Delta z)$.\nThis also enables one to recover the Gutzwiller trace formula, via \n$\\hat{A} \\to 1 \\hspace{-1.2mm} 1 \\Rightarrow \\bar{A}(\\bar{z},\\Delta z) \\to \\delta(\\Delta z)$.\n\nThings are different for the Gaussian matrix elements, which are \nwritten as a projector over a Gaussian state.\nThey are the product of two functions depending separately on $z$ and $z'$, and\n{\\em cannot} have the property of being simultaneously smooth in\n$\\bar{z}$ and localized in $\\Delta z$: either it is localized in both, or it\nis smooth in both.\nOne cannot change $\\beta_{\\bar{z}}$ and $\\beta_{\\Delta z}$ independently.\nThis fact was noted by Zobay and Alber \\cit{ZoAl93} in their work on\nFranck-Condon \nmolecular transitions, which involved very similar equations.\n\nNevertheless, it is still fruitful to consider periodic orbits for the\nRTD.\nPutting $z'=z,p_z'=p_z$ in \\rf{gammagen}, one finds\n\\begin{eqnarray}\n\\Gamma_{\\rm PO}(z^0,p_{z}^0) &=& -\\frac{\\beta}{2 {\\cal D}} \\Bigg\\{ \nz_0^2 \\left[ -2 m_{21} - i \\beta ({\\rm Tr} M-2) \\right] +\n\\frac{1}{\\beta^2}(p_z^0)^2 \\left[ -i \\beta ({\\rm Tr} M-2) + 2 \\beta^2 m_{12}\n\\right] \\nonumber \\\\\n&& \\hspace{2cm} - 2 z_0 p_z^0 \\left[ m_{22}-m_{11} \\right]\n\\Bigg\\} \\com{.} \n\t\t\\label{gammapo}\n\\end{eqnarray}\nThis formula is equivalent to equation (19) of Narimanov and Stone \\cit{NS99}.\nAn important subset of POs are TS, and have $p_z^0=0$ {\\bf (TSPOs)}.\nAs mentioned above, TSPOs are identical to TSNOs, and therefore give \nthe same contribution \\rf{gammasrno}.\n\n\n\n\nFor the hard limit, an expansion in $\\beta_{\\bar{z}}$ and\n$1\/\\beta_{\\Delta z}$ gives\n\\begin{equation}\n{\\cal D}_{\\rm HLPO} \\to \\beta ({\\rm Tr} M-2)\/2 i \\com{,} \n\\Gamma_{\\rm HLPO} \\to -\\beta_{\\bar{z}} \\bar{z}_0^2- \n\\frac{1}{\\beta_{\\Delta z}} (\\bar{p}_z^0)^2\n\\quad \\Rightarrow \\quad\n{\\cal I}_{\\rm HLPO}=\\sum_{{\\rm POs} \\atop (z_0 \\to z'_0)} \n2 \\pi \\hbar \\sqrt{\\frac{ 2 \\pi i \\hbar p_x p_x'}{{\\rm Tr} M-2}} \ne^{i S\/ \\hbar} W(\\bar{z_0}, \\bar{p}_z^0)\n\\com{.}\t\t\\label{hlpo}\n\\end{equation}\nThis corresponds to the formula for semiclassical matrix elements \nproposed in Eckhardt {\\em et al.} \\cit{EFMW92}, that was derived for an\nobservable which is \nsmooth in phase space.\nThis formula is basically the Gutzwiller trace formula (GTF) weighted\nby the Wigner transform \ncalculated for each PO.\\footnote{The result in Eckhardt {\\em et al.}\n\\cit{EFMW92} contains \nthe {\\em average} of the Wigner function taken along the path of the PO. \nIn our case, the Bardeen cut at $x=x'=0$ means that we need to\nevaluate the Wigner function only at the starting and final points of the PO.}\nTo get the hard limit directly from the integration \\rf{integral2},\none neglects the quadratic variation of $S$ due to $\\Delta z$ \n, and neglects the variations of $e^{-\\beta \\bar{z}^2\/\\hbar}$ over the integral\n(i.e., one uses its value at the PO).\nThe integration of $e^{-\\beta \\Delta z^2\/(4 \\hbar)}$ with\nthe linear term in\n$S$ due to $\\Delta z$ is a Fourier transform, which gives the Wigner\npart $e^{-(\\bar{p}_z^0)^2\/(\\beta \\hbar)}$.\nThe integration over the variations of the action due to $\\bar{z}$\ngives the ${\\rm Tr} M-2$ prefactor, as in the GTF.\nAlternatively, one can work in phase space and \napply the SPA to \\rf{phspint}, neglecting the\nvariations of the Wigner function $W$ in the integral.\nNote finally that the hard limit formula for POs expresses in some\nsense the heuristic approach that was first used to interpret\nsemiclassically the current in the RTD, where one considered the\neffects of the stability of POs on the density of states\n given by the GTF, while taking\ninto account the accessibility of the PO to the tunneling electrons\n\\cit{From94,Mul95}.\n\n\t\\subsection{Minimal orbits (MOs)}\n\t\\label{miniorb}\n\nNarimanov and Stone \\cit{NS99} proposed the CCOs in order to extend the PO formula to\nregions where one has no real POs (also called ``ghost regions'', see\nsection \\ref{comp}), \nwhile avoiding the problem of complex dynamics raised by SOs.\nThey also proposed an extension of the CCO formula in terms of\ntime-symmetric\norbits which had a minimal momentum transfer $\\Delta p_z= -2 p_z$, \ni.e., $\\partial(\\Delta p_z)\/\\partial z=0$.\nThe argument was that the Wigner transform in the PO formula\nhas a Gaussian damping\nthat kills the contribution of trajectories with $p_z$ that are not\nsmall.\nThis is the only proposed semiclassical formula for the RTD that we\nhave not tested by comparison with quantum calculations.\nInstead, we will propose and test another formula which is based on a similar\nidea.\n\nThe SPA method applied on \\rf{varphi} prescribes\nfinding an expansion point\nwhich makes the function $\\varphi_2(z,z')$ stationary.\nThis can be achieved if one can find a point $(z_0,z'_0)$ such that\nthe linear term $\\bfm{\\xi}^{\\rm T} \\bfm{\\gamma}$ in \\rf{lint}\nvanishes {\\em for all} $\\bfm{\\gamma} = (z-z_0,z'-z'_0)$, i.e.,\n$\\bfm{\\xi}(z_0,z'_0) = (-\\beta z_0 - i p_z^0, -\\beta z'_0 + i\np_{z'}^0) = (0,0)$.\nAs already mentioned, this requires complex trajectories (the SOs).\nThe idea here is to find the {\\em real} trajectory which \n{\\em minimizes} $\\bfm{\\xi}^{\\rm T}(z_0,z'_0)$ and therefore should\ngives the {\\em minimal linear term} in \\rf{varphi2}.\nThis is in some\nsense the best {\\em real approximation} of the complex saddle point.\nDefining\n\\begin{equation}\n{\\cal L}(z,z') = \\left| \\bfm{\\xi}^{\\rm T} \\bfm{\\xi}(z,z') \\right|^2 =\n\\beta^2 \\left[ z^2 + z'^2 \\right] + p_z^2 + p_{z'}^2 \\com{,} \n\\end{equation}\none will look for\n\\begin{eqnarray}\n0 &=& \\frac{1}{2} \\pder{\\cal L}{z} = \n\\beta^2 z - \\frac{p_z m_{11} + p_z'}{m_{12}} \\\\\n0 &=& \\frac{1}{2} \\pder{\\cal L}{z'} = \n\\beta^2 z' + \\frac{p_z + p_z' m_{22}}{m_{12}} \\com{.}\n\\end{eqnarray}\nThis prescription defines {\\bf minimal orbits (MOs)}:\n\\begin{equation} \\bom{MO}\n \\left\\{ \\begin{array}{ll}\n\\displaystyle & z'_0 = - m_{22}z_0 + \\frac{m_{21}}{\\beta^2} p_z^0 \\\\\n\\displaystyle & p_{z'}^0 = \\beta^2 m_{12}z_0 - m_{11} p_z^0 \\com{.}\n\\end{array} \\right. \n\t\t\\label{condmo}\n\\end{equation}\nThe contribution of MOs to the current will be given by using \\rf{gammagen}\nwith the $\\{z_0,p_z^0;z'_0,p_{z'}^0\\}$ of the MO.\nAgain, one finds that the most important MOs are TS.\n{\\bf TSMOs} have\n$p_z^0 = z_0 \\beta^2 \\frac{m_{12}}{m_{11}-1} = z_0 \\beta |\\delta|$.\nTheir contribution will be calculated with \\rf{gammasrco}.\n\n\t\\subsection{Summary of the formulae}\n\nFor the sake of completeness, we mention here the last possibility for\nneglecting one element in \\rf{condso}.\nOne considers the case of a Wigner transform that is very localized\nin $\\bar{z}$ and $\\bar{p}_z$.\nThis corresponds to $\\beta_{\\bar{z}} \\to \\infty$ and \n$\\beta_{\\Delta z} \\to 0 $, which gives for \\rf{condsovar}:\n\\begin{equation} \\bom{AO}\n\\left\\{ \\begin{array}{ll}\n& \\bar{z}_0 =0 \\\\\n& \\bar{p}_z^0= 0 \\com{.} \\end{array} \\right. \n\t\t\\label{condao}\n\\end{equation}\nWe call such trajectories {\\bf average orbits (AOs)}, but shall\nnot write nor study their contribution.\nIt is interesting to note that for TSAOs, one has \n$z_0=z_0'=0$, which means that TSAOs are identical to TSCCOs.\n\nWe show in Fig. \\ref{conc} a schematic representation of the different \norbits and their related formulae.\nWe classify them according to the level of approximation of the SPA\nmethod used in the Gaussian integrations, that is: \n$(i)$ no approximation, which gives the saddle orbits (SOs)\nand the related formula eq. \\rf{gammaso}; \n$(ii$) approximation in the SPA condition (but none in the\nintegration), which gives the normal [NO, eq. \\rf{gammasrno}], \ncentral closed [CCO, eq. \\rf{gammacco}],\nperiodic [PO, eq. \\rf{gammapo}] or average (AO) orbits [and their\nrelated formulae];\n$(iii)$ approximation in both the SPA condition and integrations, \nwhich give the hard limit formula (HLPO, HLNO, HLCCO).\nThen we classify them according to the underlying hypotheses regarding \nthe predominance of the Green's function $\\hat{G}$ or the observable\n$\\hat{A}$ in determining the contributing trajectories.\nThis is linked to their relative smoothness in position or \nphase space.\nNote that the SOs correspond in this classification to the angle orbits found\nfor the conductance of microcavities \\cit{BJS93}, in the sense that\nboth types of orbits \nare derived without any approximation of the SPA condition.\n\n\\begin{figure}[htb!]\n\\centerline{\\psfig{figure=fig2.eps,angle=0.,height=8.cm,width=9cm}}\n\\vskip 1cm \\caption{Schematic representation of the different\nsemiclassical formulae.\nThe vertical axis describes the three different levels of\napproximation: one can neglect one function ($A$ or $G$) \nin the saddle point\ncondition [COND], also in the integrations [COND + INT], or in none [NONE].\nThe horizontal plane describes the relative localization\/oscillations\nlength scales of $\\hat{A}$ and $\\hat{G}$ (e.g., the left end means\nthat the oscillations of $\\hat{G}$ \nin position space are on a much smaller scale than the localization of\n$\\hat{A}$, etc.). \n}\n\t\t\\label{conc}\n\\end{figure}\n\n\n\\section{Comparison between semiclassical results\n and quantum calculations and experiments;analysis of results.}\n\\label{comp}\n\n\n\t\\subsection{Scaling the classical dynamics}\n\nIn our comparisons between classical\/semiclassical\/quantal \ndynamics we exploit an\n important property of the RTD : its Hamiltonian\n can be scaled with respect to the magnetic field \\cit{MD96}.\nThen, the classical dynamics depends only on the ratio $\\epsilon=F\/B^2$ instead of\nthe independent values of $F$ and $B$ (the ratio $R=E\/V$ is roughly constant\nin the experiments) .\nThe experimental regime \\cit{Mul95} corresponds to the interval \n$1000 < \\epsilon <100000$. The classical dynamics in this range\nevolve from chaotic (low $\\epsilon$) to regular dynamics\n(high $\\epsilon$) \\cit{SM98}.\nIt is preferable to scale the action not with respect to $B$, but\nrather with respect the action of the $ \\theta=0^\\circ, B=0, R \\gg 1$ problem: \n$S_0 \\simeq 2 L \\sqrt{2 m V (R+1\/2) }$.\nIn this integrable case, the number of oscillations of the wave\nfunction approximated by the WKB method \\cit{Gut90} is given by \n\\begin{equation}\n{\\cal N} := \\frac{S_0}{2 \\pi} = \\frac{L}{\\pi} \\sqrt{2 m V (R+1\/2)}\n\\com{,}\t\t\\label{nofv}\n\\end{equation}\nwhich we shall consider as a measure of the ``effective $\\hbar^{-1}$'' in \nthe general case as well. \nWe define the scaled action by $\\hat{S}(\\epsilon) := S\/S_0$.\nThis definition is convenient as the three types of experimental\noscillations \\cit{Mul95} then correspond to trajectories with $\\hat{S}\n\\simeq 1, 2$ or $3$. We term these period-one, period-two and \nperiod-three trajectories respectively.\nAlso, $\\hat{S}$ depends only on $\\epsilon$, but is roughly constant as\n$\\epsilon$ varies.\nWe called the important period-one trajectories $t$ and the most important \nprimitive period-two trajectories $s$ \\cit{SM98}.\n\nWe can obtain the period of the voltage oscillations generated by a\ngiven trajectory.\nThe frequency of the oscillations of the semiclassical current \nis given by the imaginary part of the argument of the exponential in\neq. \\rf{generalformula}:\n$\\Sigma = {\\rm I} \\hspace{-1.0mm} {\\rm Re} \\: S + {\\rm I} \\hspace{-1.0mm} {\\rm Im} \\: \\Gamma$.\nWe can also define its scaled version $\\hat{\\Sigma} = \\Sigma\/ 2 \\pi {\\cal N}$.\nThen one has two consecutive \nmaxima $\\{V, V+\\Delta V\\}$ in the current-voltage trace \n(neglecting the variation of $\\arg{\\cal D}$) when \n\\begin{equation}\n2 \\pi = \\Sigma(V+\\Delta V)- \\Sigma(V) = \\Delta [\\Sigma] = \n\\Delta \\left[2 \\pi \\hat{\\Sigma}(\\epsilon) {\\cal N}(V) \\right]\n= 2 \\pi \\hat{\\Sigma}\\frac{\\cal N}{2 V} \\Delta V\n\\quad \\Rightarrow \\quad\n\\Delta V = \\frac{2 V}{{\\cal N} \\hat{\\Sigma}}\n \\com{.} \n\t\t\\label{2pi}\n\\end{equation}\nThis can be contrasted to the heuristic interpretation based on the\nDoS that was used before \\cit{From94,Mul95}, where the voltage oscillations\nwere directly obtained from the energy oscillations given by the GTF:\n$\\Delta V = \\Delta E\/ R = 2 \\pi \\hbar\/ R T$, where $T$ is the period\nof the contributing PO. \nThis relation is not exactly correct for two reasons: the current oscillations \nare not given by the action $S$ but by $\\Sigma$, and the period $T$\narises in the GTF from $\\partial S(E,B,V)\/\\partial E =T$ taken\nat constant $B$ and $V$, while in our case $V$ varies with $E$ \nthrough the constant $R$.\n\n\n\\subsection{The different types of orbits}\n\nBecause of the decoherence induced by phonon scattering, only the\nshortest trajectories contribute to the current in the RTD we analyzed\n(which has a width $L=120 \\:{\\rm nm} = 2267 \\:{\\rm a.u.} $).\nWe compare in Fig. \\ref{plot} the shape of the different types of\ncontributing (all of the $t$-type).\nExamples of plots of the other important class of orbits,\nthe $s$-type orbits may be found elsewhere eg ~\\cite{SM98b,NS99}.\nIn (a) we present the traversing periodic orbit (PO) $t_0$, which\nmakes one bounce on each wall, and is responsible for the broad\nexperimental voltage oscillations \\cit{SM98,From94}.\nIt is perpendicular on the left wall ($p_z^0=0$), and is therefore\ntime-symmetric (TS) and also a TS normal orbit (NO).\nThere are two period-two POs, born in two pitchfork bifurcations\naround $\\epsilon=13000$.\n$t_v$ is self-retracing but non-perpendicular;\nhence it is not TS: reversing the momentum\nat the end of the trajectory (on the left wall), \none does not find oneself on the portion of \nthe trajectory on which the orbit started.\n$t_u$ is TS and therefore also a TS NO; there is also another non-TS\nNO hidden: it is ``half'' the PO $( z_0 = -80 \\to z_0' =550)$ and\nwill be\ndenoted by $t_u\\!-\\!\\! NO$.\n\n\nAs $\\epsilon$ decreases towards the chaotic regime, $t_0$ disappears with\nan unstable partner $t_0^-$ in a tangent bifurcation at $\\epsilon=6500$.\nThey are replaced by a pair of POs which are complex conjugates of each other; \nthe one giving a physical contribution is called ``ghost''\nand is denoted by $t_{\\rm gh}$.\nAt $\\epsilon=3000$ (b), a new real PO $t_1$ has appeared.\nWe also show the saddle orbit (SO) and minimal orbit (MO) that are\nrelated to the $t$-type trajectories; they do not disappear in any\nbifurcations and are linked to all the three POs ($t_0, t_{\\rm gh}$) and \n$t_1$ as $\\epsilon$ decreases.\nIn this region the SO and MO are between $t_{\\rm gh}$ and $t_1$.\n\nIn (c) we illustrate the non-periodicity of SOs.\nWe show $t_0 \\!-\\!\\! SO$, related to the primitive PO $t_0$, and the very\ndifferent $2t_0 \\!-\\!\\! SO$, that is related to the second repetition of\nthe PO.\nBecause SOs (as well as MOs and CCOs) are not periodic, one cannot continue\nthe propagation of a primitive orbit, but one has to look for another\norbit with the adequate action, period and number of bounces so that it \ncorresponds to the repetition of a primitive PO.\nNote that in some cases (e.g., $t_0\\!-\\!\\! SO$ at $ \\theta=27^\\circ, \\epsilon < 17000$),\none cannot find an SO corresponding to the second repetition of a PO,\nalthough one has the SO linked to the primitive PO.\n\n\\begin{figure}[htb!]\n\\centerline{\\psfig{figure=fig3.eps,angle=270.,height=8.cm,width=12.cm}}\n\\vskip 5mm \\caption{\nShape of different types of orbits of the $t$-type\nat $ \\theta=11^\\circ$.\nThe right wall is at $x= L = 120 \\: {\\rm nm} = 2267 \\:{\\rm a.u.} $.\nFor complex trajectories, we show the real part $( {\\rm I} \\hspace{-1.0mm} {\\rm Re} \\: x, {\\rm I} \\hspace{-1.0mm} {\\rm Re} \\: z)$.\n(a) $\\epsilon=14000$; the periodic orbit (PO) \n$t_v$ is self-retracing but not time-symmetric (TS), as\nit not perpendicular on the left wall ($p_z^0 \\neq 0$); \n$t_0$ and $t_u$ are TS POs and therefore also TS NOs;\nthe ``half PO'' $t_u \\!-\\!\\! NO$ $( z_0 = -80 \\to z_0' =550)$ is a\nnon-TS normal orbit (NO).\n(b) $\\epsilon=3000$; here we have the complex ghost $t_{\\rm gh}$ which has \nappeared in the tangent bifurcation of $t_0\\!-\\!\\! PO$ at $\\epsilon=6500$;\nwe show the related saddle orbit $t_0\\!-\\!\\! SO$ and minimal\norbit $t_0 \\!-\\!\\! MO$;\nwe also have a real PO $t_1$.\n(c) $\\epsilon=18000$; we show the saddle orbit $t_0 \\!-\\!\\! SO$ corresponding\nto the primitive $t_0\\!-\\!\\! PO$, as well as the saddle orbit $2t_0\\!-\\!\\! SO$ \ncorresponding to the second repetition of the PO.\n(d) $\\epsilon=10000$; the central closed orbit $t_0\\!-\\!\\! CCO$ is defined by $z_0=0$. \n}\n\\label{plot}\n\\end{figure}\nAn example of a central closed orbit (CCO) is shown in (d): \nit looks very different from the related PO.\n\n\n\t\\subsection{ $ \\theta=11^\\circ$ theory and experiments: ghost regions and torus\nquantization} \n\nThe method used to compare semiclassical, quantum and experimental\nresults was explained by Saraga and Monteiro \\cit{SM98}.\nFor each $\\epsilon$, we generate a scaled quantum (QM) and semiclassical\ncurrent that oscillates with ${\\cal N}$, in the range \n$12 < {\\cal N} <42 $ ,corresponding to the experimental $B$ range read at constant $V=0.5\n \\:{\\rm V} $.\nWe Fourier transform the current with respect to the pair ${\\cal N}, \\hat{S}$\nand get a power spectrum that has peaks at certain values of the action.\nThe height of the peaks gives the amplitude of the oscillation,\nwhile their position indicates their type: period-one oscillations\nwhen $\\hat{S} \\simeq 1$, period-two when $\\hat{S} \\simeq 2$, etc.\nFor the experiments, we read the amplitude of the oscillations \ndirectly from $I-V$ traces provided by G Boebinger \\cit{Mul95}, \nthat we analyzed and presented in \\cit{SM98}. \nWe correct the experimental $\\epsilon$ by $30 \\%$ to take into account\neffect of the voltage dependence of the mass. \nTo allow for damping due to phonon scattering, we scale the\namplitudes by $\\exp[T\/\\tau]$, where $T$ is the (real part of the) \ntotal time of\nthe contributing classical orbit and $\\tau \\sim 0.1\\:{\\rm ps}$ \\cit{From94}\nis a decoherence\ntime. We use here the value $\\tau \\simeq 0.115 \\:{\\rm ps}$ \nestimated by comparing the maximal values of the \nquantum and experimental amplitudes. \nWe normalize all amplitudes to the\namplitudes at $ \\theta=0^\\circ$, where for the semiclassics we have $ \\Gamma =0$ and \n$|{\\cal D} ( \\theta=0^\\circ)|= 2 B, \\forall \\epsilon$.\n\n\nFig. \\ref{amp11} presents the amplitudes of the different\nsemiclassical formulae at $ \\theta=11^\\circ$.\nHere we study period-one oscillations, which correspond to the broad\nvoltage oscillations seen in the experiments\\cit{Mul95} and called\n``$t$'' series \\cit{From94}.\nThey arise from trajectories making one bounce on each wall;\n( the $t_0$ orbit at high $\\epsilon$ and $t_1$ at low $\\epsilon$).\n\nIn (a) we see that the quantum calculation based on the Bardeen model\n reproduces quantitatively the experimental behaviour, over a\nlarge range of parameters (corresponding to $3 \\:{\\rm T} < B < 12 \\:{\\rm T} $).\nWe did not read experimental amplitudes for $\\epsilon>25000$, because of\nthe presence of period-two oscillations.\n\n\\begin{figure}[htb!]\n\\centerline{\\psfig{figure=fig4.eps,angle=270.,height=10.cm,width=14.cm}}\n\\vskip 5mm \\caption\n{Amplitude of the semiclassical formulae for period-one oscillations\nat $ \\theta=11^\\circ$, compared with quantum mechanical calculations [QM, dotted\nline with squares] as the dynamical parameter $\\epsilon$ varies.\n(a) Comparison between experimental results [EXP] and quantum\nmechanical results [QM].\n(b)Quantal results and semiclassical results for the PO\/NO formula, \nto which three POs contribute: $t_0$, the complex\nghost $t_{\\rm gh}$ and $t_1$.\nWe also show the contribution of $t_u\\!-\\!\\! NO$, which is not a PO.\n(c) Quantal results and the semiclassical CCO formula.\n(d) Quantal results and the semiclassical SO and MO formulae.\nThe figure shows that while both the SO and the MO formulae\ngive good agreement over the whole range, the PO\/NO\/CCO formulae\ngive agreement only over a partial range.}\n\\label{amp11}\n\\end{figure}\n\nThe periodic orbit theory is compared to quantum calculations in (b).\nFor $t_0, t_{\\rm gh}$ and $t_1$ we use the PO\/NO formula\n\\rf{gammasrno}, which is the common formula given by POs and NOs.\nThe semiclassical formula is accurate in both the chaotic\n($\\epsilon<3000$) and regular ($\\epsilon>10000$) regions.\nIn the latter, the semiclassical contribution can be understood by\nMiller torus quantization \\cit{Mil75}.\nThe large stable island of $t_0$ supports quantum states that are\napproximately harmonic oscillators (HO) functions in the plane\nperpendicular to the orbit; this will be discussed in more detail below.\nWe also see that the contribution of $t_u\\!-\\!\\! NO$ to the NO formula\nseems unrelated to the quantum behavior.\nNote the spike at $\\epsilon=6500$; this corresponds to the tangent\nbifurcation where $t_0$ and $t_0^-$ disappear.\nIt is {\\em not} a divergence, as the complex ${\\cal D}$ does not vanish.\nThe spike is due to the rapid variation of $\\Gamma$ near the bifurcation.\n\n\nThe most interesting region is $3000 < \\epsilon <6500$, where there is no\nreal contributing PO (the ``ghost'' region).\nWe included the contribution of the complex ghost PO $t_{\\rm gh}$, but \nwe see that its contribution is too small \\cit{SMR98}.\nA detailed view is shown in Fig. \\ref{det11} (a).\nThere we see that the ghost contribution is too small by roughly a\nfactor three compared to the QM results.\nThe saddle orbit (SO), on the other hand, describes accurately the QM\namplitudes all across the tangent bifurcation, the ghost region and\nthe region where $t_1\\!-\\!\\! PO$ takes over [see also Fig. \\ref{amp11} (d)].\nFinally, we see in Fig. \\ref{det11} (a) that the unstable partners of\nthe tangent bifurcations $t_0^-$ and $t_1^-$ do not contribute to the\ncurrent.\nThis is a general feature that we observed at any angle $\\theta$: only a\nvery small subset of trajectories give a significant semiclassical\ncontribution.\nA study of the amplitudes of period-two oscillation (not shown here) shows \nthat other POs like $t_v$ and $t_u$ are not related to the QM results\nalthough their contribution to the PO\/NO formula is significant.\n\n\n\\begin{figure}[htb!]\n\\centerline{\n\\psfig{figure=fig5a.eps,angle=270.,height=5.cm,width=7.cm}\n\\psfig{figure=fig5b.eps,angle=270.,height=5.cm,width=7.cm}}\n\\vskip 5mm \\caption\n{\n(a) Details of the period-one amplitudes at $ \\theta=11^\\circ$ in the \n low $\\epsilon$ region.\nThis is the region where no real PO exist, and where the contribution\nfrom the complex ghost PO $t_{\\rm gh}$\nis too small, while the SO contribution is\naccurate and joins up the contribution from the POs $t_0$ and $t_1$.\nWe also show the contribution of the unstable POs $t_0^-$ and $t_1^-$, \nwhich are not seen in the QM behaviour.\n(b) Comparison of experimental voltage periods (line with\ncrosses) with the semiclassical period\ngenerated by the SO $t_0$ (solid line).\n}\n\\label{det11}\n\\end{figure}\n\nThe contributions of central closed orbits (CCOs) are shown in\nFig. \\ref{amp11} (c).\nThe main objective of the CCO theory as presented by Narimanov and Stone \\cit{NS99} was to\ncomplement their PO theory in the absence of real PO (the ``ghost''\nregion).\nIt partially succeeds, as its amplitude for $3000< \\epsilon < 5000$\ncorresponds to the quantum one.\nHowever, it is clearly inaccurate for $\\epsilon>5000$ and gets worse in the \nregular region, where one could have expected the \nassumptions underlying the theory to be respected (in this regular\nregime, the oscillations of the Green's function should be smooth\ncompared to the localization of the initial state).\nSimilarly, the CCO theory is not very accurate in the chaotic region\n(low $\\epsilon$).\n\nWe show in (d) the result of the saddle (SO) and\nminimal orbit (MO) formulae.\nThere is only one SO and one MO corresponding to the three POs $t_0,\nt_{\\rm gh}$ and $t_1$.\nBoth theories are very accurate and reproduce the quantum amplitudes\nacross the whole transition from regular to chaotic dynamics.\nActually, the MO contribution is even more precise than the SO at very \nlow $\\epsilon$.\n\nFinally, we study in Fig. \\ref{det11} (b) the frequencies of the\noscillations via their voltage period $\\Delta V$.\nWe show the semiclassical period calculated with eq. \\rf{2pi} from the \nsaddle orbit $t_0$.\nWe do not show quantum periods, which are accurately described by the\nsemiclassics.\nThe theoretical periods underestimate the experimental values by some $10 \n\\%$.\nThis however is a confirmation of the fact that $t_0$ orbits are indeed\nlinked to the broad experimental oscillations.\n\n\nTorus quantization is illustrated in Fig. \\ref{torquant}, where we\npresent Wigner and wave functions of quantum states\ncontributing to the current.\nThe wave functions are approximately separable into a HO\n state and a WKB-type wave function along the trajectory.\n${\\cal N}_i$ gives roughly the number of longitudinal oscillations,\nwhile the number of perpendicular oscillations corresponds to the\npseudo quantum number $k$ of the HO state.\nOne can use this assumption to build a current as the overlap between\nthe initial state and the HO state \\cit{SM98}.\nThis is valid for stable POs, and it has been shown to be equivalent to \nthe PO \\cit{NSB98} and NO \\cit{BR98} formula in the case of\ntime-symmetric orbits.\nThe Wigner distributions show the ring structure associated with HO states.\n\n\n\n\\begin{figure}[tb!]\n\\centerline{\n\\psfig{figure=fig6a.eps,angle=0.,height=4.3cm,width=3.75cm}\n\\psfig{figure=fig6b.eps,angle=0.,height=4.3cm,width=3.75cm}\n\\psfig{figure=fig6c.eps,angle=0.,height=4.3cm,width=3.75cm}\n\\psfig{figure=fig6d.eps,angle=0.,height=4.4cm,width=3.752cm}\n}\n\\vspace{5mm}\n\\centerline{\n\\psfig{figure=fig6e.eps,angle=0.,height=4.3cm,width=3.75cm}\n\\psfig{figure=fig6f.eps,angle=0.,height=4.3cm,width=3.75cm}\n\\psfig{figure=fig6g.eps,angle=0.,height=4.3cm,width=3.75cm}\n\\psfig{figure=fig6h.eps,angle=0.,height=4.3cm,width=3.75cm}\n}\n\\vskip 5mm \\caption{\nQuantum state contributing to the current \nin the torus quantization regime at $ \\theta=11^\\circ, \\epsilon=15000$, labelled by\ntheir eigenvalue ${\\cal N}_i$ . \nFor the Wigner distributions on the emitter barrier (top row),\n the vertical axis is $z$ and the horizontal axis is $-p_z$; \nthe range is adapted to the size of the (classically allowed) surface\nof section. \nFor the wave functions (bottom rows) the vertical axis is \n$x \\in [0,L=2267] \\:{\\rm a.u.} $ and the horizontal axis is $-z \\in [-2000,2000] \\:{\\rm a.u.} $.\nFor ${\\cal N}_i=20.894$ we also show in solid lines the classical\nstructures: $t_0$ for the wave function, and the main features of\nthe Poincar\\'e surface of section (points representing trajectories\nhitting the left wall)\nfor the Wigner distribution.\nIn the first two rows, the torus numbers are (left to right):\n$k=1,3,2$ and $0$.\n}\n\\label{torquant}\n\\newpage\n\\end{figure}\n\n\nThe hard limit formulae for POs and NOs (not shown here; see\n\\cit{SM98} for a test of the HLNO formula)\ngreatly underestimate the\ncontribution of off-center POs, because they cannot take into account torus \nquantization, which gives some accessibility to the PO via outer tori\nwhich can extend to the $z=0$ region at the center of the initial\nstate.\nFinally, torus quantization effects can explain ``jumps'' in the\nexperimental current, when the dominant torus changes with the\nmagnetic field \\cit{SM98}.\n\t\t\\subsection{Comparison at $ \\theta=27^\\circ$: non-isolated orbits}\n\nWe present in Fig. \\ref{amp27} \namplitudes of period-two oscillations at $ \\theta=27^\\circ$ (they were called\n``peak-doubling'' regions by M\\\"uller \\al \\cit{Mul95} as there is a secondary\nvoltage oscillation compared to the broad period-one oscillation).\nThe quantum model describes qualitatively the rather broad\nexperimental period-two signal (a). \nHowever, it overestimates it by some $20 \\%$.\nThis would be consistent with uncertainties in the\nestimate of the decoherence time. \n$\\tau$.\n\n\\begin{figure}[htb!]\n\\centerline{\n\\psfig{figure=fig7.eps,angle=270.,height=10.cm,width=14.cm}}\n\\vskip 5mm \\caption\n{Amplitude of the semiclassical formulae for period-two oscillations\nat $ \\theta=27^\\circ$, compared with quantum mechanical calculations [QM, dotted\nline with squares] as the dynamical parameter $\\epsilon$ varies.\n(a) Experimental results [EXP]; they have been multiplied by\n$\\exp(T\/\\tau)$, where $T$ is the (real part of the) total time of\nthe contributing classical orbit and $\\tau \\simeq 0.115 \\:{\\rm ps}$ \nis a decoherence time associated with phonon scattering. \n(b) PO\/NO formula,\nto which three POs contribute: $s'$, $2t_0$ and $s_1$; \nnote the gap $3500 < \\epsilon <7700$ (``ghost'' region)\n between the take over of $s''$ and\nthe tangent bifurcation where $s'$ appears ($s'$ disappears at\n$\\epsilon=18000$ in a ``cusp bifurcation''.\n(c) CCO formula; note the extension of the semiclassical amplitude\ndown to $\\epsilon=4400$ in the ghost region, and the inaccurate\ncontribution of $2t_0$.\n(d) SO and MO formula; note how both formulae describe accurately the\nquantum amplitude from regular to chaotic across the ghost region\n(there is only one SO\/MO related to both POs $s'$ and $s''$).\n}\n\\label{amp27}\n\\end{figure}\n\nTwo types of orbit contribute to the period-two current, which have\nroughly double action and period than $t$-type orbits.\nFirst, we have the second repetition of $t$, that is a $2\\!\\!:\\!\\! 2$ orbit\nmaking two bounces on each barrier.\nSecondly, there are orbits of the $s$-type $(1\\!\\!:\\!\\! 2$), making one\nbounce on the left wall and two bounces on the right wall, with a\nturning point (where the particle runs out of kinetic energy)\nin-between.\n\nFirst we test the PO\/NO formula in (b).\nThe peak of the contribution of $2t_0$ corresponds to two successive\nperiod-doubling pitchfork bifurcations ($\\epsilon=12600$ and $14000$) of\nthe primitive PO $t_0$.\nThis peak can be understood via torus quantization effects: at the\nbifurcations,\nthe winding angle of the stable $t_0$ reaches the value $\\pi$; hence\ntwo period-one torus series corresponding to two successive $k$ numbers become\nexactly $\\pi$ out-of-phase and create a strong period-two signal.\n\nThere is a very large contribution from another PO ($s'$) in the same\nregion; it also describes qualitatively well the\nquantum behavior, as does $2t_0$.\nBoth POs give a current with very similar scaled frequencies:\n$\\hat{\\Sigma} \\simeq 1.9054 $ \nfor $s'$ and $\\hat{\\Sigma} \\simeq 1.9154$ for $2t_0$.\nHence, one should not consider their independent amplitudes, as is\ndone in Fig. \\ref{amp27} (b), but instead consider the coherent superposition\nof their oscillatory current.\nThis will be done below in Fig. \\ref{det27}; however it seems that\nthese two competing contributions cannot be easily separated.\n\n$s'$ appears at $\\epsilon=18000$ in a ``cusp bifurcation'' \\cit{NS98b}, due to \nthe discontinuity of the hard bounce on the left wall (increasing\n$\\epsilon$ makes the trajectory hit the left wall instead of having a\nturning point on the energy surface).\nThen it\nundergoes a synchronous pitchfork bifurcation at $\\epsilon=13600$, with\nthe non self-retracing PO $s_*$; there is no \neffect on the semiclassical current.\nFinally it disappears at $\\epsilon=7700$ in a tangent bifurcation, below\nwhich there is no real PO able to explain the quantum and experimental \nsignal until $\\epsilon=4000$, where another PO of the same type ($s''$)\ngives a significant contribution.\nHence one has another ``ghost'' region between $\\epsilon=4000$ and $7700$.\nThe low $\\epsilon$ quantum peak ($\\epsilon \\simeq 2000$) is well described by\n$s_1$ (a $1\\!\\!:\\!\\! 2$ PO which has one more cyclotron rotation than $s'$\nand $s''$).\nNote that the large spikes are not divergences.\n\nThe central closed orbit (CCO) formula is shown in Fig. \\ref{amp27}\n(c).\nThe low $\\epsilon$ is well described by the CCO $s_1$, with no spike\nas the one found in the PO\/NO formula.\nThe $s'$ contribution to the high $\\epsilon$ peak is significantly\nextended by the CCO formula, and\nfollows accurately the quantum amplitude down to $\\epsilon=4700$, where it has an\nunphysical spike.\nThe CCO theory also separates the contribution of $s'$ and $2t_0$, as\nthe latter appears where the former disappears, at $\\epsilon=21000$.\nHowever, the contribution of $2t_0$ is inaccurate.\n\nFinally, we show the saddle and minimal orbit formulae in (d).\nFor both theories, $s'$ describes accurately the quantum results all the way\nthrough the ghost region down to $\\epsilon=4000$, where there is no such spike\nas the one found with the CCO.\nThey also separate $2t_0$ and $s'$ in an accurate way, as $2t_0$\ndescribes precisely the quantum results for high $\\epsilon$ too.\nAs for $ \\theta=11^\\circ$, the SOs and MOs provide the best semiclassical\ndescription.\nWhile the success of the SOs is expected (as they are the correct\nsaddle point of the SPA integrations), the efficiency of the MOs is\nagain rather surprising.\nNote also the consequences of the non-periodicity of SOs, MOs and\nCCOs, for which the $2t_0$ orbit only appears around $\\epsilon \\simeq\n20000$, while the $t_0$ orbits (not shown here) exist at lower $\\epsilon$.\n\n\nBoth POs $2t_0$ and $s'$ give important contributions for the peak of the\nperiod-two signal at $ \\theta=27^\\circ$.\nTo build the coherent superposition\nof their oscillatory current, one needs to take into account the constant phase\n$(- i\\mu \\pi\/2)$ given by the number $\\mu$ of conjugate points [shown in\nFig. \\ref{det27} (a)].\nThe amplitude of their\ncollective contribution is given by the height of the peak of the Fourier\ntransform of the current, and is shown in Fig. \\ref{det27} (b).\n\n\n\\begin{figure}[htb!]\n\\centerline{\n\\psfig{figure=fig8a.eps,angle=270.,height=5.cm,width=7.cm}\n\\psfig{figure=fig8b.eps,angle=270.,height=5.cm,width=7.cm}}\n\\vskip 5mm \\caption\n{\n(a) Maslov index and (b) coherent superposition\n(``$2t_0+s'$'',solid line with dots, )\nof the $2t_0$ and $s'$ contributions. \nWe also show the contribution of the non self-retracing $s_*$ PO, and the \nQM results. \n(c) Experimental (line with crosses) and semiclassical (solid line) \nvoltage periods.\n}\n\\label{det27}\n\\end{figure}\n\nThe coherent superposition [$2t_0+s'$] is much larger than the\nquantum amplitudes, because the individual isolated contributions\nare already as high as the QM results.\nNote the discontinuous change at $\\epsilon=13600$; it is due to the\ndiscontinuous change of $\\mu$ for $s'$ at the pitchfork bifurcation.\nIt is clear that the coherent superposition of $2t_0$ and $s'$ cannot, \nwhatever their relative phases, describe accurately the quantum\nresults.\nNote that these POs are {\\em not} involved together in a bifurcation;\nthis is not the usual breakdown of semiclassics near a bifurcation,\nthat one could solve with the use of normal forms (cubic expansions of \nthe action).\nAlso, $2t_0$ and $s'$ seem to be well separated in position space\n(their starting \npositions are $z_0 \\simeq 0$ for $s'$ and $z_0 \\simeq 600$ for\n$2t_0$).\n\nLooking at quantum states contributing to the current reveals that\nthe two POs are, in some sense, not isolated.\nA Wigner distribution and the related wave function are shown in \nFig. \\ref{noniso-27}.\n\n\n\\begin{figure}[htb!]\n\\centerline{\n\\psfig{figure=fig9a.eps,angle=0.,height=6.cm,width=6.cm}\n\\psfig{figure=fig9b.eps,angle=0.,height=6.cm,width=6.cm}\n}\n\\vskip 5mm \\caption{ \nLeft: Wigner distribution (color plot) at $ \\theta=27^\\circ,\n\\epsilon=16000$ and classical Poincar\\'e surface of section (dots), in the\n$(-p_z,z)$ plane on the emitter wall.\nThe square on lower-right corner represents $\\hbar$.\nRight: corresponding wave function in the $(-z,x)$ plane, \nwith the POs $t_0$ and $s'$.\nThe scales are the same as in Fig. \\ref{torquant}.}\n\\label{noniso-27}\n\\end{figure}\n\nThe Wigner distribution has the familiar ring structure of a $k=2$ quantized \ntorus in the stable island surrounding $t_0$.\nThe ring is nevertheless distorted in some way and is also localized\non the (here stable) PO $s'$.\nSimilarly, both POs are within the region of the\nlocalization of the wave function in position space.\nWe can conclude that the quantum state cannot ``distinguish''\nthe two POs, and that the POs are hence\nnon-isolated: they contribute collectively to the quantum state and to \nthe current. The use of either SO or MO orbits, however, circumvents \nthis problem and yields good results throughout.\n\t\t\\subsection{Comparison at $ \\theta=20^\\circ$: divergence of the\nsaddle orbit formula} \n\nFig. \\ref{amp20} presents amplitudes for the period-two signal at $ \\theta=20^\\circ$.\nThe situation is similar to $ \\theta=27^\\circ$; we do not present here the CCO formula.\nThe low $\\epsilon$ quantum peak can be described rather well by the PO\n(a), SO (b) or MO (c) formulae.\n\nThe quantum model can describe qualitatively the shape of the\nexperimental peak at high $\\epsilon$ (c), over a large range of\nparameters ($10000 < \\epsilon <40000$).\nThe $15 \\%$ discrepancy is probably due to a small inaccuracy in the\nestimate of the decoherence time $\\tau$.\nSemiclassically, we have the same two competing orbits as at $ \\theta=27^\\circ$.\nAlthough its contribution is important, it seems that $s'$ does not\ninfluence the quantum amplitudes.\nNote the difference between the $s'$ contribution given by the PO\/NO\nformula from the one given by the SO and MO formulae.\n\n\n\\begin{figure}[htb!]\n\\centerline{\n\\psfig{figure=fig10.eps,angle=270.,height=10.cm,width=14.cm}}\n\\vskip 5mm \\caption\n{The different semiclassical theories for P2 amplitudes at $ \\theta=20^\\circ$.\nThe broad maximum is related to $2t_0$ and $s'$ orbits,\nwhile the lower maximum is given by $s_1$ orbits.\nWe show QM results and experimental readings [EXP].\n(a) PO\/NO formula.\n(b) SO formula.\nThe inset is in a larger scale.\n(c) MO formula and experimental results.\n}\n\\label{amp20}\n\\end{figure}\nThe contribution of $2t_0$ to the PO\/NO formula is good, but the\nposition of the peak is not very accurate.\nThe SO formula yields unexpected results: it has a very large peak (see \ninset), which we shall investigate below.\nThe MO formula gives the correct position for the peak, but the height \nis not as\nprecise as could be expected.\n\n\n\n\nWe investigate in Fig. \\ref{det20} the large spike of the amplitude of\nthe contribution of $2t_0\\!-\\!\\! SO$ to the saddle orbit formula.\nWe see in (a) that the reason for this is the fact that the\ndeterminant $\\cal D$ [eq. \\rf{cald}] of the quadratic expansion used\nin the SPA integration almost vanishes around $\\epsilon =17500 - 17800$.\nBoth its real (solid line) and imaginary (dashed line)\npart simultaneously approach zero.\nThis is a remarkable coincidence, as $\\cal D$ is a complex\nfunction, that one expects to vanish only if one can vary a parameter in the \n{\\em complex} plane (i.e., two real parameters).\nIn this case, varying $\\epsilon$ only on the real axis approaches very\nclosely the zero of $\\cal D$, which should be reached for\na value of $\\epsilon$ with a small imaginary component.\n\n\\begin{figure}[htb!]\n\\centerline{\n\\psfig{figure=fig11.eps,angle=270.,height=8.cm,width=12.cm}}\n\\vskip 5mm \\caption\n{\nClassical characteristics of the SO $2t_0$ around the spike at $\\epsilon\n\\sim 17000-18000, \\theta=20^\\circ$. The solid line indicates the real component\nand the dashed line indicates the imaginary component.\n(a) Scaled determinant $\\tilde{\\cal D}={\\cal D}\/B$ [eq. \\rf{cald}] \nof the quadratic\nexpansion used in the SPA integration.\n(b) Scaled action $\\hat{S}$.\n(c) Starting position $z_0$.\n(d) Real shape $( {\\rm I} \\hspace{-1.0mm} {\\rm Re} \\: x, {\\rm I} \\hspace{-1.0mm} {\\rm Re} \\: z)$ of $2t_0 \\!-\\!\\! SO$ at $\\epsilon=17476$;\n for fig. (d) the dashed line indicates\nthe limit of the region allowed by ({\\em real dynamics}) \nenergy conservation. \n}\n\\label{det20}\n\\end{figure}\n\nThe classical characteristics of $2t_0\\!-\\!\\! SO$ in that region are not\nsmooth.\nThe real part of the action (b) reaches a maximum\nvalue at $\\epsilon=17588$, while the imaginary part changes abruptly over\nthe range $17000 < \\epsilon <18000$.\nThe imaginary part of the starting position (c) behaves similarly.\nThe real part of the starting position reaches the minimum value\n$ {\\rm I} \\hspace{-1.0mm} {\\rm Re} \\: z_0=-89 \\:{\\rm a.u.} $ at $\\epsilon=17476$.\n\nWe show in (d) the real shape of $2t_0\\!-\\!\\! SO$ at $\\epsilon=17476$.\nThe outer leg (with high $ {\\rm I} \\hspace{-1.0mm} {\\rm Re} \\: z$) hits the left wall perpendicularly \nat a point which is very close (less than $20 \\:{\\rm a.u.} $) to the limit\nsurface of\nthe region accessible by trajectories defined by real dynamics.\nThis limit is where some self-retracing trajectories (like $s'$ and\n$s_1$) have a turning point (i.e., have zero momentum). \nIt is impossible to find trajectories with the same bounce structure\n($2 \\!\\!:\\!\\! 2$) for $ {\\rm I} \\hspace{-1.0mm} {\\rm Re} \\: z_0 < -100$, because they would miss the\nintermediate bounce and go back to the right wall via the limit surface.\nThis is similar to the ``cusp bifurcation'' of the PO $s'$; it is also \nthe only observed mechanism that can remove an SO, MO or CCO as $\\epsilon$ \nchanges.\n\nAs $\\epsilon$ increases, $ {\\rm I} \\hspace{-1.0mm} {\\rm Re} \\: z_0$ decreases until it reaches $-89 \\:{\\rm a.u.} $.\nIf it evolved smoothly, one would expect the SO at higher $\\epsilon$ to\nstart with a lower $ {\\rm I} \\hspace{-1.0mm} {\\rm Re} \\: z_0$ --which seems impossible as we have\nseen that such starting condition did not allow the correct bounce\nstructure.\nHence, one would expect ``a cusp disappearance'' of $2t_0\\!-\\!\\! SO$.\nThis nevertheless does not happen: one can still find the SO for\nhigher $\\epsilon$, as abruptly increasing $ {\\rm I} \\hspace{-1.0mm} {\\rm Re} \\: z_0$ satisfies the SO\ncondition.\nHence, it seems that we have a ``failed'' cusp disappearance of $2t_0\\!-\\!\\! \nSO$, precisely in the region where the quadratic expansion becomes\nalmost degenerate and where the classical characteristics of the orbit \nare not smooth.\n\nThis is altogether reminiscent of a bifurcation of periodic orbits,\nwhere two POs coalesce as the dynamical parameter ($\\epsilon$) is varied, \nand where the quadratic expansion of the action (used e.g. in the GTF) \nbecomes degenerate.\nHowever, PO bifurcations always involve more than one POs, while in\nthis case we have not observed any neighboring SO that could coalesce \nwith $2t_0\\!-\\!\\! SO$.\n\nNote that, strictly, one should not integrate \\rf{integral2} on the\nwhole $(z,z')$ plane, but only on the domain $\\Omega$\n where one does find the proper type of trajectories ($2\\!\\!:\\!\\! 2$).\nHence one should cut the integral for $z < -100$; this would yield\nError functions. \n\nThis situation raises very interesting questions about saddle\norbits.\n{\\em Do they undergo a type of bifurcation?} Apparently no, as a\nbifurcation should involve several SOs, which we have not seen.\n{\\em What is the origin the quasi-degeneracy?}\nIt comes from a failed cusp disappearance. \n{\\em Does it has an effect on the semiclassical current?}\nYes, the current shows a strong enhancement.\n{\\em Do quantum results show any sign of it?} Apparently no, as\nthe QM results are smooth in that region.\n{\\em What techniques can be employed to solve that problem?}\nWe tried a cubic expansion of the action, which removed most of the\nenhancement but did not show good agreement with quantum calculations. \nOne could try a cut-off in the integral.\nThe first task would be to locate the complex value of $\\epsilon$ for which\n${\\cal D} =0$.\n\n\n\n\t\t\\section{Conclusion}\n\nThe general semiclassical formula \\rf{generalformula} that we have\ndeveloped summarizes in a compact way all the theories that have been\nproposed for the current in the RTD (excluding the changes required by\nthe inclusion of excited Landau states and a shift of the Gaussian\n\\cit{NSB98,Sara99}).\nIt also shows clearly how the different assumptions on the smoothness\nof the Gaussian affect which type of orbits contribute to the current.\nWe found the types giving the best semiclassical description: the\ncomplex saddle orbits (SOs) and the real minimal orbits (MOs).\nIt appears that the more standard periodic or closed orbits do not\nsucceed in all the situations, in particular in the ghost regions and\nwhere one has non-isolated contributions.\n\nThe (near) divergence of the saddle orbit formula at $ \\theta=0^\\circ$ raises interesting\nquestions about the SOs, namely the existence of\na bifurcation-type phenomenon for trajectories defined by a pair of\ncondition \\rf{condso} as restrictive as the one defining POs (i.e.,\ngiving a discrete number of orbits).\nThe complex dynamics we implemented here\nare very strongly restricted by the definition of the hard\nbounces on the barriers.\nOne could obviously avoid this point by\nmodeling the barriers by soft exponential walls.\nA preliminary study shows that the ``cusp'' bifurcations are replaced\nby standard tangent bifurcations; however the search for complex POs\nand SOs with soft barriers appears to be more difficult and again\nraises interesting questions about the high-dimensional search for\ncomplex POs and SOs.\n\nThe techniques used in this work could easily be applied to other\nsystems involving Gaussian matrix elements, in particular Franck-Condon\ntransitions and the conductance of microcavities with parabolic leads\n(where one further assumes that the lowest sub-band is occupied \\cit{Nari98}).\nThe importance of saddle or minimal orbits would then depend on the\nlocalization scale of the Gaussian.\n\nThe authors would like to thank E Bogomolny and D Rouben for invaluable\nhelp with the semiclassics and the complex dynamics, E Narimanov for\ncommunication of results prior to publication, and G Boebinger\nfor unpublished experimental data.\nT S M acknowledges financial support from the EPSRC. D S S was\nsupported by a TMR scholarship from the Swiss National Science Foundation.\n\n\t\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nGiven a multiset $M$ of $n$ positive integers, a word on $M$ is a\nsequence of positive integers $w = w_1w_2 \\cdots w_n$ that reorders\n$M$. A {\\em statistic on words} is an association of an element of\n$\\mathbb{N}$ to each word. A fundamental statistic that has been\nrediscovered in many guises is the {\\em inversion number} of a word,\ndefined as the number of pairs of indices $(i\nw_j$. A {\\em descent} of a word is an index $i$ such that $w_i >\nw_{i+1}$. In 1913, Major P. MacMahon \\cite{MacMahon1913} introduced an\nimportant statistic, now called the {\\em major index} in his honor,\ndefined as the sum over the descents of a word. Using generating\nfunctions, MacMahon \\cite{MacMahon1916} proved the remarkable fact\nthat the major index has the same distribution as the inversion\nnumber. Precisely, he showed that for $W_M$ the set of words on a\nfixed multiset $M$,\n\\begin{displaymath}\n \\sum_{w \\in W_M} q^{\\ensuremath\\mathrm{maj}(w)} = \\sum_{w \\in W_M} q^{\\ensuremath\\mathrm{inv}(w)},\n\\end{displaymath}\nwhere $\\ensuremath\\mathrm{maj}(w)$ denotes the major index of $w$ and $\\ensuremath\\mathrm{inv}(w)$ denotes\nthe inversion number of $w$. Any statistic that is equidistributed\nwith the major index, i.e. a statistic satisfying the above equation,\nis called {\\em Mahonian}. MacMahon then raised the question to find a\nbijective proof that the inversion number is Mahonian. This question\nwas first resolved by Foata \\cite{Foata1968}, who constructed a\nbijection on words with the property that the major index of a word\nequals the inversion number of its image.\n\nIn this paper, we introduce a statistic called the $k$-major index\nwhich interpolates between the major index and inversion number. More\nprecisely, the $1$-major index is MacMahon's major index, and the\n$n$-major index of a word of length $n$ is the inversion number. By\nconstructing bijections on words with a recursive structure similar to\nFoata's bijection, we give a bijective proof that the $k$-major index\nis Mahonian for all $k$. Looking back through the literature, this\nsame statistic was discover by Kadell \\cite{Kadell1985} who also gave\na bijective proof that the distribution is Mahonian. Whereas Kadell's\nbijections in fact refine Foata's original bijection, the family of\nbijections defined herein is not the same as Kadell's and, when taking\nthe major index to the inversion number, give a bijection different\nfrom that of Foata. The $k$-major index statistic is defined in\n\\refsec{stats}, and the bijections and proof that the distribution is\nMahonian are given in \\refsec{bijections}.\n\nIt is also natural to define a major index statistic on standard Young\ntableaux, which are central objects in the study of symmetric\nfunctions. Recently, Haglund and Stevens \\cite{HaSt2006} defined an\ninversion number on tableaux. Their construction generalizes Foata's\nbijection to tableaux and shows that the inversion number and major\nindex are equidistributed over standard Young tableaux of a fixed\nshape. Motivated by this, we use the bijections presented here to\nextend the notion of the $k$-major index to standard Young tableaux,\nfor $k \\leq 3$. The hope is that this method might be used to build a\ncomplete family of statistics interpolating between major index and\ninversion number on tableaux. This exploration takes place in\n\\refsec{tableau}.\n\nOur discovery of the $k$-major index and the family of bijections\npresented here came about through the study of Macdonald polynomials\n\\cite{Assaf2007-2}. In \\refsec{macdonald}, we elaborate on this\nconnection and present a conjecture for yet another family of\nbijections sharing many of the same properties that would have the\nfurther consequence of providing a remarkably simple combinatorial\nproof of Macdonald positivity.\n\n\n\\section{Definitions and notation}\n\\label{sec:stats}\n\nAt times it will be convenient to consider a slightly more general\ndefinition for a word $w$, where $w_i$ is allowed to be either a\npositive integer or an $\\emptyset$. In this case, $\\emptyset$'s should\nbe regarded as incomparable to other letters, so that they are simply\na way of spacing out the nonempty letters of $w$. This idea will be\nespecially important in connection with Macdonald polynomials\ndiscussed in \\refsec{macdonald}.\n\n\\begin{definition}\n For $w$ a word, $k$ a positive integer, define the {\\em $k$-descent\n set of $w$}, denoted $\\ensuremath\\mathrm{Des}_k(w)$, by\n \\begin{displaymath}\n \\ensuremath\\mathrm{Des}_k(w) = \\{ (i,i+k) \\; | \\; w_i > w_{i+k} \\} ,\n \\end{displaymath}\n and define the {\\em $k$-inversion set of $w$}, denoted $\\ensuremath\\mathrm{Inv}_k(w)$,\n by\n \\begin{displaymath}\n \\ensuremath\\mathrm{Inv}_k(w) = \\{ (i,j) \\; | \\; k > j-i > 0 \\; \\mbox{and} \\; w_i >\n w_j \\} .\n \\end{displaymath}\n\\end{definition}\n\nFor example, for $w = 986173245$ and $k=3$ we have\n\\begin{eqnarray*}\n \\ensuremath\\mathrm{Des}_3(9 \\; 8 \\; 6 \\; 1 \\; 7 \\; 3 \\; 2 \\; 4 \\; 5) & = & \\{\n (1,4), (2,5), (3,6), (5,8)\\}, \\\\ \n \\ensuremath\\mathrm{Inv}_3(9 \\; 8 \\; 6 \\; 1 \\; 7 \\; 3 \\; 2 \\; 4 \\; 5) & = & \\{ (1,2),\n (1,3), (2,3), (2,4), (3,4), (5,6), (5,7), (6,7)\\}.\n\\end{eqnarray*}\n\nIn fact, it is enough to define $k$-descents since $k$-inversions may\nbe recovered from the observation\n\\begin{equation}\n \\ensuremath\\mathrm{Inv}_k(w) \\; = \\; \\bigcup_{j < k} \\ensuremath\\mathrm{Des}_j(w) .\n\\label{eqn:alt-Invk}\n\\end{equation}\n\nNote that when $k=1$, $\\ensuremath\\mathrm{Des}_k$ gives the usual descent set for a\nword. Similarly, when $N \\geq n$, $\\ensuremath\\mathrm{Inv}_N$ gives the usual set of\ninversion pairs for a word of length $n$. We interpolate between the\ncorresponding statistics, $\\ensuremath\\mathrm{maj}$ and $\\ensuremath\\mathrm{inv}$, with the following\nstatistic depending on the parameter $k$.\n\n\\begin{definition}\n Given a word $w$ and a positive integer $k$, define the {\\em\n $k$-major index of $w$} by\n \\begin{displaymath}\n \\ensuremath\\mathrm{maj}_k (w) \\; = \\; \\left| \\ensuremath\\mathrm{Inv}_k(w) \\right| + \\sum_{(i,i+k) \\in\n \\ensuremath\\mathrm{Des}_k(w)} i .\n \\end{displaymath}\n\\end{definition} \n\nFor the same example, we have $\\ensuremath\\mathrm{maj}_3(9 \\; 8 \\; 6 \\; 1 \\; 7 \\; 3 \\; 2\n\\; 4 \\; 5) = 8 + 1 + 2 + 3 + 5 = 19$. For a word $w$ of length $n \\leq\nN$, the previous observations show that\n\\begin{eqnarray*}\n \\ensuremath\\mathrm{maj}_1(w) & = & \\ensuremath\\mathrm{maj}(w), \\\\\n \\ensuremath\\mathrm{maj}_N(w) & = & \\ensuremath\\mathrm{inv}(w).\n\\end{eqnarray*}\nThe statistic $\\ensuremath\\mathrm{maj}_k$ was first defined by Kadell \\cite{Kadell1985},\nwho gives a bijective proof that this statistic is Mahonian. Kadell's\nbijections take $\\ensuremath\\mathrm{inv}$ to $\\ensuremath\\mathrm{maj}_k$, with the extreme case from $\\ensuremath\\mathrm{inv}$\nto $\\ensuremath\\mathrm{maj}$ corresponding precisely to the inverse of Foata's bijection\n\\cite{Foata1968}. In \\refsec{bijections}, we give a different family\nof bijections, taking $\\ensuremath\\mathrm{maj}_{k-1}$ to $\\ensuremath\\mathrm{maj}_k$, which, when composed\nappropriately, give a different bijection from $\\ensuremath\\mathrm{maj}$ to $\\ensuremath\\mathrm{inv}$.\n\nIn the case when $w$ is a permutation (possibly with $\\emptyset$s), we\nwill also be interested in the descent set of the inverse permutation,\ndenoted $\\ensuremath\\mathrm{iDes}$, defined by\n\\begin{equation}\n \\ensuremath\\mathrm{iDes}(w) = \\ensuremath\\mathrm{Des}(w^{-1}) = \\{ i \\ | \\ \\mbox{$i$ appears to the left\n of $i+1$ in $w$} \\} .\n\\end{equation}\nFor example, $\\ensuremath\\mathrm{iDes}(9 \\; 8 \\; 6 \\; 1 \\; 7 \\; 3 \\; 2 \\; 4 \\; 5) =\n\\{2,5,7,8\\}$. \n\nRecall that a {\\em partition} $\\lambda$ is a weakly decreasing\nsequence of positive integers:\n$\\lambda=(\\lambda_1,\\lambda_2,\\ldots,\\lambda_m)$, $\\lambda_1 \\geq\n\\lambda_2 \\geq \\cdots \\geq \\lambda_m > 0$. A partition $\\lambda$ may\nbe identified with its {\\em Young diagram}: the set of points $(i,j)$\nin the $\\mathbb{Z}_+ \\times\\mathbb{Z}_+$ lattice quadrant such that $1\n\\leq i \\leq \\lambda_j$. We draw the diagram so that each point $(i,j)$\nis represented by the unit cell southwest of the point. A {\\em\n standard Young tableau of shape $\\lambda$} is a labelling of the\ncells of the Young diagram of $\\lambda$ with the numbers $1$ through\n$n$, where $n = \\sum_i \\lambda_i$, such that the entries increase\nalong rows and up columns. For example, see \\reffig{tableau}.\n\n\\begin{figure}[ht]\n \\begin{center}\n \\begin{displaymath}\n \\tableau{8 \\\\ 2 & 5 & 6 \\\\ 1 & 3 & 4 & 7}\n \\end{displaymath}\n \\caption{\\label{fig:tableau}A standard Young tableau of shape\n $(4,3,1)$.}\n \\end{center}\n\\end{figure}\n\nFor a standard Young tableau $T$, recall the {\\em descent set of $T$},\ndenoted $\\ensuremath\\mathrm{Des}(T)$, defined by\n\\begin{equation}\n \\ensuremath\\mathrm{Des}(T) = \\{ (i,i+1) \\; | \\; i \\; \\mbox{lies strictly south of} \\;\n i+1 \\; \\mbox{in} \\; T \\}.\n\\label{eqn:DesT}\n\\end{equation}\nCompletely analogous to the case with words, define the {\\em major\n index of $T$}, denoted $\\ensuremath\\mathrm{maj}(T)$, by\n\\begin{equation}\n \\ensuremath\\mathrm{maj}(T) = \\sum_{(i,i+k) \\in \\ensuremath\\mathrm{Des}(T)} i .\n\\label{eqn:majT}\n\\end{equation}\nFor the example in \\reffig{tableau}, $\\ensuremath\\mathrm{Des} = \\{(1,2), (4,5), (7,8)\\}$\nand so $\\ensuremath\\mathrm{maj} = 1+4+7 = 12$. The descent set for tableaux corresponds\nto the descent set of permutations in the sense that for a fixed set\n$D$,\n\\begin{displaymath}\n \\# \\{ w \\in \\mathcal{S}_n \\; | \\; \\ensuremath\\mathrm{Des}(w) = D \\} =\n \\sum_{\\lambda} f^{\\lambda} \\cdot \\# \\{ T \\in \\ensuremath\\mathrm{SYT}(\\lambda) \\; | \\;\n \\ensuremath\\mathrm{Des}(T) = D \\},\n\\end{displaymath}\nwhere $\\ensuremath\\mathrm{SYT}(\\lambda)$ denotes the set of standard Young tableaux of\nshape $\\lambda$ and $f^{\\lambda} = |\\ensuremath\\mathrm{SYT}(\\lambda)|$. This identity can\nbe proved using the Robinson-Schensted-Knuth correspondence which\nbijectively associates each permutation $w$ with a pair of standard\ntableaux $(P,Q)$ of the same shape such that $\\ensuremath\\mathrm{iDes}(w) = \\ensuremath\\mathrm{Des}(Q)$. We\npostpone the definition of $\\ensuremath\\mathrm{Des}_k$ and $\\ensuremath\\mathrm{maj}_k$ for \\refsec{tableau}.\n\n\n\\section{A family of bijections on words}\n\\label{sec:bijections}\n\nFor $k \\geq 2$, we will construct bijections $\\phi^{(k)}$ on words of\nlength $n$ such that \n\\begin{equation}\n \\ensuremath\\mathrm{maj}_{k-1}(w) = \\ensuremath\\mathrm{maj}_k(\\phi^{(k)}(w)) .\n\\label{eqn:dist}\n\\end{equation}\nAs noted earlier, these bijections are not equivalent to those defined\nby Kadell, and the appropriate composition does not give Foata's\nbijection. That said, the construction below follows the idea of\n\\cite{Foata1968} in that $\\phi^{(k)}$ will be defined recursively\nusing an involution $\\gamma_{j}^{(k)}$ which permutes the letters of a\ngiven word. \n\nLet $x,a,b$ be (not necessarily distinct) integers. Say that $x$\n{\\em splits} the pair $a,b$ if $a \\leq x < b$ or $b \\leq x < a$. Let\n$w$ be a word of length $n$. For $k \\geq 2$ and $j \\leq n$, define a\nset of indices $\\Gamma_{j}^{(k)}$ of $w$ by\n\\begin{equation}\n j-k \\in \\Gamma_{j}^{(k)}(w) \\;\\; \\mbox{if} \\;\\; w_j\n \\;\\mbox{splits the pair} \\; w_{j-k}, w_{j-k+1},\n\\label{eqn:Gamma-i}\n\\end{equation}\n and if $i \\in \\Gamma_{j}^{(k)}(w)$, then\n\\begin{equation}\n i-k \\in \\Gamma_{j}^{(k)}(w) \\;\\; \\mbox{if exactly one of} \\; w_{i}\n \\;\\mbox{or}\\; w_{i+1} \\;\\mbox{splits the pair}\\; w_{i-k},w_{i-k+1}.\n\\label{eqn:Gamma-r}\n\\end{equation}\nFor our running example, we have $\\Gamma_{8}^{(3)}(9 \\; 8 \\; 6 \\; 1 \\;\n7 \\; 3 \\; 2 \\; 4 \\; 5) = \\{ 5, 2\\}$.\n\nLet permutations act on words by permuting the indices, i.e. $\\tau\n\\cdot w \\; = \\; w_{\\tau(1)} w_{\\tau(2)} \\cdots w_{\\tau(n)}$. Define a\nmap $\\gamma_{j}^{(k)}$ by\n\\begin{equation}\n \\gamma_{j}^{(k)}(w) \\; = \\; \\left( \\prod_{i \\in \\Gamma_{j}^{(k)}(w)} (i,\n i+1) \\right) \\cdot w .\n\\label{eqn:gammak}\n\\end{equation}\nThat is to say, $\\gamma_{j}^{(k)}(w)$ is the result of interchanging\n$w_{i}$ and $w_{i+1}$ for all $i \\in \\Gamma_{j}^{(k)}(w)$. Back to our\nrunning example, we have $\\gamma_{8}^{(3)}(9 \\; \\mathbf{8} \\;\n\\mathbf{6} \\; 1 \\; \\mathbf{7} \\; \\mathbf{3} \\; 2 \\; 4 \\; 5) = 9 \\;\n\\mathbf{6} \\; \\mathbf{8} \\; 1 \\; \\mathbf{3} \\; \\mathbf{7} \\; 2 \\; 4 \\;\n5$.\n\nFor $w$ a word of length $n$, define $\\phi^{(k)}$ by\n\\begin{equation}\n \\phi^{(k)} (w) \\; = \\; \\gamma_{n}^{(k)} \\circ \\gamma_{n-1}^{(k)}\n \\circ \\cdots \\circ \\gamma_{1}^{(k)} (w).\n\\label{eqn:phik}\n\\end{equation}\nSince $\\gamma_j^{(k)}$ is the identity for $j \\leq k$, these terms may\nbe omitted from equations \\ref{eqn:phik} and \\ref{eqn:psik}.\n\nFor example, for $w=6 \\; 9 \\; 3 \\; 8 \\; 1 \\; 7 \\; 2 \\; 4 \\; 5$,\n$\\phi^{(3)}(w)$ is computed as follows.\n\\begin{displaymath}\n \\begin{array}{rcccccccccc}\n w \\phantom{)} & = & \n 6 & 9 & 3 & 8 & \\rnode{m3}{1} & 7 & 2 & 4 & 5 \\\\\n \\gamma_4^{(3)}(w) & = & \n 9 & 6 & 3 & 8 & \\rnode{m4}{1} & 7 & 2 & 4 & 5 \\\\\n \\gamma_{5}^{(3)}\\gamma_4^{(3)}(w) & = & \n 9 & 6 & 3 & 8 & \\rnode{m5}{1} & 7 & 2 & 4 & 5 \\\\\n \\gamma_{6}^{(3)}\\gamma_{5}^{(3)}\\gamma_4^{(3)}(w) & = & \n 9 & 6 & 8 & 3 & \\rnode{m6}{1} & 7 & 2 & 4 & 5 \\\\\n \\gamma_{7}^{(3)}\\gamma_{6}^{(3)}\\gamma_{5}^{(3)}\\gamma_4^{(3)}(w) & = & \n 9 & 6 & 8 & 1 & \\rnode{m7}{3} & 7 & 2 & 4 & 5 \\\\\n \\gamma_{8}^{(3)}\\gamma_{7}^{(3)}\\gamma_{6}^{(3)}\\gamma_{5}^{(3)}\\gamma_4^{(3)}(w) & = & \n 9 & 8 & 6 & 1 & \\rnode{m8}{7} & 3 & 2 & 4 & 5 \\\\\n \\phi^{(3)}(w) = \\gamma_{9}^{(3)}\\gamma_{8}^{(3)}\\gamma_{7}^{(3)}\\gamma_{6}^{(3)}\\gamma_{5}^{(3)}\\gamma_4^{(3)}(w) & = & \n 9 & 8 & 6 & 1 & \\rnode{m9}{7} & 3 & 2 & 4 & 5\n \\end{array}\n \\psset{nodesep=3pt,linewidth=.1ex}\n \\everypsbox{\\scriptstyle}\n\\end{displaymath}\nNotice that for this example $\\ensuremath\\mathrm{maj}_2(w) = 19 =\n\\ensuremath\\mathrm{maj}_3(\\phi^{(3)}(w))$. Before proving \\refeq{dist} in general, we\ntake note of a few important properties that $\\phi^{(k)}$ shares with\nFoata's bijection (for Foata, properties (i) and (ii) are shown in\n\\cite{Foata1968}, and property (iii) is shown in \\cite{FoSc1978}).\n\n\\begin{proposition}\n For each $k \\geq 2$, we have\n \\begin{itemize}\n \\item[(i)] the map $\\phi^{(k)}$ is a bijection on words on $M$ with\n fixed $\\emptyset$ positions;\n \\item[(ii)] for $w$ a word of length $n$, $w_{n-k+1} > w_n$ if and\n only if $\\phi^{(k)}(w)_{n-k} > \\phi^{(k)}(w)_n = w_n$;\n \\item[(iii)] for $w$ a permutation, $\\ensuremath\\mathrm{iDes}(w) = \\ensuremath\\mathrm{iDes}(\\phi^{(k)}(w))$.\n \\end{itemize}\n\\label{prop:props}\n\\end{proposition}\n\n\\begin{proof}\n Since $\\Gamma_{j}^{(k)}(\\gamma_{j}^{(k)}(w)) = \\Gamma_{j}^{(k)}(w)$,\n $\\gamma_{j}^{(k)}$ is an involution on words of length $n$ for all\n $j \\leq n$ and $k \\geq 2$. Therefore $\\phi^{(k)}$ is a bijection on\n words of length $n$ for all $k \\geq 2$ with inverse given by\n \\begin{equation}\n \\psi^{(k)} (w) \\; = \\; \\gamma_{1}^{(k)} \\circ \\cdots \\circ\n \\gamma_{n-1}^{(k)} \\circ \\gamma_{n}^{(k)} (w) .\n \\label{eqn:psik}\n \\end{equation}\n It is clear from the definition of $\\gamma_j^{(k)}$ that\n $\\phi^{(k)}$ in fact fixes the last $k-1$ letters of a word, so\n indeed the last letter is fixed for every $k$. Let $u =\n \\gamma_{n-1}^{(k)} \\cdots \\gamma_{1}^{(k)}(w)$, $u_{j} = w_{j}$ for\n $j \\geq n-k+1$. If $u_{n-k}$ and $u_{n-k+1}$ compare the same with\n $u_{n}$, then $u = \\phi^{(k)}(w)$ and (ii) clearly holds; otherwise,\n these two letters are interchanged by $\\gamma_{n}^{(k)}$, again\n showing that (ii) is satisfied. Also note that $\\phi^{(k)}$ may be\n defined recursively by\n \\begin{equation}\n \\phi^{(k)}(wx) = \\gamma_{n+1}^{(k)} \\left( \\phi^{(k)}(w) \\right) x ,\n \\label{eqn:recursivePhi}\n \\end{equation}\n which completely parallels Foata's original construction. Finally,\n since consecutive letters cannot be split, in the sense of\n $\\Gamma_j^{(k)}$, they may never be interchanged by\n $\\gamma_j^{(k)}$. Thus the inverse descent set is preserved.\n\\end{proof}\n\nTo prove \\refeq{dist}, we follow the strategy of \\cite{Foata1968}. The\nkey, therefore, lies in the following lemma.\n\n\\begin{lemma}\n For $k \\geq 2$, $w$ a word of length $n$ and $j \\leq n$,\n \\begin{displaymath}\n \\ensuremath\\mathrm{maj}_k \\left( \\gamma_{j}^{(k)}(w_1 \\cdots w_{j-1}) \\right) \n \\; = \\; \\ensuremath\\mathrm{maj}_k(w_1 \\cdots w_{j-1}) + \n \\left\\{ \\begin{array}{rl}\n 1 & \\mbox{if} \\;\\; w_{j-k} > w_j \\geq w_{j-k+1}, \\\\\n -1 & \\mbox{if} \\;\\; w_{j-k+1} > w_j \\geq w_{j-k}, \\\\\n 0 & \\mbox{otherwise}.\n \\end{array} \\right.\n \\end{displaymath}\n\\label{lem:gamma}\n\\end{lemma}\n\n\\begin{proof}\n If neither of the first two cases holds, then $j-k \\not\\in\n \\Gamma_{j}^{(k)}(w)$, so $\\gamma_{j}^{(k)}(w) = w$ and the result is\n immediate. Assume, then, that $j-k \\in \\Gamma_{j}^{(k)}(w)$, and set\n $u = w_{j-k} w_{j-k+1} \\cdots w_{j-1}$. Then\n \\begin{equation}\n \\ensuremath\\mathrm{maj}_k\\left(\\gamma_{j}^{(k)}(u)\\right) = \n \\ensuremath\\mathrm{maj}_k(u) + \\left\\{ \\begin{array}{rl}\n 1 & \\mbox{if} \\;\\; w_{j-k} > w_j \\geq w_{j-k+1} , \\\\\n -1 & \\mbox{if} \\;\\; w_{j-k+1} > w_j \\geq w_{j-k} .\n \\end{array} \\right.\n \\label{eqn:majk-u}\n \\end{equation}\n For $i \\in \\Gamma_{j}^{(k)}(w)$, let $u = w_{i} w_{i+1} \\cdots\n w_{j-1}$, and, by induction, assume that \\refeq{majk-u} holds for\n $u$. Let $u' = w_{i-k} w_{i-k+1} \\cdots w_{j-1}$. We will show that\n $u'$ also satisfies \\refeq{majk-u} by considering the contribution\n to $\\ensuremath\\mathrm{maj}_k$ of $w_{i-k},w_{i-k+1}, \\ldots, w_{i-1}$. For $i-k+1 < h\n < i$, $k$-inversions and $k$-descents involving $w_h$ are the same\n for $u'$ and $\\gamma_{j}^{(k)}(u')$, so we need only consider\n contributions from the potential $k$-inversions $(i-k,i-k+1)$ and\n $(i-k+1,i)$, and the potential $k$-descents $(i-k,i)$ and\n $(i-k+1,i+1)$.\n \n First suppose that $i-k \\in \\Gamma_{j}^{(k)}(w)$. \n In all eight possible scenarios for $w_{i-k},w_{i-k+1},w_{i},w_{i+1}$,\n we have\n \\begin{eqnarray*}\n (i-k,i-k+1) \\in \\ensuremath\\mathrm{Inv}_k(w) & \\Leftrightarrow & (i-k,i-k+1) \\not\\in\n \\ensuremath\\mathrm{Inv}_k\\left(\\gamma_{j}^{(k)}(w)\\right), \\\\\n (i-k,i) \\in \\ensuremath\\mathrm{Des}_k(w) & \\Leftrightarrow & (i-k+1,i+1) \\in\n \\ensuremath\\mathrm{Des}_k\\left(\\gamma_{j}^{(k)}(w)\\right), \\\\\n (i-k+1,i+1) \\in \\ensuremath\\mathrm{Des}_k(w) & \\Leftrightarrow & (i-k,i) \\in\n \\ensuremath\\mathrm{Des}_k\\left(\\gamma_{j}^{(k)}(w)\\right).\n \\end{eqnarray*}\n \n If both or neither of $(i-k,i)$ and $(i-k+1,i+1)$ are $k$-descents\n of $w$, then the same holds for $u'$ and $\\gamma_{j}^{(k)}(u')$. In\n this case \n exactly one of $(i-k,i-k+1)$ and $(i-k+1,i)$ is a $k$-inversion for\n $w$, and\n \\begin{eqnarray*}\n (i-k+1,i) \\in \\ensuremath\\mathrm{Inv}_k(u') & \\Leftrightarrow & \n (i-k+1,i) \\not\\in \\ensuremath\\mathrm{Inv}_k\\left(\\gamma_{j}^{(k)}(u')\\right).\n \\end{eqnarray*}\n The lemma now follows. On the other hand, if exactly one of\n $(i-k,i)$ and $(i-k+1,i+1)$ is a $k$-descent of $w$, \n then the difference in the contribution to $\\ensuremath\\mathrm{maj}_k$ from the\n potential $k$-descents is offset by the difference from the\n potential $k$-inversion $(i-k,i-k+1)$. Furthermore,\n \\begin{eqnarray*}\n (i-k+1,i) \\in \\ensuremath\\mathrm{Inv}_k(u') & \\Leftrightarrow & \n (i-k+1,i) \\in \\ensuremath\\mathrm{Inv}_k\\left(\\gamma_{j}^{(k)}(u')\\right),\n \\end{eqnarray*}\n thereby establishing the result.\n \n To complete the proof, note that when $i-k \\not\\in\n \\Gamma_{j}^{(k)}(w)$, either $w_{i}$ and $w_{i+1}$ compare the same\n with $w_{i-k}$ and also with $w_{i-k+1}$ and so the $k$-inversions\n and $k$-descents beginning with $i-k$ or $i-k+1$ are unchanged, or\n the $k$-descent at $(i-k+1,i+1)$ is exchanged for a $k$-descent at\n $(i-k,i)$ along with a $k$-inversion at $(i,i+1)$. In both cases the\n contribution to the $k$-major index is preserved.\n\\end{proof}\n\n\\begin{proposition}\n For $k \\geq 2$ and $w$ a word, $\\displaystyle{\\ensuremath\\mathrm{maj}_{k-1} (w) =\n \\ensuremath\\mathrm{maj}_k \\left( \\phi^{(k)}(w) \\right)}$.\n\\label{prop:majk}\n\\end{proposition}\n\n\\begin{proof}\n The result is clear for a words of length $\\leq k$. We proceed by\n induction, assuming the result for words of length $n-1$. Let $w$\n be a word of length $n-1$ and $x$ a letter. To simplify notation,\n let\n $$\n u = \\gamma_{n}^{(k)} \\left( \\phi^{(k)}(w) \\right).\n $$\n By expanding the definition of $\\ensuremath\\mathrm{maj}_k$ and applying\n \\reflem{gamma}, we have\n \\begin{displaymath}\n \\begin{array}{l}\n \\ensuremath\\mathrm{maj}_k \\left(\\phi^{(k)}(wx)\\right) \\\\\n \\hspace{2em} = \\ensuremath\\mathrm{maj}_k(ux) \\\\ \n \\hspace{2em} = \\ensuremath\\mathrm{maj}_k(u) + \\# \\{ i > n-k \\; | \\; u_i > x \\}\n + \\left\\{ \\begin{array}{rl}\n 0 & \\mbox{if} \\; x \\geq u_{n-k} \\\\\n n\\!-\\!k & \\mbox{if} \\; u_{n-k} > x \\end{array} \\right. \\\\\n \\hspace{2em} = \\ensuremath\\mathrm{maj}_k(u) + \\# \\{i>n-k+1 \\; | \\; u_i > x\\} \n + \\left\\{ \\begin{array}{rcrl}\n n\\!-\\!k + 1 & \\mbox{if} & u_{n-k} > x, & u_{n-k+1} > x \\\\\n 0 + 0 & \\mbox{if} & x \\geq u_{n-k}, & x \\geq u_{n-k+1} \\\\\n n\\!-\\!k + 0 & \\mbox{if} & u_{n-k} > x, & x \\geq u_{n-k+1} \\\\\n 0 + 1 & \\mbox{if} & x \\geq u_{n-k}, & u_{n-k+1} > x\n \\end{array} \\right. \\\\\n \\hspace{2em} = \\ensuremath\\mathrm{maj}_k \\left( \\gamma_{n}^{(k)}(u) \\right)\n + \\# \\{i>n-k+1 \\; | \\; u_i > x\\} \n + \\left\\{ \\begin{array}{rl}\n n\\!-\\!k\\!+\\!1 + 0 & \\mbox{if} \\; u_{n-k},u_{n-k+1} > x \\\\\n 0 + 0 & \\mbox{if} \\; x \\geq u_{n-k},u_{n-k+1} \\\\\n n\\!-\\!k\\ + 1 & \\mbox{if} \\; u_{n-k} > x \\geq u_{n-k+1} \\\\\n 1 - 1 & \\mbox{if} \\; u_{n-k+1} > x \\geq u_{n-k}\n \\end{array} \\right. \\\\\n \\hspace{2em} = \\ensuremath\\mathrm{maj}_{k-1} \\left( \\phi^{(k)}(w) \\right) \n + \\# \\{i>n-k+1 \\; | \\; u_i > x\\}\n + \\left\\{ \\begin{array}{rl}\n 0 & \\mbox{if} \\; x \\geq u_{n-k} \\\\\n n\\!-\\!k\\!+\\!1 & \\mbox{if} \\; u_{n-k} > x\n \\end{array} \\right.\n \\end{array}\n \\end{displaymath}\n\n Recall from \\refprop{props} that for $i \\geq n-k+2$, $u_i = w_i$,\n and so\n $$\n \\{i>n-k+1 \\; | \\; u_i > x\\} \\; = \\; \\{i>n-k+1 \\; | \\; w_i > x\\} .\n $$\n Furthermore, since $\\phi^{(k)}(w)_{n-k+1} = w_{n-k+1}$, we also have\n $$\n u_{n-k} \\leq x \\; \\Leftrightarrow \\; w_{n-k+1} \\leq x .\n $$\n\n Continuing from the above equation using these two facts and the\n inductive hypothesis, we have\n \\begin{displaymath}\n \\ensuremath\\mathrm{maj}_k \\left(\\phi^{(k)}(wx)\\right) \n = \\ensuremath\\mathrm{maj}_{k-1}(w) \n + \\# \\{i>n-k+1 \\; | \\; w_i > x\\} + \\left\\{ \\begin{array}{rl}\n 0 & \\mbox{if} \\; x \\geq w_{n-k+1} \\\\\n n\\!-\\!k\\!+\\!1 & \\mbox{if} \\; w_{n-k+1} > x\n \\end{array} \\right.\n \\end{displaymath}\n which is exactly $\\ensuremath\\mathrm{maj}_{k-1} (wx)$, as desired.\n\\end{proof}\n\nFor $1 \\leq h < i$, we can compose these bijections to form the\nbijection\n\\begin{equation}\n \\phi^{[i,h]} = \\phi^{(i)} \\circ \\cdots \\circ \\phi^{(h+1)}\n\\end{equation}\nsatisfying $\\ensuremath\\mathrm{maj}_h (w) = \\ensuremath\\mathrm{maj}_i \\left( \\phi^{[i,h]}(w) \\right)$. In\nparticular, $\\phi^{[k,1]}$ provides a bijective proof of the\nfollowing.\n\n\\begin{theorem}\n Let $W_M$ be the set of words on a multiset $M$ with a fixed\n $\\emptyset$ positions. Then for $k \\geq 1$,\n \\begin{eqnarray*}\n \\sum_{w \\in W_M} q^{\\ensuremath\\mathrm{maj}(w)} & = &\n \\sum_{w \\in W_M} q^{\\ensuremath\\mathrm{maj}_k(w)} .\n \\end{eqnarray*}\n That is to say, the $k$-major index has Mahonian distribution.\n\\label{thm:mahonian}\n\\end{theorem}\n\n\n\\section{Extending the $k$-major index to tableaux}\n\\label{sec:tableau}\n\nIn \\cite{HaSt2006}, Haglund and Stevens define an inversion number for\nstandard tableaux which is equidistributed with the major\nindex. Therefore it is natural to try to extend the $k$-major index\nstatistic to tableaux in a similar manner. However, to do this, we\nmust first define $\\ensuremath\\mathrm{Des}_k$ for standard Young tableaux.\n\nConsider the possible relative positions of $i$ and $i+k$ in a\nstandard Young tableau $T$. Since $i < i+k$, $i$ must lie strictly\nwest or strictly south of $i+k$. If $i$ lies strictly west and weakly\nnorth of $i+k$, then the pair $(i,i+k)$ should not count as a\n$k$-descent. Conjugately, if $i$ lies strictly south and weakly east\nof $i+k$, then the pair $(i,i+k)$ should count as a $k$-descent. The\ndifficulty arises in how to resolve the situation where $i$ lies\nstrictly southwest of $i+k$. The approach given in \\cite{HaSt2006} is\nquite involved as it is based on {\\em inversion paths} which must be\ncomputed iteratively. In most cases, interchanging even two\nconsecutive entries in a tableau completely alters the inversion paths\nin an opaque way. Therefore we begin at the other extreme, though\nbelow we succeed only up to $k=3$.\n\nFor $k=2$, the ambiguous case when $i$ lies strictly southwest of\n$i+2$ cannot arise in a standard tableaux. However, for $k=3$ we must\ndecide whether $(i-3,i)$ is a $3$-descent when $i-3,i-2,i-1,i$ appear\nin a $2\\times 2$ box in $T$. For reasons that will be made clear, we\nresolve the situations as indicated in \\reffig{3Des}.\n\n\\begin{figure}[ht]\n \\begin{center}\n \\begin{displaymath}\n \\begin{array}{\\ccii\\ccii}\n \\bigtableau{i\\!\\!-\\!\\!1 & i \\\\ i\\!\\!-\\!\\!3 & i\\!\\!-\\!\\!2} & \n \\bigtableau{i\\!\\!-\\!\\!2 & i \\\\ i\\!\\!-\\!\\!3 & i\\!\\!-\\!\\!1} \\\\[2\\vsp]\n (i-3,i) \\in \\ensuremath\\mathrm{Des}_3 & (i-3,i) \\not\\in \\ensuremath\\mathrm{Des}_3\n \\end{array}\n \\end{displaymath} \n \\caption{\\label{fig:3Des} Ambiguous cases for whether $(i-3,i)$\n should constitute a $3$-descent.}\n \\end{center} \n\\end{figure}\n\nTo simplify notation, we introduce the following terminology. For\n$i} {l}{c} \\naput{\\gamma^{(2)}_{4}}\n \\ncline{->} {c}{r} \\naput{\\gamma^{(2)}_{6}}\n \\end{displaymath}\n \\caption{\\label{fig:Phi2}An example of $\\Phi^{(2)}$; here\n $\\gamma^{(2)}_{j} = \\mathrm{id}$ for $j\\neq 4,6$.}\n \\end{center} \n\\end{figure}\n\nSimilar to before, the inverse of $\\Phi^{(k)}$ is given by composing\nthe maps $\\gamma^{(k)}_j$ in the reverse order. This establishes the\nanalogue of the property (i) of \\refprop{props}, and the analogue of\nproperty (ii) is that the largest letter of $T$ is fixed by\n$\\Phi^{(k)}$. As property (iii) has no real analogue in this setting, we\nmove on to the more important statement observed in the example,\nnamely the analogue of \\refprop{majk} below.\n\n\\begin{proposition}\n For $T$ a standard Young tableau and $k=2,3$, we have\n $\\displaystyle{\\ensuremath\\mathrm{maj}_{k-1} (T) \\; = \\; \\ensuremath\\mathrm{maj}_k \\left( \\Phi^{(k)}(T)\n \\right)}$.\n\\label{prop:majk-T}\n\\end{proposition}\n\n\\begin{proof}\n We use the proofs of \\reflem{gamma} and \\refprop{majk}. For this to\n make sense, we make the substitution that for $i < n$, $w_i > w_n$\n should be interpreted as ``i attacks n'' and similarly $w_i \\leq\n w_n$ should be interpreted as ``i does not attack n''. In order for\n the arguments to remain valid under this translation, interchanging\n entries using $\\gamma_j^{(k)}$ may not change $k$-inversions or\n $k$-descents between unmoved entries. The only potential violation\n of this is the potential $3$-descent between $i-3$ and $i$ in the\n situations depicted in \\reffig{3Des}. However, in either case $i-2\n \\not\\in \\Gamma^{(3)}_j$ since neither $i+1$ nor $i+2$ can split the\n pair $i-2, i-1$. Therefore, with this translation, the proofs carry\n through verbatim.\n\\end{proof}\n\n\\begin{theorem}\n For $\\lambda$ a partition, we have\n \\begin{equation}\n \\sum_{T \\in \\ensuremath\\mathrm{SYT}(\\lambda)} q^{\\ensuremath\\mathrm{maj}(T)} =\n \\sum_{T \\in \\ensuremath\\mathrm{SYT}(\\lambda)} q^{\\ensuremath\\mathrm{maj}_{2}(T)} =\n \\sum_{T \\in \\ensuremath\\mathrm{SYT}(\\lambda)} q^{\\ensuremath\\mathrm{maj}_{3}(T)} .\n \\end{equation}\n \\label{thm:mahonian-T}\n\\end{theorem}\n\nUnfortunately, \\refthm{mahonian-T} is the best we can do towards\nextending \\refthm{mahonian} using this direct analogue of\n$\\phi^{(k)}$. This technique breaks down at $k=4$ for the shape\n$(2,2,2)$. In this case, the $6$ must lie in the northeast corner and\nwill necessarily interchange the $2$ and $3$ if they both lie in the\nfirst two rows. Then if $1,2,3,4$ occupy the first two rows, this\nchanges whether $(1,4)$ is a $3$-descent ($4$-inversion). In order to\novercome this shortfall, either we must adopt a more dynamic notion of\n$k$-inversions as in the Haglund-Stevens approach or a more\ncomplicated bijection. \n\n\n\\section{Connections with Macdonald polynomials}\n\\label{sec:macdonald}\n\nThe $k$-major index statistic was rediscovered in the author's study\nof Macdonald polynomials. In this section we connect the results of\n\\refsec{bijections} back to Macdonald polynomials.\n\nIn \\cite{HHLRU2005}, Bylund and Haiman introduced the $k$-inversion\nnumber of a $k$-tuple of tableaux to be the number of inversions\nbetween certain entries, and it is shown that this statistic may be\nused to give an alternative definition for Lascoux-Leclerc-Thibon\npolynomials. In particular, when each shape of the $k$-tuple is a\nribbon, i.e. contains no $2 \\times 2$ block, the $k$-inversion number\nof the $k$-tuple is exactly $|\\ensuremath\\mathrm{Inv}_k(w)|$ where $w$ is a certain\nreading word of the $k$-tuple. In further study of these objects\n\\cite{Assaf2007-2}, it became natural to associate to each $k$-tuple\nnot only the $k$-inversion number, but also a $k$-descent set. Again,\nwhen the shapes of the $k$-tuple in question are all ribbons, this is\nexactly given by $\\ensuremath\\mathrm{Des}_k(w)$ for the same reading word $w$. Here it is\nessential that $w$ be allowed to contain $\\emptyset$'s in order to\ncorrectly space the entries of the $k$-tuple.\n\nThe case when the $k$-tuple consists entirely of ribbons is an\nimportant special case in light of \\cite{Haglund2004,HHL2005} where it\nis shown that the Macdonald polynomials are in fact positive sums of\nLLT polynomials where the shapes are ribbons. In this context, the\nindex $k$ is given by the number of columns of the indexing partition\nof the Macdonald polynomial. The Macdonald Positivity Theorem,\nconjectured by Macdonald in 1988 \\cite{Macdonald1988}, was first\nproved by Haiman using algebraic geometry \\cite{Haiman2001}, and more\nrecently by Grojnowski and Haiman using Kazhdan-Lusztig theory\n\\cite{GrHa2007} and the author using a purely combinatorial argument\n\\cite{Assaf2007-2}. This latter proof, while purely combinatorial,\nrelies on new combinatorial machinery, namely {\\em dual equivalence\n graphs}, involving rather technical proofs of the main\ntheorems. Below we suggest how Macdonald positivity may be recovered\nin a completely elementary way using bijections similar to\n$\\phi^{(k)}$.\n\nThe main idea behind \\cite{Assaf2007-2} is to group together terms of\na Macdonald polynomial which contribute to a single Schur function and\nhave the same associated statistics. This is done in three steps; for\ncomplete details, see \\cite{Assaf2007-3}. First, quasisymmetric\nfunctions are used to reduce to standard words, i.e. permutations, and\nit is here that the inverse descent set of a permutation is\nrelevant. Next, for a given $k$, the permutations are divided into\nequivalence classes (in the language of \\cite{Assaf2007-2}, connected\ncomponents of a graph) using the following involutions.\n\nFor $i \\geq 2$, define involutions $d_i$ and $\\tilde{d}_i$ on permutations\nwhere $i$ does not lie between $i-1$ and $i+1$ by\n\\begin{eqnarray}\n d_i (\\cdots\\; i \\;\\cdots\\;i\\pm 1\\;\\cdots\\;i\\mp 1\\;\\cdots ) \n & = & \\cdots\\;i\\mp 1\\;\\cdots\\;i\\pm 1\\;\\cdots\\; i \\;\\cdots \\; ,\n \\label{eqn:d} \\\\\n \\tilde{d}_i(\\cdots\\; i \\;\\cdots\\;i\\pm 1\\;\\cdots\\;i\\mp 1\\;\\cdots ) \n & = & \\cdots\\;i\\pm 1\\;\\cdots\\;i\\mp 1\\;\\cdots\\; i \\;\\cdots \\; ,\n \\label{eqn:dwig}\n\\end{eqnarray}\nwhere all other entries remain fixed. Combining these, define\n$D^{(k)}_i$ by\n\\begin{equation}\n D^{(k)}_i(w) \\; = \\; \\left\\{\n \\begin{array}{ll}\n d_i(w) & \\mbox{if} \\;\\; \\mathrm{dist}(i-1,i,i+1) > k \\\\\n \\tilde{d}_i(w) & \\mbox{if} \\;\\; \\mathrm{dist}(i-1,i,i+1) \\leq k\n \\end{array} \\right. ,\n \\label{eqn:Dk}\n\\end{equation}\nwhere $\\mathrm{dist}(i-1,i,i+1)$ is the maximum distance between the positions\nof $i-1,i,i+1$ in $w$. The $\\emptyset$'s, or spacers, in $w$ are\nessential for this step as they adjust the relative distance of the\nletters of $w$.\n\n\\begin{definition}\n Call two permutations $w$ and $u$ {\\em $k$-equivalent}, denoted\n $w \\sim_{k} u$, if $w = D^{(k)}_{i_1} D^{(k)}_{i_2} \\cdots\n D^{(k)}_{i_m}(u)$ for some sequence $i_1,i_2,\\ldots,i_m \\geq 2$.\n\\end{definition}\n\n\\begin{figure}[ht]\n \\begin{center}\n \\begin{displaymath}\n \\begin{array}{\\ccr \\ccr \\ccr \\ccr c}\n \\mbox{$1$-classes:} &\n \\left\\{ 1 \\ 2 \\ 3 \\right\\}; &\n \\left\\{ 2 \\ 1 \\ 3 \\ , \\ 3 \\ 1 \\ 2 \\right\\}; &\n \\left\\{ 2 \\ 3 \\ 1 \\ , \\ 1 \\ 3 \\ 2 \\right\\}; &\n \\left\\{ 3 \\ 2 \\ 1 \\right\\} \\\\[.3\\vsp]\n \\mbox{$2$-classes:} &\n \\left\\{ 1 \\ 2 \\ 3 \\right\\}; &\n \\left\\{ 2 \\ 1 \\ 3 \\ , \\ 1 \\ 3 \\ 2 \\right\\}; &\n \\left\\{ 2 \\ 3 \\ 1 \\ , \\ 3 \\ 1 \\ 2 \\right\\}; &\n \\left\\{ 3 \\ 2 \\ 1 \\right\\}\n \\end{array}\n \\end{displaymath} \n \\caption{\\label{fig:classes} Equivalence classes of permutations\n of length $3$.}\n \\end{center}\n\\end{figure}\n\n\\begin{remark}\n Note that the $1$-equivalence classes are exactly the {\\em dual\n equivalence classes} for partitions; see \\cite{Haiman1992}. In\n particular, the sum of the quasisymmetric functions associated to\n the permutations in a $1$-equivalence class is a Schur function.\n\\label{rmk:dec}\n\\end{remark}\n\nA key observation in \\cite{Assaf2007-2} is that $\\ensuremath\\mathrm{Des}_k(w) =\n\\ensuremath\\mathrm{Des}_k\\left( D^{(k)}_{i} (w) \\right)$ and $|\\ensuremath\\mathrm{Inv}_k(w)| = |\\ensuremath\\mathrm{Inv}_k\\left(\n D^{(k)}_{i} (w) \\right)|$. In particular, $\\ensuremath\\mathrm{Des}_k$ and $|\\ensuremath\\mathrm{Inv}_k|$\nare constant on $k$-equivalence classes. Therefore, the third and\nfinal step toward establishing the Macdonald Positivity Theorem is to\nprove that the sum over the quasisymmetric functions associated to a\ngiven $k$-equivalence class is Schur positive. By \\refrmk{dec}, a\nnatural approach is to relate $k$-classes to $1$-classes. Indeed, the\nproof presented in \\cite{Assaf2007-2} does this by showing that a\nconnected component of the graph for $k$-columns (a $k$-equivalence\nclass) may be broken into a union of connected dual equivalence graphs\n($1$-equivalence classes). It is for this step that the proof becomes\nquite technical and involved, and so the idea is to bypass the\nmachinery of dual equivalence graphs altogether. The following\nproposition achieves this for the $2$-column\/$2$-equivalence class\ncase.\n\n\\begin{proposition}\n For $w$ a permutation such that $i$ does not lie between $i-1$ and\n $i+1$, we have\n \\begin{equation}\n \\phi^{(2)} \\left( D^{(1)}_i(w) \\right) = D^{(2)}_i \\left(\n \\phi^{(2)}(w) \\right) .\n \\end{equation}\n\\label{prop:2equiv}\n\\end{proposition}\n\n\\begin{proof}\n First note that $D^{(1)}_i = d_i$. Furthermore, $D^{(2)}_i(w) =\n d_i(w)$ unless $i-1,i,i+1$ are adjacent in $w$. Without loss of\n generality, we may assume that $w_r = i+1$, $w_s = i-1$ and $w_t =\n i$ for some indices $r 2$ in\n $\\gamma_t \\cdots \\gamma_1 (w)$. In the affirmative case,\n $\\Gamma_{t}(w) = \\{t-2\\}$ since both or neither $i-1,i+1$ splits any\n pair of preceding letters and $\\Gamma_{t}(\\widetilde{w}) =\n \\emptyset$. Therefore $\\gamma_t \\cdots \\gamma_1 (w) =\n \\tilde{d}_i \\left( \\gamma_t \\cdots \\gamma_1 (\\widetilde{w})\n \\right)$ as desired since $\\mathrm{dist}(i-1,i,i+1) = 2$ in $\\gamma_t \n \\cdots \\gamma_1 (w)$.\n\n Now consider the effect of $\\gamma_j$ for $j > t$. For the same\n reasons as before, $\\Gamma_{j}(w) \\neq \\Gamma_{j}(\\widetilde{w})$ if and only\n if $i-1,i,i+1$ are adjacent either before or after $\\gamma_j$ is\n applied. For $\\widetilde{w}$, the relative positions of $i-1,i,i+1$ will never\n change. Moreover, the position of $i+1$ in $\\widetilde{w}$ tracks the position\n of $i$ in $w$, and the positions of $i,i-1$ in $\\widetilde{w}$ are the\n positions of $i-1,i+1$ in $w$ (though not necessarily\n respectively). For $w$, each time $i$ moves between adjacent and\n nonadjacent to $i-1,i+1$, the difference between $\\Gamma_j$ for $w$\n and $\\widetilde{w}$ is exactly that the former contains the index of the\n leftmost of $i-1,i+1$ and the latter does not. Comparing $d_i$ with\n $\\tilde{d}_i$, this is exactly the difference between the two\n involutions, i.e. $i-1$ and $i+1$ interchange positions. Therefore\n in the end, $\\tilde{d}_i(\\phi^{(2)}(w)) = \\phi^{(2)}(\\widetilde{w})$ if\n $i-1,i,i+1$ are adjacent in $\\phi^{(2)}(w)$, and $d_i(\\phi^{(2)}(w))\n = \\phi^{(2)}(\\widetilde{w})$ otherwise.\n\\end{proof}\n\nRecall that the sum over of an equivalence class is determined by the\nquasisymmetric functions associated to the permutation of the\nclass. Since the quasisymmetric function associated to a permutation\nis determined by the inverse descent set of the permutation,\n\\refprop{props} (iii) and \\refrmk{dec} establish the following\ncorollary to \\refprop{2equiv}.\n\n\\begin{corollary}\n Macdonald polynomials indexed by partitions with $2$ columns are\n Schur positive.\n\\label{cor:2pos}\n\\end{corollary}\n\nFor $k \\geq 3$, it is not possible for $\\phi^{(k)} \\left(\n D^{(k-1)}_i(w) \\right) = D^{(k)}_i \\left( \\phi^{(k)}(w) \\right)$ in\ngeneral. The reason for this is that the sizes of the $k$-equivalence\nclasses increase with $k$. For permutations of length $n$, the\n$n$-equivalence classes have a nice description given in\n\\cite{Assaf2007-2} which allows us to prove the following.\n\n\\begin{proposition}\n For $w$ a permutation such that $i$ does not lie between $i-1$ and\n $i+1$, we have\n \\begin{equation}\n \\phi^{[1,n]} (w) \\sim_{n} \\phi^{[1,n]} \\left( D^{(1)}_i(w) \\right) .\n \\end{equation}\n\\label{prop:nequiv}\n\\end{proposition}\n\n\\begin{proof}\n For this case, $\\ensuremath\\mathrm{maj}_n = \\ensuremath\\mathrm{inv}$ in the usual sense and there are no\n $n$-descents to consider. As already noted, $\\ensuremath\\mathrm{inv}$ is constant on\n $n$-equivalence classes and, since $D^{(n)}_i \\equiv \\tilde{d}_i$,\n $w_1 > w_n$ for some $w$ in an $n$-class if and only if $w_1 > w_n$\n for every $w$ in an $n$-class. Furthermore, it is not difficult to\n show that these two properties completely characterize $n$-classes.\n Since $D^{(1)}_i \\equiv d_i$, $w_{n-1} > w_n$ for some $w$ in a\n $1$-class if and only if $w_{n-1} > w_n$ for every $w$ in a\n $1$-class. By \\refprop{majk}, $\\ensuremath\\mathrm{maj}(w) = \\ensuremath\\mathrm{inv}(\\phi^{[1,n]} (w))$,\n and by \\refprop{props} (ii), $w_{n-1} > w_n$ if and only if\n $\\phi^{[1,n]} (w)_1 > \\phi^{[1,n]} (w)_n$. Therefore if $w \\sim_1\n u$, then $\\ensuremath\\mathrm{inv}(\\phi^{[1,n]} (w)) = \\ensuremath\\mathrm{inv}(\\phi^{[1,n]} (u))$ and\n $\\phi^{[1,n]} (w)_1 > \\phi^{[1,n]} (w)_n$ if and only if\n $\\phi^{[1,n]} (u)_1 > \\phi^{[1,n]} (u)_n$. The result now follows.\n\\end{proof}\n\n\\begin{corollary}\n Macdonald polynomials indexed by a single row are Schur positive.\n\\label{cor:npos}\n\\end{corollary}\n\nGiven this, one might still hope to express each $k$-equivalence class\nas a union of the images of certain $k-1$-equivalence classes under an\nappropriate map. However, for $k \\geq 3$, neither $\\phi^{(k)}$ nor the\ncorresponding composition of Kadell's bijections accomplishes\nthis. There is, however, considerable evidence suggesting that such a\nfamily of bijections does exist, and so we conclude with the following\nconjecture which, as a corollary, would yield a simple proof of\nMacdonald positivity.\n\n\\begin{conjecture}\n There exists a family of bijections $\\theta^{(k)}$ on permutations\n satisfying Propositions \\ref{prop:props} and \\ref{prop:majk} such\n that if $w \\sim_{k-1} u$ then $\\theta^{(k)}(w) \\sim_{k}\n \\theta^{(k)}(u)$.\n\\label{conj:theta}\n\\end{conjecture}\n\n\n\n\n\\bibliographystyle{amsalpha}\n\n\n\n\n\n\n\\bibliographystyle{abbrv} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nHD~140283 (V = 7.21; Casagrande et al. 2010) is a subgiant, metal-poor \nstar in the solar neighbourhood, which is analysed extensively in the literature, \nand has historical importance in the context of the existence of \nmetal-deficient stars (Chamberlain \\& Aller 1951). \nIn recent decades, element abundances in HD~140283 were studied \nby several authors, and a bibliographic compilation up to 2010 can be found \nin the PASTEL catalogue (Soubiran et al. 2010). Most recently, Frebel \\& Norris (2015) \nstressed the importance of this star to the history of the discovery of the most metal-poor \nstars in the halo. More recently, hydrodynamical \n3D models and NLTE computations were applied for lithium lines, for example, by \nLind et al. (2012) and Steffen et al. (2012). \nThe fractions of odd and even barium isotopes in HD~140283 have been the subject of \nintense debate, given that the even-Z isotopes are only produced by the \nneutron capture s-process, whereas the odd-Z isotopes are produced by both \nthe s- and r-processes (Gallagher et al. 2010, 2012, 2015). \nBased on UV spectra, the molybdenum abundance in HD~140283 was derived by \nPeterson (2011). Roederer (2012) also used UV lines to obtain abundances of \nzinc (\\ion{Zn}{II}), arsenic (\\ion{As}{I}), and selenium (\\ion{Se}{I}), \namong other elements, in addition to upper limits for germanium (\\ion{Ge}{I}) \nand platinum (\\ion{Pt}{I}). Siqueira-Mello et al. (2012), hereafter Paper I, \nanalysed the origin of heavy elements in HD~140283 deriving the europium \nabundance and making the case for an r-process contribution in this star. \n\nBond et al. (2013) derived an age of 14.46$\\pm$0.31 Gyr for HD~140283, using \na trigonometric parallax of 17.15$\\pm$0.14 mas measured with the Hubble \nSpace Telescope, making this object the oldest known star for which a \nreliable age has been determined. Bond et al. employed evolutionary tracks \nand isochrones computed with the University of Victoria code (VandenBerg et \nal. 2012), with an adopted helium abundance of Y$=$0.250, and including \neffects of diffusion, revised nuclear reaction rates, and enhanced \noxygen abundance. More recently, VandenBerg et al. (2014) presented a \nrevised age of 14.27$\\pm$0.38 Gyr. This age is slightly larger than \nthe age of the universe of 13.799$\\pm$0.038 Gyr based on the cosmic \nmicrowave background (CMB) radiation as given by the \nPlanck collaboration (Adam et al. 2015). \nAccording to VandenBerg et al. (2014), uncertainties, particularly in the oxygen \nabundance and model temperature to observed colour relations, can explain \nthis difference, but the remote possibility that this object is older than \n14 Gyr cannot be excluded. HD~140283 is therefore a very old \nstar that must have formed soon after the Big Bang. In the future, \nasteroseismology could help to verify the age of HD~140283.\n\nIn this work, we carry out a detailed analysis and abundance derivation \nfor HD~140283. The main motivation for this study was triggered by the \ncontroversy discussed above about the interpretation of barium isotopic abundances, \nand the possibility of testing whether heavy elements \nare produced by the r- or s-process. With this purpose, we \nobtained a seven-hour exposure high-S\/N spectrum, with a wavelength coverage \nin the range 3700~$<\\lambda({\\rm \\AA})<$~10475 for this star.\n\nIn Sect. 2 the observations are reported. \nIn Sect. 3 the atmospheric parameters are derived. \nIn Sect. 4 the abundances computed in LTE and NLTE are presented. \nIn Sect. 5 the results are discussed, and final conclusions are drawn in Sect. 6.\n\n\\section {Observations and reductions}\n\nHD~140283 was observed in programme 11AB01 (PI: B. Barbuy) \nat the CFHT telescope with the spectrograph \nESPaDOnS in Queue Service Observing (QSO) mode, to obtain a spectrum \nin the wavelength range 3700-10475~{\\AA} with a resolving power of \nR~$=$~81 000. The observations were carried out in 2011, June 12, 14, 15, \nand 16, and July 8. The total number of 23 individual spectra with 20 min \nexposure each produced a total exposure time of more than seven hours. \nThe data reduction was performed using the software \nLibre-ESpRIT, a new release of ESpRIT (Donati et al. 1997), running within \nthe CFHT pipeline Upena\\footnote{http:\/\/www.cfht.hawaii.edu\/Instruments\/Upena\/index.html}. \nThis package facilitates reducing all exposures automatically, and further fits continua \nand normalizes to 1. The co-added spectrum was obtained after radial velocity correction and \na S\/N ratio of 800~$-$~3400 per pixel was obtained. Three spectra were \ndiscarded because of their lower quality as compared with the average.\n\n\\section{Atmospheric parameters}\n\n\n\\subsection{Measurement of equivalent widths}\n\\label{EW}\n\nTo derive the atmospheric parameters and abundances, \nwe measured the equivalent widths (EWs) of several iron and titanium \nlines in their neutral and ionized states using a semi-automatic code, \nwhich traces the continuum and uses a Gaussian profile to fit the \nabsorption lines, as described in Siqueira-Mello et al. (2014). The \ncode can deal with blends on the wings, excluding the parts of the \nline from the computations. \n\nTo check the reliability of the implemented code, the results were \ncompared with those obtained using the routine for the automatic \nmeasurement of line equivalent widths in stellar spectra ARES \n(Sousa et al. 2007), and only the lines identified by ARES were used to achieve \nthe best confidence in the final results. A very good agreement between \nthe two measurements is shown in Fig. \\ref{EW_compara}. We find a mean \ndifference of $\\hbox{EW(this work)}-\\hbox{EW(ARES)}=-0.22\\pm0.43$~m{\\AA}, \nwhich can be considered negligible in terms of EWs.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\hsize]{EW.eps}\n\\caption{Comparison of EWs measured for a set of \\ion{Fe}{I} and \n\\ion{Fe}{II} lines in HD 140283 using the code by Siqueira-Mello et al.\n(2014) and EWs measured using ARES.}\n\\label{EW_compara}\n\\end{figure}\n\nThe complete list of Fe and Ti lines used is shown in Table \\ref{EW_measurements}, \nwhich includes wavelength ({\\AA}), excitation potential (eV), \\loggf~values from \nthe VALD and NIST\\footnote{http:\/\/physics.nist.gov\/PhysRefData\/ASD\/lines$_-$form.html} \ndatabases, the measured EWs (m{\\AA}), and the derived abundances. Using the errors \ngiven for the parameters of the Gaussian profile obtained from the fitting \nprocedure, the uncertainties $\\sigma$EW of the equivalent widths were \ncomputed based on standard error propagation. \n\n\n\\subsection{Calculations}\n\nIron and titanium abundances were derived using equivalent widths, \nas usual. All other element abundances were derived from fits of \nsynthetic spectra to the observed spectrum of HD~140283. The OSMARCS \n1D model atmosphere grid was employed (Gustafsson et al. 2008). \n\nWe used the spectrum synthesis code Turbospectrum (Alvarez \\& Plez 1998), \nwhich includes treatment of scattering in the blue and UV domain, \nmolecular dissociative equilibrium, and collisional broadening by H, \nHe, and H$_{2}$, following Anstee \\& O'Mara (1995), Barklem \\& O'Mara (1997), \nand Barklem et al. (1998). The calculations used the Turbospectrum molecular \nline lists (Alvarez \\& Plez 1998), and atomic line lists from the VALD \ncompilation (Kupka et al. 1999)\\footnote{http:\/\/vald.astro.univie.ac.at\/~vald3\/php\/vald.php}. \nWhen available, new experimental oscillator strengths were adopted from \nliterature. In addition, hyperfine structure (HFS) splitting and isotope \nshifts were implemented when needed and available.\n\nWe also performed NLTE calculations for the following ions: \\ion{C}{I}, \n\\ion{O}{I}, \\ion{Na}{I}, \\ion{Mg}{I}, \\ion{Al}{I}, \\ion{K}{I}, \\ion{Ca}{I}, \n\\ion{Sr}{II}, and \\ion{Ba}{II}. Atomic models for these species were used \nin combination with the NLTE MULTI code (Carlsson 1986; Korotin et al. 1999), \nwhich facilitates a very good description of the radiation field. The updated \nversion of the MULTI code includes opacities from ATLAS9 (Castelli \\& Kurucz 2003), \nwhich modify the intensity distribution in the UV region.\n\nThe lines studied in NLTE are often blended \nwith lines of other species. Proper comparison of the synthesized and \nobserved profiles thus requires a multi-element synthesis. To accomplish this, \nwe fold the NLTE (MULTI) calculations into the LTE synthetic spectrum \ncode SYNTHV (Tsymbal 1996). With these two codes, we calculate synthetic \nspectra for each region in the vicinity of the line of interest taking \n(in LTE) all the blending lines located in this region and listed in \nthe VALD database into account. \nFor the lines of interest (treated in NLTE), the corresponding departure \ncoefficients (so-called $b$-factors: $b = n_{\\rm i}\/n^{*}_{\\rm i}$, the \nratio of NLTE to LTE atomic level populations) obtained with MULTI code are \nthe input to SYNTHV code, where they are used in the calculation of the line \nsource function, and then for the NLTE line profile. Other possible blending \nlines are treated in LTE. \n\n\n\\subsection{Stellar parameters}\n\nFollowing Paper I, we adopted the stellar \nparameters \\Teff~$=5750\\pm100$~K, $\\hbox{[Fe\/H]}\\footnote{\n[X\/H]~$=$~A(X)$_{star}-$~A(X)$_{\\odot}$}=-2.5\\pm0.2$ \nand $\\xi=1.4\\pm0.1$~\\kms~from Aoki et al. (2004) and \n\\logg~$=3.7\\pm0.1$~[g in cgs] from Collet et al. (2009). \nUsing the newly measured EWs, we obtained the iron \nabundances A(\\ion{Fe}{I})\\footnote{We adopted the notation \nA(X) = log~(X) = log~n(X)\/n(H)~+~12, with n = number \ndensity of atoms.}~$=+4.91\\pm0.07$ and A(\\ion{Fe}{II})~$=+4.96\\pm0.07$, \nor $\\hbox{[Fe I\/H]}=-2.59\\pm0.08$ and $\\hbox{[Fe II\/H]}=-2.54\\pm0.08$, \nusing the solar abundance A(Fe)$_{\\odot}=+7.50\\pm0.04$ \nfrom Asplund et al. (2009). The results are in very good agreement with the \nliterature, as in Gallagher et al. (2010), where [Fe\/H]$=-2.59\\pm0.06$ was obtained. \nFor titanium, we found A(\\ion{Ti}{I})~$=+2.71\\pm0.01$ \nand A(\\ion{Ti}{II})~$=+2.69\\pm0.03$, or $\\hbox{[Ti I\/H]}=-2.24\\pm0.05$ \nand $\\hbox{[Ti II\/H]}=-2.26\\pm0.06$, using the solar abundance of \ntitanium A(Ti)$_{\\odot}=+4.95\\pm0.05$ from Asplund et al. (2009). \nIn Table \\ref{summary} we summarize the atmospheric parameters \nadopted, together with the iron and titanium abundances.\n\nFigure \\ref{avalia} shows the dependence of [\\ion{Fe}{I}\/H], [\\ion{Fe}{II}\/H], \n[\\ion{Ti}{I}\/H], and [\\ion{Ti}{II}\/H] on log~$(EW\/\\lambda)$, and on the \nexcitation potential of the lines obtained for HD~140283. \nThe blue solid lines represent the average \nabundances in each case. The excitation and ionization equilibria of \nFe and Ti lines, resulting from the set of atmospheric parameters adopted \nfor HD~140283, confirm the robustness of our choice. However, it should be \nnoted that the surface gravity value may be affected by NLTE effects and \nby uncertainties in the oscillator strengths of the Fe and Ti lines.\n\nThe broadening parameters in HD~140283 were carefully analysed by \nseveral authors. We adopted a Gaussian profile \nto take the effects of macroturbulence, rotational, and \ninstrumental broadening into account.\n\n\\begin{table}\n\\caption{Atmospheric parameters adopted for HD~140283.} \n\\label{summary} \n\\scalefont{1.0}\n\\centering \n\\begin{tabular}{cc} \n\\hline\\hline \n\\noalign{\\smallskip}\n\\hbox{} & \\hbox{Values} \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\hbox{\\Teff} & $5750\\pm100$~K \\\\\n\\hbox{\\logg} & $3.7\\pm0.1$~[g in cgs] \\\\\n\\hbox{[Fe\/H]$_{model}$}& $-2.5\\pm0.2$ \\\\\n\\hbox{$\\xi$} & $1.4\\pm0.1$~\\kms \\\\\n\\hbox{[\\ion{Fe}{I}\/H]} & $-2.59\\pm0.08$ \\\\\n\\hbox{[\\ion{Fe}{II}\/H]}& $-2.54\\pm0.08$ \\\\\n\\hbox{[\\ion{Ti}{I}\/H]} & $-2.24\\pm0.05$ \\\\\n\\hbox{[\\ion{Ti}{II}\/H]}& $-2.26\\pm0.06$ \\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\hsize]{avalia.eps}\n\\caption{Excitation and ionization equilibria of Fe and Ti lines, \nresulting from the set of atmospheric parameters for HD~140283. \nThe black dots are the abundances obtained from \\ion{Ti}{I} and \n\\ion{Fe}{I} lines, the red squares are those from \\ion{Ti}{II} \nand \\ion{Fe}{II} lines, and the blue solid lines represent the \naverage abundances.}\n\\label{avalia}\n\\end{figure}\n\n\\subsection{Uncertainties in the derived abundances}\n\nAs described in Paper I, the adopted atmospheric parameters present \ntypical errors of $\\Delta$T$_{\\rm eff}=\\pm100$~K, $\\Delta$\\logg~$=\\pm0.1$~[g in cgs], \nand $\\Delta\\xi=\\pm0.1$~\\kms. Since the stellar parameters are not \nindependent from each other, the quadratic sum of the various sources \nof uncertainties is not the best way to estimate the total error budget, \notherwise it is mandatory to include the covariance terms in this calculation, \nand an estimated correlation matrix may introduce uncontrollable error sources.\n\nTo compute the total error budget in the abundance analysis arising \nfrom the stellar parameters, we created a new atmospheric model with a 100 K \nlower temperature, determining the corresponding surface gravity and \nmicroturbulent velocity with the traditional spectroscopic method. \nRequiring that the iron abundance derived from \\ion{Fe}{I} and \\ion{Fe}{II} \nlines be identical, we determined the respective \\logg~value, and the microturbulent \nvelocity was found requiring that the abundances derived for individual \\ion{Fe}{I} \nlines be independent of the equivalent width values. \nThe result is a model with T$_{\\rm eff}=5650$~K, \\logg~$=3.3$~[g in cgs], \nand $\\xi=1.2$~\\kms. The abundance analysis was carried out with this new model, \nand the difference in comparison with the nominal model should represent the \nuncertainties from the atmospheric parameters. \n\nObservational errors were estimated using the standard deviation of the abundances \nfrom the individual lines for each element, and taking into account the uncertainties \nin defining the continuum, fitting the line profiles, and in the oscillator strengths. \nFor the elements with only one line available, we adopted the observational error from \niron as a representative value. The adopted total error budget is the quadratic sum of \nuncertainties arising from atmospheric parameters and observations. \n\nTo estimate the uncertainties in the individual abundances due to the uncertainties \nin the EWs $\\sigma$EW, we recomputed the abundances using $\\hbox{EW}+\\sigma\\hbox{EW}$, \nand the differences with respect to the nominal values were adopted as the errors. \nThe errors line-by-line are also shown in Table \\ref{EW_measurements}. \n\n\n\\begin{table}\n\\caption{Abundance uncertainties due to stellar parameters $\\Delta_{par}$, \nobservational errors $\\Delta_{obs}$, and adopted total error budget $\\Delta_{total}$ \nfor LTE abundances.} \n\\label{finalabund} \n\\scalefont{1.0}\n\\centering \n\\begin{tabular}{ccrr} \n\\hline\\hline \n\\noalign{\\smallskip}\n\\hbox{Element} & \\hbox{$\\Delta_{par}$} & \\hbox{$\\Delta_{obs}$} & \\hbox{$\\Delta_{total}$}\\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\hbox{[Fe\/H]} & $-$0.07 & 0.07 & 0.10 \\\\\n\\hbox{[Li\/H]} & $-$0.08 & 0.07 & 0.11 \\\\\n\\hbox{[C\/Fe]} & $+$0.01 & 0.07 & 0.07 \\\\\n\\hbox{[N\/Fe]} & $-$0.10 & 0.07 & 0.12 \\\\\n\\hbox{[O\/Fe]} & $-$0.15 & 0.07 & 0.17 \\\\\n\\hbox{[Na\/Fe]} & $-$0.04 & 0.08 & 0.09 \\\\\n\\hbox{[Mg\/Fe]} & $-$0.04 & 0.07 & 0.08 \\\\\n\\hbox{[Al\/Fe]} & $-$0.01 & 0.07 & 0.07 \\\\\n\\hbox{[Si\/Fe]} & $-$0.08 & 0.01 & 0.08 \\\\\n\\hbox{[K\/Fe]} & $-$0.06 & 0.07 & 0.09 \\\\\n\\hbox{[Ca\/Fe]} & $-$0.03 & 0.04 & 0.05 \\\\\n\\hbox{[Sc\/Fe]} & $-$0.07 & 0.06 & 0.09 \\\\\n\\hbox{[Ti\/Fe]} & $-$0.07 & 0.03 & 0.08 \\\\\n\\hbox{[V\/Fe]} & $-$0.07 & 0.08 & 0.11 \\\\\n\\hbox{[Cr\/Fe]} & $-$0.08 & 0.04 & 0.09 \\\\\n\\hbox{[Mn\/Fe]} & $-$0.07 & 0.04 & 0.08 \\\\\n\\hbox{[Co\/Fe]} & $-$0.07 & 0.11 & 0.13 \\\\\n\\hbox{[Ni\/Fe]} & $-$0.07 & 0.05 & 0.09 \\\\\n\\hbox{[Zn\/Fe]} & $-$0.07 & 0.07 & 0.10 \\\\\n\\hbox{[Sr\/Fe]} & $-$0.02 & 0.07 & 0.07 \\\\\n\\hbox{[Y\/Fe]} & $-$0.10 & 0.06 & 0.12 \\\\\n\\hbox{[Zr\/Fe]} & $-$0.11 & 0.05 & 0.12 \\\\\n\\hbox{[Ba\/Fe]} & $-$0.09 & 0.10 & 0.13 \\\\\n\\hbox{[Ce\/Fe]} & $-$0.07 & 0.18 & 0.19 \\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\section{Abundance derivation}\n\nTable \\ref{linelist} shows the line list of elements other than Fe and \nTi used in this work, including wavelength ({\\AA}), excitation \npotential (eV), adopted oscillator strength, and abundance derived \nfrom each line. The final LTE abundances derived in HD~140283 for all \nthe analysed species are shown in Table \\ref{finalabund}. The adopted \nsolar abundances from Asplund et al. (2009) are also listed in \nTable \\ref{finalabund}. We discuss below each element in terms of \nlines used, HFS splitting, and abundances adopted. In Table \n\\ref{fractions} we report the solar isotopic fractions adopted \nfrom Asplund et al. (2009) that are relevant for HFS computations, \nand in Table \\ref{HFSconstants} we summarize the hyperfine coupling \nconstants adopted for the lines retained in this analysis.\n\n\\begin{table}\n\\caption{LTE abundances. Column 2 gives the solar abundances from Asplund et al.\n(2009), columns 3, 4, and 5 give the absolute abundance\nwith respect to A(H)=12.0 and the usual logarithmic ratio notation\n with respect to H and to Fe, respectively.} \n\\label{finalabund} \n\\scalefont{1.0}\n\\centering \n\\begin{tabular}{ccrrr} \n\\hline\\hline \n\\noalign{\\smallskip}\n\\hbox{Ion} & \\hbox{A(X)$_{\\odot}$} & \\hbox{A(X)} & \\hbox{[X\/H]} & \\hbox{[X\/Fe]} \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\hbox{\\ion{Fe}{I}} & 7.50$\\pm$0.04 & $+$4.91 & $-$2.59 & --- \\\\\n\\hbox{\\ion{Fe}{II}} & 7.50$\\pm$0.04 & $+$4.96 & $-$2.54 & --- \\\\\n\\hbox{\\ion{Li}{I}} & 1.05$\\pm$0.10 & $+$2.14 & $+$1.09 & --- \\\\\n\\hbox{C(CH)} & 8.43$\\pm$0.05 & $+$6.30 & $-$2.13 & $+$0.46 \\\\\n\\hbox{\\ion{C}{I}} & 8.43$\\pm$0.05 & $+$6.44 & $-$1.99 & $+$0.60 \\\\\n\\hbox{N(CN)} & 7.83$\\pm$0.05 & $+$6.30 & $-$1.53 & $+$1.06 \\\\\n\\hbox{[\\ion{O}{I}]} & 8.69$\\pm$0.05 & $+$6.95 & $-$1.74 & $+$0.85 \\\\\n\\hbox{\\ion{O}{I}} & 8.69$\\pm$0.05 & $+$7.11 & $-$1.58 & $+$1.00 \\\\\n\\hbox{\\ion{Na}{I}} & 6.24$\\pm$0.04 & $+$3.62 & $-$2.62 & $-$0.04 \\\\\n\\hbox{\\ion{Mg}{I}} & 7.60$\\pm$0.04 & $+$5.27 & $-$2.33 & $+$0.26 \\\\\n\\hbox{\\ion{Mg}{II}} & 7.60$\\pm$0.04 & $+$5.66 & $-$1.95 & $+$0.64 \\\\\n\\hbox{\\ion{Al}{I}} & 6.45$\\pm$0.03 & $+$2.96 & $-$3.50 & $-$0.91 \\\\\n\\hbox{\\ion{Si}{I}} & 7.51$\\pm$0.03 & $+$5.30 & $-$2.21 & $+$0.38 \\\\\n\\hbox{\\ion{Si}{II}} & 7.51$\\pm$0.03 & $+$5.35 & $-$2.16 & $+$0.43 \\\\\n\\hbox{\\ion{K}{I}} & 5.03$\\pm$0.09 & $+$2.98 & $-$2.05 & $+$0.54 \\\\\n\\hbox{\\ion{Ca}{I}} & 6.34$\\pm$0.04 & $+$4.03 & $-$2.31 & $+$0.27 \\\\\n\\hbox{\\ion{Ca}{II}} & 6.34$\\pm$0.04 & $+$4.43 & $-$1.91 & $+$0.68 \\\\\n\\hbox{\\ion{Sc}{I}} & 3.15$\\pm$0.04 & $+$0.58 & $-$2.58 & $+$0.01 \\\\\n\\hbox{\\ion{Sc}{II}} & 3.15$\\pm$0.04 & $+$0.75 & $-$2.40 & $+$0.18 \\\\\n\\hbox{\\ion{Ti}{I}} & 4.95$\\pm$0.05 & $+$2.71 & $-$2.24 & $+$0.34 \\\\\n\\hbox{\\ion{Ti}{II}} & 4.95$\\pm$0.05 & $+$2.69 & $-$2.26 & $+$0.32 \\\\\n\\hbox{\\ion{V}{I}} & 3.93$\\pm$0.08 & $+$1.44 & $-$2.49 & $+$0.09 \\\\\n\\hbox{\\ion{V}{II}} & 3.93$\\pm$0.08 & $+$1.70 & $-$2.23 & $+$0.36 \\\\\n\\hbox{\\ion{Cr}{I}} & 5.64$\\pm$0.04 & $+$2.95 & $-$2.69 & $-$0.11 \\\\\n\\hbox{\\ion{Cr}{II}} & 5.64$\\pm$0.04 & $+$3.32 & $-$2.32 & $+$0.26 \\\\\n\\hbox{\\ion{Mn}{I}} & 5.43$\\pm$0.04 & $+$2.56 & $-$2.87 & $-$0.29 \\\\\n\\hbox{\\ion{Co}{I}} & 4.99$\\pm$0.07 & $+$2.69 & $-$2.30 & $+$0.29 \\\\\n\\hbox{\\ion{Ni}{I}} & 6.22$\\pm$0.04 & $+$3.76 & $-$2.46 & $+$0.12 \\\\\n\\hbox{\\ion{Ni}{II}} & 6.22$\\pm$0.04 & $+$3.88 & $-$2.34 & $+$0.25 \\\\\n\\hbox{\\ion{Zn}{I}} & 4.56$\\pm$0.05 & $+$2.22 & $-$2.34 & $+$0.25 \\\\\n\\hbox{\\ion{Sr}{II}} & 2.87$\\pm$0.07 & $+$0.10 & $-$2.77 & $-$0.18 \\\\\n\\hbox{\\ion{Y}{II}} & 2.21$\\pm$0.05 & $-$0.78 & $-$2.99 & $-$0.40 \\\\\n\\hbox{\\ion{Zr}{II}} & 2.58$\\pm$0.04 & $-$0.07 & $-$2.65 & $-$0.07 \\\\\n\\hbox{\\ion{Ba}{II}} & 2.18$\\pm$0.09 & $-$1.22 & $-$3.40 & $-$0.81 \\\\\n\\hbox{\\ion{La}{II}} & 1.10$\\pm$0.04 & $<-$1.85 & $<-$2.95 & $<-$0.36 \\\\\n\\hbox{\\ion{Ce}{II}} & 1.58$\\pm$0.04 & $-$0.83 & $-$2.41 & $+$0.18 \\\\\n\\hbox{\\ion{Eu}{II}} & 0.52$\\pm$0.04 & $-$2.35 & $-$2.87 & $-$0.28 \\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\end{table}\n\n\nIn Table \\ref{finalabundNLTE} we present the complete line list analysed in NLTE, \nwith the respective individual abundances. Below we also give element-by-element \ninformation about sources of the NLTE atomic models we used, and we also present \nthe graphical results of the NLTE line synthesis. \n\n\\begin{table}\n\\caption{Isotopic abundance fractions in the solar system from \nAsplund et al. (2009), relevant for HFS computations adopted in \nthe present work.} \n\\label{fractions} \n\\scalefont{1.0}\n\\centering \n\\begin{tabular}{ccr} \n\\hline\\hline \n\\noalign{\\smallskip}\n\\hbox{Elements} & \\hbox{Isotopes} & \\hbox{\\%} \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\hbox{Sodium} & \\hbox{$^{23}$Na} & 100.000 \\\\\n\\hbox{Aluminum} & \\hbox{$^{27}$Al} & 100.000 \\\\\n\\hbox{Potassium} & \\hbox{$^{39}$K} & 93.132 \\\\\n\\hbox{} & \\hbox{$^{40}$K} & 0.147 \\\\\n\\hbox{} & \\hbox{$^{41}$K} & 6.721 \\\\\n\\hbox{Scandium} & \\hbox{$^{45}$Sc} & 100.000 \\\\\n\\hbox{Vanadium} & \\hbox{$^{50}$V} & 0.250 \\\\\n\\hbox{} & \\hbox{$^{51}$V} & 99.750 \\\\\n\\hbox{Manganese} & \\hbox{$^{55}$Mn} & 100.000 \\\\\n\\hbox{Zinc} & \\hbox{$^{64}$Zn} & 48.630 \\\\\n\\hbox{} & \\hbox{$^{66}$Zn} & 27.900 \\\\\n\\hbox{} & \\hbox{$^{67}$Zn} & 4.100 \\\\\n\\hbox{} & \\hbox{$^{68}$Zn} & 18.750 \\\\\n\\hbox{} & \\hbox{$^{70}$Zn} & 0.620 \\\\\n\\hbox{Barium} & \\hbox{$^{130}$Ba}& 0.106 \\\\\n\\hbox{} & \\hbox{$^{132}$Ba}& 0.101 \\\\\n\\hbox{} & \\hbox{$^{134}$Ba}& 2.417 \\\\\n\\hbox{} & \\hbox{$^{135}$Ba}& 6.592 \\\\\n\\hbox{} & \\hbox{$^{136}$Ba}& 7.854 \\\\\n\\hbox{} & \\hbox{$^{137}$Ba}& 11.232 \\\\\n\\hbox{} & \\hbox{$^{138}$Ba}& 71.698 \\\\\n\\hbox{Lanthanum} & \\hbox{$^{138}$La}& 0.091 \\\\\n\\hbox{} & \\hbox{$^{139}$La}& 99.909 \\\\\n\\hbox{Europium} & \\hbox{$^{151}$Eu}& 47.81 \\\\\n\\hbox{} & \\hbox{$^{153}$Eu}& 52.19 \\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\begin{table}\n\\caption{NLTE abundances derived in HD~140238 using NLTE equivalent \nwidth fitting (EW) and NLTE line profile fitting (PF).} \n\\label{finalabundNLTE} \n\\scalefont{1.0}\n\\centering \n\\begin{tabular}{ccrrr} \n\\hline\\hline \n\\noalign{\\smallskip}\n\\hbox{Ion} & \\hbox{$\\lambda$~(\\AA)} & \\hbox{(X\/H)+12} & \\hbox{[X\/H]} & \\hbox{Method} \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\hbox{\\ion{C}{I}} & 8335.14 & $+$6.00 & $-$2.43 & PF \\\\ \n\\hbox{\\ion{C}{I}} & 9061.43 & $+$5.97 & $-$2.46 & PF \\\\\n\\hbox{\\ion{C}{I}} & 9062.48 & $+$6.03 & $-$2.40 & PF \\\\\n\\hbox{\\ion{O}{I}} & 7771.94 & $+$7.04 & $-$1.67 & PF \\\\\n\\hbox{\\ion{O}{I}} & 7774.16 & $+$6.98 & $-$1.73 & PF \\\\\n\\hbox{\\ion{O}{I}} & 7775.39 & $+$7.00 & $-$1.71 & PF \\\\\n\\hbox{\\ion{O}{I}} & 8446.36 & $+$7.04 & $-$1.67 & PF \\\\\n\\hbox{\\ion{Na}{I}} & 5889.95 & $+$3.37 & $-$2.88 & PF \\\\\n\\hbox{\\ion{Na}{I}} & 5895.92 & $+$3.37 & $-$2.88 & PF \\\\\n\\hbox{\\ion{Na}{I}} & 8194.82 & $+$3.37 & $-$2.88 & PF \\\\\n\\hbox{\\ion{Mg}{I}} & 4167.27 & $+$5.38 & $-$2.16 & PF \\\\\n\\hbox{\\ion{Mg}{I}} & 4571.10 & $+$5.38 & $-$2.16 & PF \\\\\n\\hbox{\\ion{Mg}{I}} & 4702.99 & $+$5.38 & $-$2.16 & PF \\\\\n\\hbox{\\ion{Mg}{I}} & 5172.68 & $+$5.38 & $-$2.16 & PF \\\\\n\\hbox{\\ion{Mg}{I}} & 5183.60 & $+$5.38 & $-$2.16 & PF \\\\\n\\hbox{\\ion{Mg}{I}} & 5528.40 & $+$5.38 & $-$2.16 & PF \\\\\n\\hbox{\\ion{Mg}{I}} & 5711.09 & $+$5.38 & $-$2.16 & PF \\\\\n\\hbox{\\ion{Mg}{I}} & 8806.76 & $+$5.38 & $-$2.16 & PF \\\\\n\\hbox{\\ion{Al}{I}} & 3944.01 & $+$3.70 & $-$2.73 & PF \\\\\n\\hbox{\\ion{Al}{I}} & 3961.52 & $+$3.65 & $-$2.78 & PF \\\\\n\\hbox{\\ion{K}{I}} & 7698.96 & $+$2.78 & $-$2.33 & PF \\\\\n\\hbox{\\ion{Ca}{I}} & 4226.73 & $+$4.04 & $-$2.27 & PF \\\\\n\\hbox{\\ion{Ca}{I}} & 4283.01 & $+$4.21 & $-$2.10 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 4289.37 & $+$4.14 & $-$2.17 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 4302.53 & $+$4.15 & $-$2.16 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 4318.65 & $+$4.09 & $-$2.22 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 4425.44 & $+$4.18 & $-$2.13 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 4434.96 & $+$4.07 & $-$2.24 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 4435.68 & $+$4.17 & $-$2.14 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 4454.78 & $+$4.04 & $-$2.27 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 5188.84 & $+$4.20 & $-$2.11 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 5261.70 & $+$4.15 & $-$2.16 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 5265.56 & $+$4.20 & $-$2.11 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 5349.46 & $+$4.22 & $-$2.09 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 5512.98 & $+$4.09 & $-$2.22 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 5581.96 & $+$4.22 & $-$2.09 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 5588.75 & $+$4.11 & $-$2.20 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 5590.11 & $+$4.22 & $-$2.09 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 5594.46 & $+$4.19 & $-$2.12 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 5857.45 & $+$4.17 & $-$2.14 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 6102.72 & $+$4.14 & $-$2.17 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 6122.22 & $+$4.14 & $-$2.17 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 6162.17 & $+$4.05 & $-$2.26 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 6169.56 & $+$4.17 & $-$2.14 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 6439.07 & $+$4.08 & $-$2.23 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 6462.56 & $+$4.06 & $-$2.25 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 6471.66 & $+$4.08 & $-$2.23 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 6493.78 & $+$4.10 & $-$2.21 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 6499.65 & $+$4.23 & $-$2.08 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 7148.15 & $+$4.19 & $-$2.12 & EW \\\\\n\\hbox{\\ion{Ca}{I}} & 7326.14 & $+$4.19 & $-$2.12 & EW \\\\\n\\hbox{\\ion{Ca}{II}}& 3933.68 & $+$4.15 & $-$2.16 & PF \\\\\n\\hbox{\\ion{Ca}{II}}& 3968.47 & $+$4.15 & $-$2.16 & PF \\\\\n\\hbox{\\ion{Ca}{II}}& 8498.02 & $+$4.15 & $-$2.16 & PF \\\\\n\\hbox{\\ion{Ca}{II}}& 8542.09 & $+$4.15 & $-$2.16 & PF \\\\\n\\hbox{\\ion{Ca}{II}}& 8662.14 & $+$4.15 & $-$2.16 & PF \\\\\n\\hbox{\\ion{Sr}{II}}& 4077.72 & $+$0.02 & $-$2.90 & PF \\\\\n\\hbox{\\ion{Sr}{II}}& 4215.52 & $+$0.07 & $-$2.85 & PF \\\\\n\\hbox{\\ion{Sr}{II}}&10327.31 & $+$0.00 & $-$2.92 & PF \\\\\n\\hbox{\\ion{Ba}{II}}& 4554.03 & $-$1.07 & $-$3.24 & PF \\\\\n\\hbox{\\ion{Ba}{II}}& 6141.70 & $-$1.07 & $-$3.24 & PF \\\\\n\\hbox{\\ion{Ba}{II}}& 6496.92 & $-$1.01 & $-$3.18 & PF \\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular} \n\\end{table}\n\n\\subsection{Light elements}\n\n$Lithium$. \nWe derived the LTE Li abundance A(Li)~$=+2.14$ based on \nthe \\ion{Li}{I} lines located at 6103~{\\AA} and 6707~{\\AA} \n(see Fig. \\ref{Li_fig}). The wavelength and oscillator strength values \nwere adopted from NIST, based on calculations by \nYan \\& Drake (1995). NLTE corrections are given in Asplund et al. (2006) \nfor the two Li lines in HD~140283: $+0.09$~dex for 6103~{\\AA} and \n$+0.03$ for 6707~{\\AA}. The corrected Li abundance \nA(Li)~$=+2.20$ is in excellent agreement with the NLTE \nabundance from Asplund et al. (2006).\n\n\\begin{figure}\n\\centering\n\\resizebox{80mm}{!}{\\includegraphics[angle=0]{Li.eps}}\n\\caption{LTE lithium abundance in HD~140283 from the \\ion{Li}{I} lines \nlocated at 6103~{\\AA} (upper panel) and 6707~{\\AA} (lower panel). \nObservations (crosses) are compared with synthetic spectra computed with \ndifferent abundances (blue dotted lines), as well as with the adopted \nabundances (red solid lines).}\n\\label{Li_fig}\n\\end{figure}\n\n\n$Carbon$. \nIn Paper I we analysed the CH A-X electronic transition band (G band) \nin HD~140283, showing a very good agreement between the observed spectrum \nand the computation using the LTE carbon abundance A(C)~$=+6.30$ adopted from \nHonda et al. (2004a). In fact, several CH lines were analysed and all of \nthem presented a good fit, leading us to assume that they are properly \ntaken into account.\n\nIt was also possible to use three \\ion{C}{I} lines: 8335.14~{\\AA}, \n9061.43~{\\AA}, and 9062.48~{\\AA}. They are free of telluric and \nother atomic or molecular lines. The lines of \\ion{C}{I} in the \nvisual part of the spectrum are not detectable. The LTE carbon \nabundance derived from these atomic transitions A(C)~$=+6.44$ is \nslightly higher in comparison with the result from molecular bands. \n\nThe same \\ion{C}{I} lines were analysed in NLTE (see upper left panel \nin Fig. \\ref{NLTE_O} for the 9062.48~{\\AA} line). To perform this work, \nwe used the carbon atomic model first proposed by Lyubimkov et al. (2015). \nOur calculations show that NLTE effects are very strong in the analysed lines, \nand act towards a strengthening of their equivalent widths: the ratio between NLTE \nand LTE EWs reaches a factor around 3. This means that simply analysing the program \ncarbon lines in LTE approximation results in the derived abundance overestimate \nby about $0.4-0.5$~dex (as it should be for the generally weak lines where equivalent \nwidth linearly depend upon the number of absorbing atoms). The same conclusion was \ngiven by Asplund (2005), who noted that strong NLTE effects in high-excitation neutral \ncarbon lines are due to the decrease of source function (compared to the Planck function). \nIn particular this is valid for near-infrared \\ion{C}{I} lines. Fabbian et al. (2006) \nalso predict strong NLTE corrections in the carbon abundances in extremely metal-poor \nstars of $\\sim$$0.4$~dex, in good agreement with the present result.\n\nWe find a difference between the C abundance deduced from CH and that \nfrom \\ion{C}{I} lines computed in LTE and NLTE \nof $\\Delta($C(CH)~$-$~C(\\ion{C}{I})$_{LTE})$~$=-0.14$~dex, and \n$\\Delta($C(CH)~$-$~C(\\ion{C}{I})$_{NLTE})$~$=+0.30$~dex, respectively. \nThe CH lines forming in the upper atmosphere might be affected \nin a 3D modelling calculation (Asplund 2005).\n\n$Nitrogen$. \nThe bandhead of CN(0,0) B$_2$$\\Sigma$~$-$~X$_2$$\\Sigma$ at 3883~{\\AA} gives A(N)~$=+6.30$, \nassuming the abundances A(C)~$=+6.30$ and A(O)~$=+7.00$. In Fig. \\ref{NO_fig} \n(upper panel) we present the synthetic spectra fitting to the CN bandhead. \nThis profile is located in the blue wing of the H8 line of the Balmer series, \ntaken into account in the calculations.\n\n\n$Oxygen$.\nThe forbidden [\\ion{O}{I}]~6300.31~{\\AA} line is free from NLTE effects (Kiselman 2001), \nand therefore it is the best oxygen abundance indicator. The line was inspected for telluric \nlines in each of the original exposures. From our series of observations, four of the spectra \nobserved in July had to be discarded, given that the telluric lines were masking the oxygen line. \nWe then proceeded to co-add the other 19 spectra, and were able to obtain a weak, but measurable line \n(see lower panel in Fig. \\ref{NO_fig}). We derive an abundance of A(O)~$=+6.95$, considered in \nthis calculation A(C)~$=+6.30$ (C derived from CH lines) and A(N)$=+6.30$.\n\nThe triplets at 7771-7775~{\\AA} and 8446~{\\AA} were also checked to compute the LTE O abundance \nA(O)~$=+7.11$ in HD~140283. Adopting an average of different models presented in Behara et al. (2010) \nfor the NLTE corrections to be applied on the triplet 7771-7775~{\\AA} lines \n($-0.13$~dex, $-0.10$~dex, and $-0.10$~dex, respectively), we obtain A(O)~$=+6.97$ from this triplet, \nin excellent agreement with the result derived using the forbidden line.\n\nThe triplets at 7771-7775~{\\AA} (see Fig. \\ref{NLTE_O}) and 8446~{\\AA} \nwere also analysed in NLTE. Our NLTE model of this atom \nwas first described in Mishenina et al. (2000), and then updated by \nKorotin et al. (2014). An updated model was applied to study the NLTE \nabundance of this element in cepheids, and its distribution in the \nGalactic disc (see, for instance, Korotin et al. 2014; Martin et al. 2015). \nWe obtain a NLTE oxygen abundance A(O)~$=+7.02$ (or [O\/Fe]~$=+0.90$), \nin agreement with the derived LTE abundances corrected for NLTE effects.\n\n\n\\subsection{$\\alpha$-elements: Mg, Si, S, and Ca}\n\n\n$Magnesium$. \nAfter checking the \\ion{Mg}{I} lines used by Zhao et al. (1998) in \nthe solar spectrum and the bluer lines used by Cayrel et al. (2004) \nfor metal-poor stars, we retained 12 lines from an initial list of 21 \nprofiles to derive the LTE abundance A(\\ion{Mg}{I})~$=+5.27$. \nThe oscillator strengths were adopted from the critical compilation \npresented in Kelleher \\& Podobedova (2008a). The individual abundances \nare consistent among the lines (see upper panels in Fig. \\ref{Mg_fig}), \nwith the exception of \\ion{Mg}{I}~8806.76~{\\AA}, which is $0.13$~dex higher \nthan the average abundance (see lower panel in Fig. \\ref{Mg_fig}).\n\nIn addition, it was possible to use two \\ion{Mg}{II} lines with the \nspectrum of HD~140283: 4481.13~{\\AA} and 4481.15~{\\AA}. The derived LTE \nabundance A(\\ion{Mg}{II})~$=+5.66$ is higher than the value obtained \nfrom the neutral species, but both \\ion{Mg}{II} lines are blended and \nthe result should be used with caution.\n\nSeven magnesium lines were selected for NLTE \nabundance determination. All these lines have very well-defined \nprofiles, which provide a robust NLTE abundance of this element (see\nFig. \\ref{NLTE_Mg}). To generate \\ion{Mg}{I} line profiles, \nwe used the NLTE \\ion{Mg}{I} atomic model described in detail \nby Mishenina et al. (2004). This model was used in several studies, in \nparticular, in the determination of the magnesium abundance in a sample \nof metal-poor stars (Andrievsky et al. 2010).\n\n\n\n\n$Silicon$. \nThe main Si abundance indicators are the \\ion{Si}{I} lines at 3905.52~{\\AA} \nand 4102.94~{\\AA}. The line at 3905~{\\AA} is blended with CH lines, but these \nmolecular lines are weak enough in a rather hot subgiant such that the \\ion{Si}{I} line can \nbe used. On the other hand, the line at 4102~{\\AA} is located in the red wing of the \nH$\\delta$ line, and therefore it was necessary to take the hydrogen line \nin the spectrum synthesis into account. Fig. \\ref{Si_fig} shows the LTE synthetic spectra \nused for these lines in HD~140283. It was possible to use another three weaker \n\\ion{Si}{I} lines located at 5645.61~{\\AA}, 5684.48~{\\AA}, and 7034.90~{\\AA}, with \nindividual abundances in good agreement with the previous transitions. \nThe final LTE Si abundance obtained was A(\\ion{Si}{I})$=+5.30$. \nBecause of the good quality of our data, it was possible to derive the LTE silicon abundance \nfrom the ionized species A(\\ion{Si}{II})$=+5.35$, using the \\ion{Si}{II}~6371.37~{\\AA} line. \n\n\n$Sulfur$. \nThree \\ion{S}{I} lines, which are potentially available for the abundance \ndetermination in metal-poor stars, are located at 9212.86~{\\AA}, \n9228.09~{\\AA}, and 9237.54~{\\AA}. Unfortunately, the first \nline is severely blended with telluric lines, and the other two lines \nare on a small gap in the spectrum. As a consequence, a reliable sulfur\nabundance is not presented here for HD~140283.\n\n\n\n$Calcium$. \nTo derive the LTE Ca abundance, 36 \\ion{Ca}{I} lines were used to obtain \nA(\\ion{Ca}{I})$=+4.03$. We adopted the \\loggf~values \nfrom Spite et al. (2012). Because of the high-quality spectrum, even weak and \nblended lines are useful, as shown in Fig. \\ref{Ca_fig}. \n\nWe inspected a few \\ion{Ca}{II} lines to evaluate the LTE calcium abundance \nfrom the ionized species. The presence of a strong NLTE effect, however, \ndoes not permit us to use most of these transitions. \nThe LTE abundance A(\\ion{Ca}{II})$=+4.43$ was derived based only on the \n\\ion{Ca}{II}~8927.36~{\\AA} line. We obtained a difference of \n$+0.40$~dex in comparison with the abundance derived from transitions \nof the neutral element. Mashonkina et al. (2007) analysed HD~140283 in \ntheir study of neutral and singly-ionized calcium in late-type stars, \nand the LTE results show a difference of $+0.30$~dex. Their NLTE \nabundances are [\\ion{Ca}{I}\/Fe]~$=+0.29\\pm0.06$ and [\\ion{Ca}{II}\/Fe]~$=\n+0.24$. Using similar atmospheric parameters for this star, Spite et al. \n(2012) found the NLTE abundances A(\\ion{Ca}{I})~$=+4.12\\pm0.04$ and \nA(\\ion{Ca}{II})~$=+4.08\\pm0.05$. Our LTE abundance shows that the \ndeparture from LTE is not strong for neutral calcium in this star.\n\nThe NLTE abundance A(Ca)~$=+4.14$ was determined \nfrom the average of 35 lines, including: i) the EW analysis of \n30 \\ion{Ca}{I} lines with 10~$<$~EW(m{\\AA})~$<$~40; \nii) the profile of five other lines, namely the strongest line of this \natom at 4226.73~{\\AA} (see upper \npanel in Fig. \\ref{NLTE_Ca}); the H and K \\ion{Ca}{II} lines \n(see lower panel in Fig. \\ref{NLTE_Ca}); and another three \n\\ion{Ca}{II} lines located in the red. The NLTE atomic model was \ndescribed in Spite et al. (2012), where it was used for the study of \nhalo metal-poor stars. \n\n\n\n\\subsection{Odd-Z elements: Na, Al, and K}\n\n$Sodium$. \nTo derive the LTE sodium abundance we used an initial \nline list based on Baum\\\"uller et al. (1998) with \nupdated oscillator strengths from Kelleher \\& Podobedova (2008a). After \nchecking 13 \\ion{Na}{I} lines, the final result A(\\ion{Na}{I})~$=+3.62$ \nis based on five sodium lines. $^{23}$Na is the only stable isotope \nrepresenting the sodium abundance, with nuclear spin I~$=3\/2$\n\\footnote{Adopted from the Particle Data Group (PDG) collaboration: \nhttp:\/\/pdg.lbl.gov\/} and therefore exhibiting HFS. The hyperfine \ncoupling constants are adopted from Das \\& Natarajan (2008) \nand Safronova et al. (1999). When not available, these constants were assumed to be null. \nThe HFS for each line were computed by employing a code described and \nmade available by McWilliam et al. (2013). Fig. \\ref{Na_fig} shows in \nthe upper panel the adopted synthetic spectrum for \\ion{Na}{I}~5895.92~\n{\\AA} and \\ion{Na}{I}~8194.82~{\\AA} lines as example of typical fitting \nprocedures. \n\n\n\nA NLTE atomic model of this element was presented for the first time in \nKorotin \\& Mishenina (1999) and then updated by Dobrovolskas et al. \n(2014). We analysed three \\ion{Na}{I} lines: D$_{1}$, D$_{2}$, and the \nline at 8194.82~{\\AA}. Synthesized NLTE profiles fitted to the observed \nsodium line profiles are shown in Fig. \\ref{NLTE_Na}, and the final \nNLTE abundance A(\\ion{Na}{I})~$=+3.37$ was adopted.\n\n\n\n\n$Aluminum$. \nThe only stable isotope for aluminum is $^{27}$Al and to derive \nthe Al abundance we used the resonance doublet \\ion{Al}{I}~3944.01~{\\AA} \nand \\ion{Al}{I}~3961.52~{\\AA}, taking the CH line blending \nwith the first line into account. The local continuum around the \\ion{Al}{I}~3961.52~{\\AA} \nline is defined by the blue wings of the H$\\varepsilon$ and H \\ion{Ca}{II} lines, \nwhich were taken into account in the spectrum synthesis. \n$^{27}$Al has nuclear spin I~$=5\/2$ and we adopted \nthe hyperfine coupling constants from Nakai et al. (2007) and Brown \\& Evenson (1999), \nwith updated oscillator strengths from Kelleher \\& Podobedova (2008b). \nFig. \\ref{Al_K_fig} shows in the upper panel the LTE synthetic \nspectra used for \\ion{Al}{I}~3961.52~{\\AA} as an example. The adopted LTE \nabundance is A(\\ion{Al}{I})$=+2.96$ and must be corrected for NLTE effects. \n\n\nThe NLTE \\ion{Al}{I} atomic model presented in Andrievsky et al. (2008) was \nemployed and the result of the profile fitting for \\ion{Al}{I}~3961.52~{\\AA} \nline is shown in Fig. \\ref{NLTE_AlK} (upper panel). The final NLTE abundance \nA(\\ion{Al}{I})$=+3.68$ is $0.72$~dex higher than the LTE result.\n\nAs it was stated above, in some cases the NLTE profile synthesis should \nbe combined with the LTE synthesis, which takes the blending lines in \nthe vicinity of the studied line into account. For instance, we would get \nthe wrong result if we derived the NLTE abundance from UV \\ion{Al}{I} lines \nonly using the pure MULTI NLTE profiles, since these lines are located in the \nwings of very strong H and K \\ion{Ca}{II} lines. \nTherefore it is absolutely necessary to take continuum distortion in their \nvicinity into account. This is made through a combination\nof calculations with the codes MULTI NLTE and SYNTHV LTE, which provides \na correct aluminum abundance. \n\n\n\n\n$Potassium$. \nThe final LTE potassium abundance A(\\ion{K}{I})~$=+2.98$ was only derived \nfrom the \\ion{K}{I}~7698.96~{\\AA} line, as shown in the lower \npanel of Fig. \\ref{Al_K_fig}. The other red doublet component at 7665~{\\AA} \nis strongly blended with telluric lines and was excluded. For this \nelement, we adopted the isotopic abundance fractions in the solar system, \nas described in Asplund et al. (2009) and reproduced in Table \n\\ref{fractions}. The HFS was computed taking into account the major \npotassium isotope $^{39}$K, which has nuclear spin I~$=3\/2$. We adopted \nhyperfine coupling constants from Belin et al. (1975) and Falke et al. \n(2006), with updated oscillator strength from Sansonetti (2008).\n\nA NLTE \\ion{K}{I} atomic model was presented in Andrievsky et al. (2010), \nwhere it was first employed to derive potassium NLTE abundances in a \nsample of extremely metal-poor halo stars. This model was used to \nsynthesize the profile of the \\ion{K}{I}~7698.96~{\\AA} line in HD 140283\n(see lower panel in Fig. \\ref{NLTE_AlK}), which gives \nA(\\ion{K}{I})~$=+2.78$ as the final abundance.\n\n\n\n\\subsection{Iron-peak elements: Sc, Ti, V, Cr, Mn, Fe, Co, Ni, and Zn}\n\n\n\n$Scandium$. \nWe were able to use two \\ion{Sc}{I} lines to derive the scandium \nabundance. The lines located at 4020.39~{\\AA} and 4023.68~{\\AA} present low \nexcitation potential and the average abundance A(\\ion{Sc}{I})$=+0.58$ should \nsuffer from over-ionization via NLTE effects (Zhang et al. 2008). \n$^{45}$Sc is the only stable scandium isotope, with nuclear spin I~$=7\/2$,\nand to compute the HFS we adopted hyperfine coupling constants from Childs \n(1971), Siefart (1977), and Ba\\c{s}ar et al. (2004). On other hand, the \nionized species presented 19 useful lines, covering a wider range in \nwavelengths and excitation potential values. The average result A(\\ion{Sc}{II})\n$=+0.75$ is higher in comparison with the abundance from the neutral species \nand it is not affected strongly by NLTE effects, at least at solar metallicity \n(see Asplund et al. 2009). The hyperfine coupling constants were adopted \nfrom Villemoes et al. (1992), Mansour et al. (1989), and Scott et al. \n(2015). In Fig. \\ref{Sc_fig} we show the fitting procedures used for the \n\\ion{Sc}{II}~4246.82~{\\AA} and \\ion{Sc}{II}~5526.79~{\\AA} lines as \nexamples. \n\n\n\n$Titanium$. \nFor titanium, \nwe applied equivalent widths to derive the final abundances \nA(\\ion{Ti}{I})~$=+2.71$ and A(\\ion{Ti}{II})~$=+2.69$ (Sect. \\ref{EW}).\n\n\n\n\n\n\n$Vanadium$. \nIn Paper I we analysed five \\ion{V}{I} \nlines and seven \\ion{V}{II} lines to derive the vanadium \nabundances A(\\ion{V}{I})~$=+1.35\\pm0.10$ and A(\\ion{V}{II})~$=+1.72\\pm0.10$, \nrespectively. The average result A(V)~$=+1.56\\pm0.11$ is in good \nagreement with A(V)~$=1.55$ from Honda et al. (2004a), derived in their \nanalysis from three lines (see Table 3 in Siqueira-Mello et al. 2012 \nfor details). We adopted seven \\ion{V}{I} and eight \n\\ion{V}{II} lines, including HFS based on hyperfine coupling \nconstants from Unkel et al. (1989), \nPalmeri et al. (1995), Armstrong et al. (2011), Gyzelcimen et al. (2014), \nand Wood et al. (2014), with nuclear spin I~$=7\/2$. We only used the major \nV isotope in the computations (see Table \\ref{fractions}). \nThe oscillator strengths were adopted from Whaling et al. (1985) for \\ion{V}\n{I} and from Wood et al. (2014) for \\ion{V}{II}. We obtained A(\\ion{V}{I})~$=\n+1.44$ and A(\\ion{V}{II})~$=+1.70$, in agreement with the previous results. \nIn addition, the final abundances derived from \\ion{V}{I} and \\ion{V}{II} \nlines are more consistent in the present analysis. \nIn Fig. \\ref{V_fig} we show the \n\\ion{V}{I}~4379.230~{\\AA} line (upper panel) and the \\ion{V}{II}~4023.378~\n{\\AA} line (lower panel) as examples. The differences measured between the \ntwo ionization stages should be explained by strong NLTE effects expected \nfor \\ion{V}{I} lines.\n\n\n\n\n$Chromium$. \nWe derived individual chromium abundances for 28 \\ion{Cr}{I} lines,\nbut we excluded the transitions located at 4756.11~{\\AA} and 5237.35~{\\AA} \nbecause of the higher abundances in comparison with results from other lines, \nand we obtained the abundance \nA(\\ion{Cr}{I})~$=+2.95$ based on seven \\ion{Cr}{I} lines. The fitting \nprocedure used for \\ion{Cr}{I}~4254.33~{\\AA} line is shown in Fig. \n\\ref{Cr_Mn_fig}. In addition, it was possible to use seven \n\\ion{Cr}{II} lines in HD~140283 to derive the abundance A(\\ion{Cr}{II})~$=\n+3.32$, higher by 0.37 dex than the result obtained from the neutral species.\n\n$Manganese$. \nThe only manganese stable isotope is $^{55}$Mn, \nwith nuclear spin I~$=5\/2$. \nIn addition to the three lines belonging \nto the resonance triplet (\\ion{Mn}{I}~4030.75~{\\AA}, 4033.06~{\\AA}, and \n4034.48~{\\AA}), it was also possible to measure 13 subordinate lines.\nWe took the HFS properly into account based on the hyperfine \nstructure line component patterns from Den Hartog et al. (2011), \nwhich also present the sources for hyperfine coupling constants. \nIt is well known that the abundances derived from the triplet\nresonance lines are \nsystematically lower in comparison with the results from subordinate lines \nin very metal-poor giant stars. Indeed, the result we obtained \nfor HD~140283 using the triplet A(\\ion{Mn}{I})~$=+2.31$ is lower than \nA(\\ion{Mn}{I})~$=+2.56$ derived from the subordinate lines. The resonance \ntriplet lines are more susceptible to NLTE effects, \nand for this reason we did not include them in the average\nMn abundance. The line \\ion{Mn}{I}~4041.35~{\\AA} is \nshown in Fig. \\ref{Cr_Mn_fig} (lower panel).\n\n\n$Iron$.\nWe used equivalent widths to derive the final \nabundances A(\\ion{Fe}{I})~$=+4.91$ and A(\\ion{Fe}{II})~$=+4.96$, as described\nin Sect. \\ref{EW}. \n\n$Cobalt$.\nIndividual cobalt abundances were derived from 17 \n\\ion{Co}{I} lines. The final abundance A(\\ion{Co}{I})~$=+2.69$ was \ncomputed excluding the lines located at 3861.16~{\\AA} and 3873.11~{\\AA} \nbecause of the higher differences in comparison with the average abundance. \n\n$Nickel$. \nFor this element \nwe derived the abundance A(\\ion{Ni}{I})~$=+3.76$ \nbased on 43 \\ion{Ni}{I} lines. In Fig. \\ref{Ni_Zn_fig} (upper panel) we show \nthe \\ion{Ni}{I}~3858.29~{\\AA} line. We were also able to use \nthe \\ion{Ni}{II}~3769.46~{\\AA} line in the present spectrum to derive the \nabundance A(\\ion{Ni}{II})~$=+3.88$ in this star. \n\n$Zinc$. \nThe isotopic structure of zinc is complex (see Table \\ref{fractions}), \nbut HFS is not needed to be accounted for, therefore, we assumed \nZn as having a unique isotope with wavelengths dominated by the $^{64}$Zn. \nThe abundance A(\\ion{Zn}{I})~$=+2.21$ was derived from the \\ion{Zn}{I}~\n4722.15~{\\AA} and \\ion{Zn}{I}~4810.53~{\\AA} (see Fig. \\ref{Ni_Zn_fig}, lower \npanel) lines, with oscillator strengths adopted from Roederer \\& Lawler (2012).\n\n\n\n\n\\subsection{Neutron-capture elements}\n\n\n\n$Strontium$. \nIn HD~140283 the Sr abundance was derived based on three \\ion{Sr}{II} \nlines located at 4077.72~{\\AA}, 4215.52~{\\AA}, and 10327.31~{\\AA}. \nThese transitions show HFS effects, but hyperfine coupling constants \nonly exist for $^{87}$Sr (Borghs et al. 1983), which accounts for \nless than 7\\% of Sr (Asplund et al. 2009). In addition, the atomic lines from \nthe even isotopes $^{84}$Sr, $^{86}$Sr, and $^{88}$Sr appear as single lines \ndue to the small isotopic splitting for Sr (Hauge 1972). In conclusion, we treated \neach Sr line as a single component, with oscillator strengths adopted from \nGratton \\& Sneden (1994), which gives an average abundance \nof A(\\ion{Sr}{II})~$=+0.10$. In the upper panel of Fig. \\ref{Sr_Y_fig}, \nwe show the fitting procedure used for the \\ion{Sr}{II}~4077.72~{\\AA} line.\n\n\\begin{figure}\n\\centering\n\\resizebox{80mm}{!}{\\includegraphics[angle=0]{Sr_Y.eps}}\n\\caption{LTE strontium abundance in HD~140283 from \\ion{Sr}{II}~4077.72~{\\AA} \nline (upper panel) and yttrium abundance from \\ion{Y}{II}~3774.33~{\\AA} line \n(lower panel). Symbols are the same as in Fig. \\ref{Li_fig}.}\n\\label{Sr_Y_fig}\n\\end{figure}\n\nOur NLTE strontium atomic model was described in Andrievsky et al. (2011), \nwhere it was applied to a sample of metal-poor stars. \nWe analysed the same two blue \\ion{Sr}{II} lines at 4077.72~{\\AA} and \n4215.52~{\\AA} (see upper panel in Fig. \\ref{NLTE_Sr}), and a third \nline in the near-infrared located at 10327.31~{\\AA} (see lower panel in Fig. \n\\ref{NLTE_Sr}), with a final NLTE abundance A(\\ion{Sr}{II})~$=+0.03$ \nas the adopted result.\n\nIn Andrievsky et al. (2011) the NLTE corrections were calculated for the \n\\ion{Sr}{II}~4077.72~{\\AA}, 4215.52~{\\AA}, besides near-infrared lines for different \ntemperatures and gravities. Considering Fig. 7 of Andrievsky et al. (2011),\n at \\Teff~$=5750$~K and \\logg~$=3.7$ the correction \nshould be small and positive. We obtained a NLTE strontium \nabundance that is slightly lower than the LTE abundance. We suggest that\nmain reasons for this discrepancy can be the result of: a) the LTE results \nfrom Turbospectrum may use slightly different atomic constants; and b) this \nstar is more metal-rich ([Fe\/H]~$=-2.59$) than the calculations for [Fe\/H]~$=-3.0$ \ngiven in Andrievsky et al. (2011).\n\n\n$Yttrium$. \nFor yttrium it was possible to check in LTE five \\ion{Y}{II} lines, \nusing oscillator strengths adopted from Hannaford et al. (1982) and \nGrevesse et al. (2015), with A(\\ion{Y}{II})~$=-0.78$ as the final \nabundance. The synthetic profile adopted for \\ion{Y}{II}~3774.33~{\\AA} \nis shown in Fig. \\ref{Sr_Y_fig} (lower panel), which takes the continuum \naffected by the H11 line from the Balmer series into account.\n\n\n\n$Zirconium$. \nAfter inspecting the spectrum, we decided to retain only the three best \n\\ion{Zr}{II} lines available, located at 3836.76~{\\AA}, 4208.98~{\\AA} \nand 4443.01~{\\AA}, to derive the LTE zirconium abundance. The \\loggf~values \nwere adopted from Bi\\'emont et al. (1981), with final LTE abundance of \nA(\\ion{Zr}{II})~$=-0.07$ (see Fig. \\ref{Zr_fig}). \n\n\n\\begin{figure}\n\\centering\n\\resizebox{80mm}{!}{\\includegraphics[angle=0]{Zr.eps}}\n\\caption{LTE zirconium abundance in HD~140283 from \\ion{Zr}{II}~4208.98~{\\AA} \n(upper panel) and \\ion{Zr}{II}~4443.01~{\\AA} (lower panel) lines. \nSymbols are the same as in Fig. \\ref{Li_fig}.}\n\\label{Zr_fig}\n\\end{figure}\n\n\n$Barium$. \nThe LTE barium abundance was previously analysed in Paper~I, based on \nthe \\ion{Ba}{II}~4554.03 and \\ion{Ba}{II}~4934.08~{\\AA} lines, \nwith oscillator strengths adopted from Gallagher (1967) and hyperfine structure \nline component patterns from McWilliam (1998). \nWe added two other \\ion{Ba}{II} lines, located \nat 6141.71~{\\AA} and 6496.90~{\\AA}, with \noscillator strengths and HFS from Barbuy et al. (2014). \nWith nuclear spin I~$=3\/2$, we took the Ba isotopic nuclides into account \naccording to Table \\ref{fractions}.\n\n\nFigure \\ref{Ba_fig} shows the synthetic profiles computed for the \n\\ion{Ba}{II} lines. The blue wing of the \\ion{Ba}{II}~4934.08~{\\AA} \nline is blended with \\ion{Fe}{I}~4934.01~{\\AA}, which we took properly \ninto account using \\loggf~$=-0.59$, adjusted to describe the \nobserved spectrum. The individual abundance agrees with the result \nderived from the clear \\ion{Ba}{II}~4554.03{\\AA} line. To compute the \nprofile for \\ion{Ba}{II}~6141.71~{\\AA}, it is important to include a \nblend with \\ion{Fe}{I}~6141.73~{\\AA} for which we adopted \n\\loggf~$=-1.60$ (Barbuy et al. 2014). For \\ion{Ba}{II}~6496.90~\n{\\AA} there is a telluric line located in the blue wing in the present \nspectrum. The individual abundance agrees with that derived from \n6141.71~{\\AA} line, but it is slightly higher in comparison with the \nresults from the first two Ba lines. We decided to adopt \nA(\\ion{Ba}{II})~$=-1.22$ as the final LTE abundance.\n\n\n\\begin{figure}\n\\centering\n\\resizebox{80mm}{!}{\\includegraphics[angle=0]{Ba.eps}}\n\\caption{LTE barium abundance in HD~140283 from \\ion{Ba}{II}~4554.03~{\\AA} \n(upper left panel), \\ion{Ba}{II}~4934.08~{\\AA} (upper right panel), \n\\ion{Ba}{II}~6141.71~{\\AA} (lower left panel), and \n\\ion{Ba}{II}~6496.90~{\\AA} (lower right panel) lines. \nSymbols are the same as in Fig. \\ref{Li_fig}.}\n\\label{Ba_fig}\n\\end{figure}\n\n\nA NLTE atomic model of \\ion{Ba}{II} was presented in Andrievsky et al.\n(2009). Three \\ion{Ba}{II} lines were analysed in HD~140283: \n4554.00~{\\AA}, 6141.70~{\\AA}, and 6496.92~{\\AA} (see Fig. \\ref{NLTE_Ba}). \nWe applied the odd-to-even isotopic ratio 50:50, which is applicable to old \nmetal deficient stars, to synthesize the \\ion{Ba}{II}~4554.00~{\\AA} line.\n\n\n\n$Lanthanum$. \nFor lanthanum, available transitions are too weak to enable us to derive \na robust value for the La abundance, but the upper limit of LTE \nabundance A(\\ion{La}{II})~$<-1.85$ was estimated from the \n\\ion{La}{II}~4123.22~{\\AA} line (see upper panel in Fig. \\ref{La_Ce_fig}). \nThis profile is located in the red wing of the H$\\delta$ line, \naccounted for in the spectrum synthesis. \nWe only used the major La isotope, with nuclear spin I~$=7\/2$, \nand the hyperfine coupling constants adopted basically from Lawler \net al. (2001), but also from Furmann et al. (2008) and Honle et al. \n(1982) when not in the basic reference. We also adopted experimental \noscillator strengths from Lawler et al. (2001). \n\n\n\\begin{figure}\n\\centering\n\\resizebox{80mm}{!}{\\includegraphics[angle=0]{La_Ce.eps}}\n\\caption{LTE lanthanum abundance in HD~140283 from \\ion{La}{II}~4123.22~{\\AA} \nline (upper panel) and cerium abundance in from \\ion{Ce}{II}~4083.22~{\\AA} line \n(lower panel). Symbols are the same as in Fig. \\ref{Li_fig}.}\n\\label{La_Ce_fig}\n\\end{figure}\n\n$Cerium$. \nFor cerium, we used the improved laboratory transition probabilities \npresented in Lawler et al. (2009) to derive the LTE abundance \nA(\\ion{Ce}{II})~$=-0.83$, based on two \\ion{Ce}{II} lines located \nat 4083.22~{\\AA} (lower panel in Fig. \\ref{La_Ce_fig}) and 4222.60~{\\AA}. \nThese are well known as good abundance indicators (e.g. Hill et al. 2002). \nThe local continuum around \\ion{Ce}{II}~4083.22~{\\AA} is \ndefined by the blue wing of the H$\\delta$ line. \nThese lines are weak in HD~140283, but still clearly\ndetectable as a result of the high quality of the spectrum. \nThese two lines give [Ce\/Fe]~$=+0.36$ and $+0.01$, respectively, \nand a corresponding mean overbundance of cerium [Ce\/Fe]~$=+0.18$. \nThe overabundance is therefore to be taken with caution.\n\n\n$Europium$. \nPaper~I was dedicated to derive the LTE europium abundance in HD~140283, \nusing the isotopic fractions in the solar material (Table \\ref{fractions}) \nwith nuclear spin I~$=5\/2$. The final abundance A(\\ion{Eu}{II})~$=\n-2.35\\pm0.07$ was obtained from \\ion{Eu}{II}~4129.70~{\\AA}, which is \nconsistent with the upper limit A(\\ion{Eu}{II})~$<-2.39$ estimated from \nthe \\ion{Eu}{II}~4205.05~{\\AA} line.\n\n\n\n\\section{Discussion}\n\n\\begin{table}\n\\caption{Abundances in HD~140283 computed in LTE, NLTE, and \nfinal abundances adopted. Iron is only computed in LTE.\n } \n\\label{adopted_abundances} \n\\scalefont{1.0}\n\\centering \n\\begin{tabular}{c@{}r@{}r@{}r@{}r@{}r} \n\\hline\\hline \n\\noalign{\\smallskip}\n\\hbox{Element} & \\phantom{-}\\phantom{-}\\hbox{[X\/Fe]$_{H04}$} &\n\\phantom{-}\\phantom{-}\\hbox{[X\/Fe]$_{LTE}$} & \n\\phantom{-}\\phantom{-}\\hbox{[X\/Fe]$_{NLTE}$} & \n\\phantom{-}\\phantom{-}\\hbox{[X\/Fe]$_{adopted}$}\\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\hbox{Fe} & $-$2.53 & $-$2.59 & ---- & $-$2.59 \\\\\n\\hbox{Li} & ---- & $+$2.14 & $+$2.20$^\\dagger$& $+$2.20 \\\\\n\\hbox{C(CH)}& $+$0.47 & $+$0.46 & ---- & $+$0.46 \\\\\n\\hbox{C(CI)}& ---- & $+$0.60 & $+$0.16 & $+$0.16\\\\\n\\hbox{N(CN)}& ---- & $+$1.06 & ---- & $+$1.06 \\\\\n\\hbox{O} & ---- & $+$0.97 & $+$0.90 & $+$0.90\\\\\n\\hbox{Na} & ---- & $-$0.04 & $-$0.29 & $-$0.29\\\\\n\\hbox{Mg} & $+$0.25 & $+$0.26 & $+$0.43 & $+$0.43\\\\\n\\hbox{Al} & $-$0.94 & $-$0.91 & $-$0.17 & $-$0.17\\\\\n\\hbox{Si} & $+$0.34 & $+$0.38 & ---- & $+$0.38\\\\\n\\hbox{K} & ---- & $+$0.54 & $+$0.26 & $+$0.26\\\\\n\\hbox{Ca} & $+$0.33 & $+$0.27 & $+$0.42 & $+$0.42\\\\\n\\hbox{Sc} & $+$0.10 & $+$0.10 & ---- & $+$0.10\\\\\n\\hbox{Ti} & $+$0.36 & $+$0.33 & ---- & $+$0.33\\\\\n\\hbox{V} & $+$0.21 & $+$0.22 & ---- & $+$0.22\\\\\n\\hbox{Cr} & $+$0.30 & $+$0.08 & ---- & $+$0.08\\\\\n\\hbox{Mn} & $-$0.25 & $-$0.29 & ---- & $-$0.29\\\\\n\\hbox{Co} & $+$0.25 & $+$0.29 & ---- & $+$0.29\\\\\n\\hbox{Ni} & $+$0.13 & $+$0.12 & ---- & $+$0.12\\\\\n\\hbox{Zn} & ---- & $+$0.25 & ---- & $+$0.25\\\\\n\\hbox{Ge} & ---- & ---- & ---- &$<-$0.46\\\\\n\\hbox{As} & ---- & ---- & ---- & $+$0.38$^1$\\\\\n\\hbox{Se} & ---- & ---- & ---- & $-$0.15$^1$\\\\\n\\hbox{Sr} & $-$0.31 & $-$0.18 & $-$0.30 & $-$0.30\\\\\n\\hbox{Y} & $-$0.46 & $-$0.40 & ---- & $-$0.40\\\\\n\\hbox{Zr} & $-$0.14 & $-$0.07 & ---- & $-$0.07\\\\\n\\hbox{Mo} & ---- & ---- & ---- & $+$0.19$^2$\\\\\n\\hbox{Ru} & ---- & ---- & ---- & $<+$0.99$^2$\\\\\n\\hbox{Ba} & $-$0.96 & $-$0.81 & $-$0.63 & $-$0.63\\\\\n\\hbox{La} & ---- &$<-$0.36 & ---- &$<-$0.36 \\\\\n\\hbox{Ce} & ---- & $+$0.18 & ---- & $+$0.18\\\\\n\\hbox{Eu} & ---- & $-$0.28 & ---- & $-$0.28\\\\\n\\hbox{Pt} & ---- & ---- & ---- & $<+$0.37$^1$\\\\\n\\hbox{Pb} & ---- & ---- & ---- & $<+$1.54$^1$\\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\tablebib{H04: Honda et al. (2004a,b); LTE: this work, in LTE; \nNLTE: this work, in NLTE. $\\dagger$: based on the NLTE corrections from Asplund et al. (2006). \nReferences: 1 Roederer (2012); 2 Peterson (2011).}\n\\end{table}\n\n\\begin{figure}\n\\centering\n\\resizebox{80mm}{!}{\\includegraphics[angle=0]{abgfe-oamix3.eps}}\n\\caption{Comparison of the LTE [O\/Fe] ratio obtained for the giant stars \nin the Large Programme ``First Stars'' based on the forbidden \n[\\ion{O}{I}]~6300.31~{\\AA} line (blue filled dots) and in HD~140283 (red star). \nThe blue open circles represent the two componnents of the turnoff binary \nCS~22876-032.}\n\\label{Ocompare_fig}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\resizebox{80mm}{!}{\\includegraphics[angle=0]{pattern.eps}}\n\\caption{Abundance pattern of HD~140283 based on the present \nresults (red circles) and literature (blue circles). \nThe dotted line is the abundance pattern of CS~31082-001, \nan r-II EMP star, representing the expected\nabundances for an r-rich star, \nrescaled to match the Eu abundance in HD~140283.}\n\\label{pattern}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\resizebox{80mm}{!}{\\includegraphics[angle=0]{hd122563-140283-31082.eps}}\n\\caption{Comparison of the abundance pattern of HD~140283, based on \nthe present results and literature (blue squares), with the \npattern of HD~122563 (green circles) and CS~31082-001 (black dots). \nThe solid line is the \nresidual solar r-process pattern.}\n\\label{pattern2}\n\\end{figure}\n\n\nIn Table \\ref{adopted_abundances} we present the abundances \nderived in LTE for 25 elements and an upper limit for lanthanum \nalong with abundances derived in NLTE for 9 elements, \nand the final abundances adopted. This table also includes abundances \nfrom Honda et al. (2004a) and abundances, for As, Se, Pt, Pb, from \nRoederer (2012) and, Mo and Ru, from Peterson (2011). \nAll of these are scaled to the metallicity and solar abundances \nadopted in this analysis. \n\nWe adopted as final abundances those analysed with NLTE models \n(Table \\ref{finalabundNLTE}), for the nine elements studied \nin NLTE, otherwise we adoted the LTE results. \nThe LTE abundances for Sc, V, and Cr are \nthe average from neutral and ionized lines. For \\ion{Mg}{II}, \\ion{Si}{II}, \nand \\ion{Ca}{II}, for which individual lines\nare reported in Table \\ref{linelist},\n the number of useful lines are too small in comparison with \nthe neutral species, so we did not include them in the average results.\n\n\\subsection{Comparison with the literature}\n\n\nThe forbidden [\\ion{O}{I}]~6300.31~{\\AA} line is considered \nthe best oxygen abundance indicator, given that it is not\naffected by NLTE effects. We derive an\nabundance of A(O)~$=+6.95$. The triplet lines at 7774-7~{\\AA} and 8446~{\\AA} \ncomputed in NLTE give an oxygen abundance A(O)~$=+7.02$, and [O\/Fe]~$=+0.90$ \nwas derived for HD~140238. Note that Fe was only computed in LTE. \nMel\\'endez et al. (2006) found a difference\nof $\\Delta$Fe$_{NLTE-LTE}=+0.08$~dex, which would reduce\nthe oxygen overabundance to [O\/Fe]~$=+0.82$. \nThis absolute oxygen abundance is in rather good agreement with previous NLTE derivation\nby Mel\\'endez et al. (2006) of A(O)~$=+6.97$, and [O\/Fe]~$=+0.57$, where\nthe difference in oxygen to iron is due to their\nmetallicity of [Fe\/H]~$=-2.25$ value (in NLTE),\nhigher by $0.34$~dex than the present value of [Fe\/H]~$=-2.59$. \nValues of [Fe\/H]~$=-2.3$, [O\/H]~$=-1.67$ and [O\/Fe]~$=+0.7$ were adopted \nby Bond et al. (2013).\n\nIn Fig. \\ref{Ocompare_fig} we present a comparison of the \nLTE [O\/Fe] abundance ratios obtained for the giant stars \nanalysed in the Large Programme ``First Stars'' \n(Cayrel et al. 2004; Spite et al. 2005) based on the \nforbidden [\\ion{O}{I}]~6300.31~{\\AA} line, \nwith the result derived for HD~140283. \nWe computed [O\/Fe] in HD~140283, adopting the solar oxygen abundance \nA(O)~$=+8.77$ for compatibility with the Large Programme. \nThe [O\/Fe] ratio obtained in the two components of the well-known \nspectroscopic turnoff binary CS~22876-032 \n(Gonz\\'alez Hern\\'andez et al. 2008) from the OH band, which includes \ncorrections for 3D effects, are also presented.\n\nThe high nitrogen abundance of [N\/Fe]~$=+1.06$, and a consequent \nhigh [C+N+O\/Fe] abundance ratio confirms previous findings by Barbuy (1983). \n\nThe results from Honda et al. (2004a) reported in Table \\ref{adopted_abundances} \nwere derived from a high-resolution and high-S\/N spectrum obtained with the Subaru High \nDispersion Spectrograph. The Cr abundance is the only result \nsignificantly different from ours for which the present value is $0.22$~dex lower \nin comparison with their result. In both cases the abundance is the average \nof \\ion{Cr}{I} and \\ion{Cr}{II} lines. Honda et al. (2004a) adopted a \n\\Teff~120~K lower and a gravity \\logg~0.2 dex lower with respect to \nour set of parameters, whereas they used the same value for \nmicroturbulence, which is the most important error source in Cr abundance. However, \ntheir Cr abundance was derived only based on three lines, in comparison with \n33 lines in the present work, which we consider a more reliable final \nabundance.\n\nRoederer (2012) analysed the zinc abundance in HD 140283 based on \ntwo \\ion{Zn}{I} \nlines also used in the present work, with a spectrum obtained with HARPS\/ESO,\n deriving a \nresult $0.08$~dex lower in comparison with our abundance. However, the signal-to-noise \nratio achieved in the present work is much higher than the value from Roederer (2012) \n(250 at 4500~{\\AA}), leading us to adopt the present result as the final Zn abundance. \nIn addition, Roederer (2012) was able to use the UV \\ion{Zn}{II} line located at \n2062.00~{\\AA} with a spectrum obtained with STIS\/HST to derive A(\\ion{Zn}{II})~$=+2.27$, \nor [\\ion{Zn}{II}\/Fe]~$=+0.30$, which is higher than the result from neutral transitions. \n\nFor strontium, the LTE result of [\\ion{Sr}{II}\/Fe]~$=-0.17$ derived here agrees with \nthe result from Roederer (2012), which used the same two \\ion{Sr}{II} lines, \nbut with \\loggf~$=+0.15$ for \\ion{Sr}{II}~4077.7~{\\AA}, slightly lower than the adopted value. \nOn the other hand, this value is higher in comparison with the LTE results from \nHonda et al. (2004a). Our adopted NLTE analysis gives [\\ion{Sr}{II}\/Fe]~$=-0.30$. \n\nThe yttrium abundance was derived in Honda et al. (2004a,b) \nbased on three \\ion{Y}{II} lines,\n located at 3788.70~{\\AA}, 3950.35~{\\AA}, and 4883.69~{\\AA}. \nWe added another two \\ion{Y}{II} lines and \nthe derived abundance [\\ion{Y}{II}\/Fe]~$=-0.40$ \nis slightly higher than the result from Honda et al. (2004a,b), and it is in \nvery good agreement with the abundance derived in Peterson (2011). \n\nThe zirconium abundance we derived is $0.1$~dex higher \nthan the result \nfrom Honda et al. (2004a,b), where two \\ion{Zr}{II} lines located \nat 3836.76~{\\AA} \nand 4208.98~{\\AA} were analysed. We adopted the same values of oscillator \nstrengths, and also used an extra \\ion{Zr}{II} line located at 4443.01~{\\AA}, \nwith individual abundances in agreement with the result from the 4208.98~{\\AA} \nline. Roederer (2012) also obtained a Zr abundance $0.1$~dex lower than the \npresent result, using the \\ion{Zr}{II} lines at \n3998.96~{\\AA}, 4149.20~{\\AA}, and \n4208.98~{\\AA}, in common with our analysis. For the latter line, Roederer \n(2012) obtained A(\\ion{Zr}{II})~$=-0.09$, only $0.04$~dex lower than the present \nresult. In our spectrum the lines 3998.96~{\\AA} and 4149.20~{\\AA} appear \nseverely blended and they were excluded. In addition, Peterson (2011) derived \nan intermediate Zr abundance between the present value and that from Roederer \n(2012), in agreement with our result within error bars.\n\n\n\n\\subsection{Source of neutron-capture elements}\n\nTruran (1981) suggested an early enrichment of heavy elements uniquely \nthrough the r-process, arguing that there would be no time for \nthe s-process to operate before the formation of the oldest stars. \nIn this scenario, even dominantly s-process elements\nsuch as Sr, Y, Zr, and Ba, would have been produced by the r-process in their\nrelatively reduced amounts. \n\nGiven the recent discussions on the r-process or s-process origin\nof the heavy elements in HD~140283, below we try to understand the\nresults concerning this star. Since HD~140283 is one of the oldest \nstars known so far, formed shortly after the Big Bang, \nit is a natural test star for studies of early heavy element \nformation.\n\nAs described in Paper~I, the barium isotopic abundances in HD~140283 are subject \nto intense debate in the literature. In a 1D LTE study of barium isotopes, \nGallagher et al. (2010, 2012) found isotope ratios close to the s-process-only \ncomposition. This is supported by Collet et al. (2009), using 3D hydrodynamical \nmodels, where a maximum fraction of 15$\\pm$34\\% contribution of the r-process \nto the isotopic mix in HD~140283 is derived. More recently, Gallagher et al. (2015) \ncarried out 3D calculations of the barium isotopic lines in HD~140283, \nand the authors now favour a dominant r-process signature imprinted in the \nbarium isotopes in HD~140283. Gallagher et al. (2015) suggest that further work is \nneeded to improve the line formation in 3D, and that NLTE has \nto be taken into account.\n\nThe [Eu\/Ba] ratio is the unambigous indicator as to whether the heavy elements in \nHD~140283 are dominantly r- or s-process. In Paper I we concluded that the \n[Eu\/Ba]~$=+0.58\\pm0.15$ value indicates that r-process is the dominant nucleosynthesis process. \nWe found an LTE Ba abundance A(Ba)~$=-1.22$, in agreement \nwithin errors with A(Ba)~$=-1.28$ from 1D calculations in LTE derived by \nGallagher et al. (2015). With the newly derived Ba abundance, the ratio [Eu\/Ba]~$=+0.53\\pm0.18$ \nconfirms the previous conclusion concerning the dominant r-process origin. \n\nAn NLTE Ba abundance A(Ba)~$=-1.05$, or [\\ion{Ba}{II}\/Fe]~$=-0.63$ is derived here, \nand a lower [Eu\/Ba]~$=+0.34$ ratio is obtained in this case. \nIf an NLTE Ba abundance is considered, NLTE Eu also has to be considered: \nMashonkina et al. (2012) presented NLTE abundance corrections for the \n\\ion{Eu}{II}~4129~{\\AA} line in cool stars, showing that the corrections \nare small (lower than 0.1 dex) and positive for this element and these types of stars, \nwhich would turn the present [Eu\/Ba]$\\approx$0.44.\nA further ingredient is 3D vs. 1D calculations: Gallagher et al. (2015) \nreported A(Ba)~$=-1.43$ in HD~140283 from LTE 3D calculations, therefore, with \na $0.15$~dex lower value for the 3D with respect to 1D calculations, \nand in this case LTE [Eu\/Ba(3D)]~$=+0.74$ is obtained, or else\nadding the 3D effect to [Eu\/Ba]$\\approx$0.44 above, gives [Eu\/Ba]$\\approx$0.59.\nIn Table \\ref{eubaratio} we try to summarize all [Eu\/Ba] values given above.\n\nAccording to Simmerer et al. (2004), \na pure r-process contribution gives [Eu\/Ba]$_{r}=+0.698$, whereas \n[Eu\/Ba]$_{s}=-1.621$ is the abundance pattern due to the pure s-process. \nRecent models for the solar s-process abundances (Bisterzo et al. 2014) \npredict the same contribution for Ba in comparison with Simmerer et al. (2004), \nand a slightly higher contribution for the Eu abundance: \n2.7\\% from Simmerer et al. (2004); 6.0$\\pm$0.4\\% from Bisterzo et al. (2014). \nIn conclusion, the abundance ratios [Eu\/Ba] described above do not change \nsignificantly.\n\nIf a lower [Eu\/Ba] value relative to Paper I is confirmed,\nit may indicate that the contribution from the \ns-process to the heavy elements is not so small.\n Table \\ref{eubaratio} shows that no clear conclusion can be reached,\nbut that conclusions from Paper I are still favoured.\n This discussion indicates \nthat a robust 3D+NLTE synthesis is needed to enable further conclusions.\n\n\n\\begin{table}\n\\caption{[Eu\/Ba] abundance ratios expected from the r- and s-process, \nand derived in HD~140283.} \n\\label{eubaratio} \n\\scalefont{1.0}\n\\centering \n\\begin{tabular}{cr} \n\\hline\\hline \n\\noalign{\\smallskip}\n\\hbox{Source} & \\hbox{[Eu\/Ba]} \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\hbox{$^\\dagger$pure r-process} & $+$0.698 \\\\\n\\hbox{$^\\dagger$pure s-process} & $-$1.621 \\\\\n\\hbox{Eu$_{1D+LTE}$~$+$~Ba$_{1D+LTE}$} & $+$0.53 \\\\\n\\hbox{Eu$_{1D+LTE}$~$+$~Ba$_{1D+NLTE}$} & $+$0.34 \\\\\n\\hbox{Eu$_{1D+LTE}$~$+$~$^\\triangle$Ba$_{3D+LTE}$} & $+$0.74 \\\\\n\\hbox{$^\\diamondsuit$Eu$_{1D+NLTE}$~$+$~Ba$_{1D+NLTE}$} & $+$0.44 \\\\\n\\hbox{$^\\diamondsuit$Eu$_{1D+NLTE}$~$+$~$^\\triangle$Ba$_{3D+NLTE}$} & $+$0.59 \\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\tablebib{$\\dagger$: Simmerer et al. (2004); $\\triangle$: 3D correction \nfrom Gallagher et al. (2015); $\\diamondsuit$: NLTE correction from \nMashonkina et al. (2012).}\n\\end{table}\n\n\n\n\\subsubsection{Abundance pattern}\n\nFigure \\ref{pattern} shows the abundance pattern of HD~140283, based on \nvalues derived in the present work (red circles), as well as results \nfrom the literature (blue circles), where upper limits are also indicated. \nThe upper panel presents the elements from carbon to zinc, where \ncarbon from both CH and \\ion{C}{I} lines are indicated. \nThe neutron-capture elements are shown in the lower panel. \nAs a reference abundance pattern, we used the r-element enhanced EMP (r-II) \nstar CS~31082-001 (Barbuy et al. 2011; Siqueira-Mello et al. 2013) \nfor comparison (dotted line), rescaled to match the\ndominantly r-process europium abundance in HD~140283. \nThis figure shows the overabundance of the lighter heavy elements \n(including Sr, Y, and Zr) in comparison with expected values from an r-rich \nstar. \n\nThe LTE abundances of Ba and Sr result in [Sr\/Ba]~$=+0.63$ for HD~140283. \nIf we take the LTE 3D Ba correction from Gallagher et al. (2015) into account, \nthis value would be [Sr\/Ba]~$=+0.78$, even more overabundant. The newly \nderived NLTE abundances of Ba and Sr give [Sr\/Ba]~$=+0.33$, or \n[Sr\/Ba]~$=+0.48$ if the 3D Ba correction is applied. \nOverabundances are also obtained for Y and Zr, reaching values of \n[Y\/Ba]~$=+0.41$ and [Zr\/Ba]~$=+0.74$ in the present work, \nusing only the LTE results. With the NLTE Ba abundance these ratios \ndecrease to [Y\/Ba]~$=+0.23$ and [Zr\/Ba]~$=+0.56$ in HD~140283, but the \n3D Ba correction enhances these values to [Y\/Ba]~$=+0.38$ and [Zr\/Ba]~$=+0.71$. \nIn Table \\ref{sryzrbaratio} these values are summarized, showing that it is \nimportant to account for NLTE and 3D effects to evaluate these \nabundance ratios.\n\n\n\\begin{table}\n\\caption{[Sr, Y, Zr\/Ba] abundance ratios derived in HD~140283.} \n\\label{sryzrbaratio} \n\\scalefont{1.0}\n\\centering \n\\begin{tabular}{cc} \n\\hline\\hline \n\\noalign{\\smallskip}\n\\hbox{Source} & \\hbox{Abundance} \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\hbox{Sr$_{1D+LTE}$~$+$~Ba$_{1D+LTE}$} & [Sr\/Ba]~$=+$0.63 \\\\\n\\hbox{Sr$_{1D+NLTE}$~$+$~Ba$_{1D+NLTE}$} & [Sr\/Ba]~$=+$0.33 \\\\\n\\hbox{Sr$_{1D+LTE}$~$+$~$^\\triangle$Ba$_{3D+LTE}$} & [Sr\/Ba]~$=+$0.78 \\\\\n\\hbox{Sr$_{1D+NLTE}$~$+$~$^\\triangle$Ba$_{3D+NLTE}$} & [Sr\/Ba]~$=+$0.48 \\\\\n\\hbox{Y$_{1D+LTE}$~$+$~Ba$_{1D+LTE}$} & [Y\/Ba]~$=+$0.41 \\\\\n\\hbox{Y$_{1D+LTE}$~$+$~Ba$_{1D+NLTE}$} & [Y\/Ba]~$=+$0.23 \\\\\n\\hbox{Y$_{1D+LTE}$~$+$~$^\\triangle$Ba$_{3D+NLTE}$} & [Y\/Ba]~$=+$0.38 \\\\\n\\hbox{Zr$_{1D+LTE}$~$+$~Ba$_{1D+LTE}$} & [Zr\/Ba]~$=+$0.74 \\\\\n\\hbox{Zr$_{1D+LTE}$~$+$~Ba$_{1D+NLTE}$} & [Zr\/Ba]~$=+$0.56 \\\\\n\\hbox{Zr$_{1D+LTE}$~$+$~$^\\triangle$Ba$_{3D+NLTE}$} & [Zr\/Ba]~$=+$0.71 \\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\tablebib{$\\triangle$: 3D correction from Gallagher et al. (2015).}\n\\end{table}\n\n\nSeveral authors found high abundance ratios of first peak elements\nSr, Y, Zr, with respect to Ba: [Sr,Y,Zr\/Ba]~$>0$, in very metal-poor stars \n(e.g. Honda et al. 2004a,b; Honda et al. 2006; Fran\\c{c}ois et al. 2007; \nCowan et al. 2011). According to Simmerer et al. (2004), in the solar material these \nelements are mainly produced by the s-process, with fractions of \n89\\%, 72\\%, and 81\\%, respectively, but recent results described in \nBisterzo et al. (2014) are different for some elements (Sr: 68.9$\\pm$5.9\\%; \nY: 71.9$\\pm$8.0\\%; Zr: 66.3$\\pm$7.4\\%). Therefore, \nwhile a mechanism responsible for a r-II pattern is claimed to \nexplain the \nnucleosynthesis of the heaviest trans-iron elements, an extra \nmechanism (such as truncation or other) should act to produce \nthe enhancement of the lightest heavy elements relative to the \nheaviest elements, which must occur very early in the history of the Universe. \n\nSeveral models in the literature find that supernova explosion explain the \noverabundances of the first peak elements in metal-poor stars. \nMontes et al. (2007) provided evidence for the existence \nof a light element primary process that contributes to the nucleosynthesis of \nmost elements in the Sr to Ag range, producing early high [Sr,Y,Zr\/Ba] \nratios. The astrophysical scenarios in neutrino-driven winds are claimed \nas promising sources of light trans-iron elements (Wanajo 2013, Arcones et al. \n2013). See also the LEPP (e.g. Bisterzo et al. 2014).\n\nHansen et al. (2014) show that the observed patterns may be obtained \nby combining an r- and an s-pattern, but the 's' has to be (slowly) \nproduced in another generation (AGB?), not very compatible with the great \nage of the star. Yet, a full understanding of core collapse supernova explosion \nusing 3D hydrodynamical modelling is needed.\n\n\nA possibility that HD~140283 formed in a very early dispersed cloud \ncould also lead to a previous early chemical enrichment or pollution \nby massive AGB stars, which overproduce the first peak s-process elements \n(Bisterzo et al. 2010). In this scenario, a previous enrichment in Fe seeds is needed, \nand the timescale of the whole process is too long and not \nideal for such an old star.\n\nIn Fig. \\ref{pattern2} we show the abundance pattern in HD~140283 \n(blue squares) compared with the values obtained in HD~122563 (green circles), \na metal-poor star well-known for its excesses of light neutron-capture elements. \nThe abundances of the heavy elements in HD~122563 have been taken from Honda et al. (2006), \nCowan et al. (2005), and Roederer et al. (2010). As references, we included \nthe abundances of CS~31082-001 (black dots) and the residual solar r-process \npattern, following the deconvolution by Simmerer et al. (2004). The data were rescaled \nto match the europium abundances. The overabundance \nof first peak elements derived in HD~140283 are only slighly lower than\n the values observed in HD~122563. \n\nIn conclusion, Figures \\ref{pattern} and \\ref{pattern2} show overabundances \nof the first-peak heavy elements, and, at a lower level, also of Ba, La and Ce,\n with respect to CS~31082-001 and the residual solar r-process.\nIt is not clear if the extra mechanism claimed to explain the higher \nabundances of light neutron-capture elements might also produce \nheavier elements, but less efficiently.\n\nThe overabundance of cerium [Ce\/Fe]~$=+0.18$, a \ndominantly s-process element (83.5$\\pm$5.9\\% in the solar material, \nfrom Bisterzo et al. 2014),\nwas also found in other few stars. \nIn Fran\\c{c}ois et al. (2007), the star CS~30325-094 also showed a \nhigh Ce overabundance [Ce\/Fe]~$=+0.43$, however, with deficient values \nfor other s-elements like Ba ([Ba\/Fe]~$=-1.88$) and Sr ([Sr\/Fe]~$=-2.24$). \nIn Fig. \\ref{pattern} and \\ref{pattern2} the Ce abundance in \nHD~140283 is clearly higher than the r-process pattern, and the \nsame behaviour is observed in HD~122563. Because of the difficulty in explaining \nthe abundance pattern of HD~122563 as a combination of \nthe r-process and the main s-process, Honda et al. (2006) \nsuggested a single process that is responsible for the enhancement of the \nlight neutron-capture elements and the production of heavy elements \nin this star. A truncation process in the initial supernova \n(Aoki et al. 2013) could be a possible solution. A new model of hypernova \nshows that the explosion correctly produces the abundances of the elements \nobserved in HD~122563, therefore, explaining the so-called weak r-process \n(Fujibayashi et al. 2015) as well as the similar pattern derived in HD~140283.\n\n\n\n\\section{Conclusions}\n\nWe used a high-S\/N and high-resolution spectrum of HD~140283, obtained\nwith a seven hour exposure with the ESPaDOnS spectrograph at the CFHT telescope,\nto provide a line list for metal-poor subgiant stars of which HD~140283\nis a template. We carried out a detailed derivation of abundances, using\nboth LTE and NLTE calculations, based on as many as possible reliable lines \navailable.\n\nIn Paper I we concluded that the derived europium abundance was indicative\nof an r-process origin for europium. The present LTE [Eu\/Ba]~$=+0.53\\pm0.18$ confirms \nthat conclusion. Combining the newly derived NLTE Ba abundance with NLTE \ncorrections for Eu and 3D corrections for Ba from recent literature, the \nabundance ratio [Eu\/Ba]~$=+0.59\\pm0.18$ also indicates a small contribution (if any) \nfrom the main s-process to the neutron-capture elements in HD~140283.\n\nAn extra mechanism is claimed to explain the overabundances \nof lighter heavy elements, in addition to an r-II abundance pattern \nresponsible for the heavier elements, and possible astrophysical scenarios are \ndiscussed. The high Ba, La, and Ce abundances derived \nin HD 140283 are similar to those in HD 122563,\n and these two stars may be excellent examples of abundances dominated \nby the weak r-process.\n\n\n\n\n\n\n\n\\begin{acknowledgements}\nBased on observations obtained with Brazilian time, provided by\na contract of the Laborat\\'orio Nacional de Astrof\\'{\\i}sica (LNA\/MCTI)\nand the Canada-France-Hawaii Telescope (CFHT), which is operated by the \nNational Research Council of Canada, the Institut National des Sciences \nde l'Univers of the Centre National de la Recherche Scientifique of France, \nand the University of Hawaii.\nCS and BB acknowledge grants from CAPES, CNPq, and FAPESP. \nSMA is thankful to FAPESP for financial support and IAG for \nhospitality during his visit to Universidade de S\\~ao Paulo. \nMS and FS acknowledge the support of CNRS (PNCG and PNPS). \nSAK acknowledges the SCOPES grant No. IZ73Z0-152485 for financial support. \nWe thank the referee for his\/her useful comments. \nThis work has made use of the VALD database, operated at \nUppsala University, the Institute of Astronomy RAS in Moscow, \nand the University of Vienna.\n\\end{acknowledgements}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nAutomatic meeting\/spoken-conversation analysis is one of essential fundamental technologies required \nfor realizing, e.g. communication robots that can follow and respond to our conversations. \nThe meeting analysis comprises several tasks, namely (a) diarization, i.e., determining who is speaking when, (b) source counting, i.e., estimating the number of speakers in the conversation, \n(c) source separation, and (d) automatic speech recognition (ASR), i.e., transcribing the separated streams corresponding to each person.\nWhile ideally these tasks should be jointly accomplished to realize optimal meeting analysis,\nmost studies focus on one of the aforementioned tasks,\nsince each task itself is already very challenging in general.\n\nFor example, a considerable number of research has been carried out for developing reliable diarization systems \\cite{Diarization_review, DIHARD_data, AMI_data}.\nMost of the conventional diarization approaches perform block or block-online processing with the following two steps \\cite{Diarization_review, Araki_ICASSP2007, Araki_HSCMA_2008, DIHARD_BUT, DIHARD_JHU}.\nFirst, at each block, they perform source separation (if necessary) and obtain speaker identity features concerning each speaker, \nin the form of, e.g. i-vector \\cite{i-vector}, x-vector \\cite{x-vector}, or spatial signature \\cite{Araki_ICASSP2007, Araki_HSCMA_2008, Drude_ICASSP2018}.\nThen, the correct association of speaker identity information among blocks, i.e., diarization results, is estimated by\nclustering these features by using e.g. agglomerative hierarchical clustering \\cite{DIHARD_JHU}.\nAlthough these conventional algorithms can achieve reasonable diarization performance,\nthe results are not guaranteed to be optimal,\nbecause the steps concerning speaker identity feature extraction and clustering are done independently.\nFocusing on this limitation, \\cite{Fujita_IS2019} recently proposed a neural network (NN)-based diarization approach \nthat directly outputs diarization results (without any clustering stage), \nand showed that it can outperform a conventional two-stage approach \\cite{DIHARD_JHU} in CALLHOME task \nwhere two people speak over a phone channel.\nNote that, although some diarization systems can deal with overlapped speech, it does not mean that they can perform speech enhancement, i.e., separation and denoising, which is\neventually required for the meeting analysis.\n\nAnother key challenge for automatic meeting analysis is the separation of overlapped speech. \nPerhaps surprisingly, even in professional meetings, the percentage of overlapped speech, \ni.e., time segments where more than one person is speaking, \nis in the order of 5 to 10\\%, \nwhile in informal get-togethers \nit can easily exceed 20\\% \\cite{onlineRSAN_ICASSP2019}.\nTo address the source separation problem, recently, many NN-based single-channel approaches have been proposed,\nsuch as Deep Clustering (DC)\\cite{Hershey_ICASSP16}, \nand Permutation Invariant Training (PIT)\\cite{Yu2016, Kolbaek2017}.\nDC can be viewed as two-stage algorithms, where in the first stage embedding vectors are estimated for each time-frequency (T-F) bin. \nIn the second stage, these embedding vectors are clustered to obtain separation masks, given the correct number of clusters, i.e., sources.\nPIT, on the contrary, is a single-stage algorithm, because it lets NNs directly estimate source separation masks.\nIn PIT, however, the NN architecture depends on the maximum number of sources to be extracted. \n\nTo lift this constraint on the number of sources, \nwe proposed Recurrent Selective Attention Network (RSAN) that is a purely NN-based mask estimator\ncapable of separating an arbitrary number of speakers and simultaneously counting the number of speakers in a mixture \\cite{RSAN}.\nIt extracts one source at a time from a mixture and recursively performs this process until all sources are extracted.\nThe RSAN framework is based on a recurrent NN (RNN) \nwhich can learn and determine how many computational steps\/iterations\nhave to be performed \\cite{Graves_2016_arxiv_adaptiveRNN}.\nFollowing the idea of the recursive source extraction, \\cite{Takahashi_Interspeech2019} proposed to incorporate a time-domain audio separation network (TasNet) \\cite{luo2019conv} into the RSAN framework.\n\nTo go one step further toward ideal meeting analysis that deals with aforementioned tasks (a)(b)(c) simultaneously, \nin \\cite{onlineRSAN_ICASSP2019}, RSAN was extended to an all-neural block-online approach (hereafter, online RSAN) \nthat simultaneously performs source separation, source number counting and even speaker tracking from a block to a block, \ni.e., performing the diarization-like process.\nIt was shown that online RSAN can handle well-controlled scenarios such as clean (noiseless and anechoic) simulated dialog-like data \nof 30 seconds, and outperform conventional systems in terms of source separation and diarization performance.\n\n\nHowever, it was not clear from our past studies \nwhether such all-neural approach, i.e., online RSAN, can be generalized \nto realistic meeting scenarios containing more spontaneously-speaking speakers, \nsevere noise and reverberation, and how it performs in comparison with the state-of-the-art systems.\nTo this end, this paper focuses on the application of the online RSAN model \nto the realistic situations, and its evaluation in comparison with state-of-the-art diarization algorithms.\nWe first review the online RSAN model (in Section~\\ref{sec:prop})\nand introduces practical techniques to increase robustness against real meeting data,\nsuch as a decoding scheme that mitigates over-estimation error in source counting (in Section~\\ref{sec:decode}).\nWe then carry out experiments with real and simulated meeting data involving up to 6 speakers, \ncontaining a significant amount of noise and reverberation.\nThen, finally, it will be shown that, even in such difficult scenarios, \nonline RSAN can still perform effective speech enhancement, i.e., source separation and denoising, \nand simultaneously outperform a state-of-the-art diarization system \\cite{dihard19} developed \nfor a recent challenge \\cite{DIHARD_data}.\nThis finding is the main contribution of this paper. \nBefore concluding this paper, we also discuss the challenges that remain.\n\n\\section{Overall Structure of online RSAN}\n\\label{sec:prop}\n\n\nFigure \\ref{fig:overview} summarizes how online RSAN works on the first 2 blocks of an example mixture containing three sources.\nSince the model works in a block-online manner,\nwe first split the input magnitude spectrogram $\\mat{Y}$ into $B$ consecutive blocks of equal time length, $[\\mat{Y}_1,\\ldots,\\mat{Y}_b,\\ldots,\\mat{Y}_B]$, before feeding it to the system.\n\nOnline RSAN estimates a source extraction mask $\\hat{\\mathbf{M}}_{b,i}$ in each $b$-th block recursively to extract all source signals therein, \nwhile judging at each $i$-th source extraction iteration whether or not to proceed to the next iteration to extract more source signals.\nThe same neural network ``NN'' is repeatedly used for each block and iteration.\nAt every iteration in $b$-th block, NN receives three inputs, \na residual mask $\\mat{R}_{b,i}$ (res-mask in the figure), \nan input spectrogram $\\mat{Y}_b$, \nand an auxiliary feature $\\vect{z}_{b-1,i}$ (speaker embed.~vec. in the figure) \nto estimate a mask for a specific speaker $\\hat{\\mathbf{M}}_{b,i}$ \nand a speaker embedding vector representing that specific speaker $\\vect{z}_{b,i}$ as:\n\\begin{equation}\n \\hat{\\mathbf{M}}_{b,i}, \\vect{z}_{b,i} = \\textrm{NN}(\\mat{Y}_b, \\mathbf{R}_{b,i}, \\vect{z}_{b-1,i}).\n\\end{equation}\nThe residual mask can be seen as an attention map that informs the network\nwhere to attend to extract a speaker that was not extracted in previous iterations in the current block.\nAt every first iteration in the $b$-th block, the residual mask is initialized with $\\mat{R}_{b,0}=\\mat{1}$.\n\n\\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=90mm]{.\/fig\/network_architecture_ver2.jpg}\n \\end{center}\n \\caption{Overview of online RSAN framework}\n \\label{fig:overview}\n\\end{figure}\n\n\nIn the first processing block, $b=1$, where we have two sources, online RSAN performs two source extraction iterations \nto extract all sources.\nSince it is the first block, no speaker information is available from the previous block. \nTherefore, the input speaker information is set to zero, $\\vect{z}_{0,i}=\\vect{0}$. \nWithout guidance, the network decides on its own in which order it extracts the source signals.\nFig.~\\ref{fig:overview} shows a situation where source 1 is extracted at the first iteration.\nThen, the first iteration is finished after generating another residual mask for the next iteration \nby subtracting the estimated source extraction mask from the input residual mask as $\\mathbf{R}_{1,1} = \\mathbf{R}_{1,0}-\\hat{\\mathbf{M}}_{1,1}$.\nAt the second iteration, the network receives the residual mask $\\mathbf{R}_{1,1}$ \nand input spectrogram $\\mathbf{Y}_1$ to estimate a mask for another source.\nThen, it follows the same procedure as the first iteration.\nThe network decides to stop the iteration \nby examining how empty the updated residual mask is.\nSpecifically, the source extraction process is stopped in iteration $i$ if\n$\\frac{1}{TF} \\sum_{tf} \\mathbf{R}_{i,tf} < t_{\\text{res-mask}}$ \nwhere $T$ and $F$ correspond to the total number of time frames and frequency bins in the block,\nand $t_{\\text{res-mask}}$ is an appropriate threshold.\n\n\nNote that the speaker embedding vector $\\vect{z}_{b,i}$ is passed to the next time block, $b+1$, \nand guides the $i$-th iteration on that block to extract the same speaker as in $(b,i)$. \nIn the figure, it can be seen that the green source (source 1) is always extracted in the first, \nthe blue in the second and the pink in the third iterations. \nIf a source happens to be silent in a particular block (see the 2nd iteration in block 2), \nthe estimated mask is to be filled with zeros ($\\hat{\\mat{M}}_{b,i} = \\vect{0}$), \nand the residual mask is to stay unmodified ($\\mat{R}_{b,i} = \\mat{R}_{b,i-1}$).\n\nAt the second iteration of block 2, \nthe criterion to stop the source extraction iteration is not met because the residual mask is not empty.\nIn such a case, the model increases the number of iterations to extract any new speaker\nuntil the stopping criterion is finally met.\nOverall, by having this structure, the model can perform jointly source separation, counting and speaker tracking.\n\nTo deal with background noise,\nin this study, we force online RSAN to estimate a mask for the noise always at the first source extraction iteration \nin each block (see Fig. \\ref{fig:overview}).\nWith this scheme, we can easily identify signals corresponding to background noise among separated streams \nand discard them if necessary.\n\n\\section{Summary of practical techniques \\\\ required to handle real meeting data}\n\\label{sec:practical}\nThis section summarizes techniques that we incorporated into online RSAN\nto cope with noisy reverberant real meeting data.\nThey are divided into categories of input feature, training schemes, and decoding scheme\nand summarized in the following dedicated subsections. \n\n\\subsection{Input feature: Multichannel feature}\nAs in much past literature, e.g. \\cite{Yoshioka_ICASSP2018},\nour preliminary experiments confirmed that a multichannel feature as an additional input \nhelps improve separation performance in reverberant environments.\nIn this study, therefore, the inter-microphone phase difference (IPD) feature proposed in \\cite{Yoshioka_ICASSP2018} \nis concatenated by default to the magnitude spectrogram and used as input to online RSAN.\n\n\\subsection{Training scheme}\nTo train online RSAN,\nwe used training data comprising a pair of noisy reverberant meeting-like mixtures,\nand corresponding noiseless reverberant single-speaker signals.\n\n\\subsubsection{Cost function for model training}\n\\label{sec:prop:training}\nDuring training, the network is unrolled over multiple blocks and iterations and \nwas trained with back-propagation using the following multi-task cost function:\n\\begin{equation}\n\\mathcal{L} = {\\L^{(\\text{MMSE})}} + \\alpha {\\L^{(\\text{res-mask})}} + \\beta {\\L^{(\\text{TRIPLET})}}\n\\end{equation}\nIn the following, we explain each term on the right-hand side of the above equation,\nstarting from ${\\L^{(\\text{MMSE})}}$.\n\nAt each iteration, the network is required to output a mask for a certain source, but the order of source extraction when they first appear is not predictable.\nIn such a case, a permutation-invariant loss function is required.\nOnce a source was extracted and the permutation was chosen to minimize the error on its first appearance, \nits position in the iteration process is fixed for any following blocks, as the embedding vectors are passed and thus the desired output order is known.\nConsequently, a {\\it partially} permutation-invariant utterance-level mean square error (MSE) loss can be used for online RSAN as:\n\\begin{equation}\n{\\L^{(\\text{MMSE})}} = \\frac{1}{IB} \\sum_{i,b} |\\hat{\\mat{M}}_{i,b}\\odot \\mat{Y} - \\mat{A}_{\\phi_b}|^2,\n\\end{equation}\nwhere $\\mat{A}_{\\phi_b}$ is a target reverberant single-speaker signal.\n\\footnote{To handle reverberant mixture, it was found in our preliminary investigation \nthat the target signal $mat{A}_{\\phi_b}$ has to be magnitude spectrum of reverberant (not anechoic clean) speech}.\nWhen a source was active before, but is silent on the current block, a silent signal $\\mat{A}_{b,i}=\\mat{0}$ are inserted as a target.\nThe permutation $\\phi_b$ for $b$-th block is formed by concatenating the permutation used for the last block $\\phi_{b-1}$ with the permutation $\\phi^*_b$ that minimizes the utterance-level separation error for the\nnewly discovered sources in block $b$.\nTo force the network to estimate a mask for the noise always at the first source extraction iteration,\nwe always inserted noise magnitude spectrogram as a target at the first iteration,\nand from the second iteration, we used the above partially permutation-invariant loss.\n\n${\\L^{(\\text{res-mask})}}$ is a cost function related closely to the source counting performance.\nTo meet the speaker counting and iteration stopping criterion when all sources are extracted from a mixture, \nwe minimize this cost and pushes the values of the residual mask to $0$ if no speaker is remaining (see \\cite{RSAN} for more details).\nwe minimize the following ${\\L^{(\\text{res-mask})}}$ and pushes the values of the residual mask to $0$ if no speaker is remaining.\n\\begin{equation}\n{\\L^{(\\text{res-mask})}} = \\sum_{b,tf} \\left[\\max\\left(1-\\sum_{i}\\hat{\\mat{M}}_{b,i}, 0\\right)\\right]_{tf}\n\\end{equation}\n\n${\\L^{(\\text{TRIPLET})}}$ is a triplet loss that is shown to help increase speaker discrimination capability,\nby ensuring the cosine similarity between each pair of embedding vectors for the same speaker \nis greater than for any pair of vectors of differing speakers.\nIt is formed by first choosing one anchor vector $\\vect{a}$, a positive vector $\\vect{p}$ belonging to the same speaker as $\\vect{a}$,\nand a negative vector $\\vect{n}$ belonging to a different speaker from $\\vect{a}$, \nfrom a set of speaker embedding vectors within one minibatch.\nBased on the cosine similarities between the anchor and negative vectors $s_i^{an}$,\nand the anchor and the positive vectors $s_i^{ap}$, \nthe triplet loss for $N$ triplets can be calculated as \\cite{Li2017}: \n\\begin{equation}\n{\\L^{(\\text{TRIPLET})}} = \\sum_{i=0}^{N} \\max(s_i^{an} - s_i^{ap} + \\delta, 0).\n\\end{equation}\nwhere $\\delta$ is a small positive constant.\nInterestingly, in this study, this loss was found to improve not only speaker discrimination capability\nbut also speaker tracking capability of online RSAN.\nWhen training with meeting-like data where people speaks intermittently,\none minibatch is usually formed with only a part of a meeting, in which very often certain speaker speaks only one time\nand remains silent to the end of this meeting excerpt (although he\/she may start speaking again in a later part of the meeting).\nIf we do not use the triplet loss, the network is not encouraged to keep remembering such a person to the end of the meeting,\nand eventually, it starts estimating a speaker embedding vector irrelevant to the speaker.\nTo circumvent such issue and make the network ready always for a situation when he\/she starts speaking again,\nwe can use the triplet loss and make the network always output speaker embedding vectors that are consistent over frames\nno matter whether the speaker is speaking or not.\n\n\n\n\\subsubsection{Noise mask}\nTo deal with background noise in the mixture, we let the network to estimate a mask for background noise \nalways at the first source extraction iteration in each block.\nTo force such behavior to the network, we used the following permutation variant loss (as oppose to permutation\ninvariant loss) for the mask for background noise:\n\\begin{equation}\n{\\L^{(\\text{MMSE})}} = \\frac{1}{B} |\\hat{\\mat{M}}_{1,b}\\odot \\mat{Y} - \\mat{A}_b^{(\\textrm{noise})}|^2,\n\\end{equation}\nwhere $\\mat{A}_b^{(\\textrm{noise})}$ is the magnitude spectrum of background noise.\nWith this scheme, we can easily find and discard signals corresponding to background noise from enhancement results.\n\n\\subsubsection{Teacher forcing for iterative source extraction}\nDuring training, when calculating residual mask $\\mathbf{R}_{b,i+1}$ at $b$-th block for the $(i+1)$~-th iteration,\ninstead of using the estimated source extraction mask at the $i$-th iteration,\nwe can calculate it by using an oracle source extraction mask $\\hat{\\mathbf{M}}_{b,i}^{\\textrm{(oracle)}}$ \nbased on the estimated permutation~\\cite{RSAN},\ni.e., $\\mathbf{R}_{b,i+1} = \\mathbf{R}_{b,i}-\\mathbf{M}_{b,i}^{\\textrm{(oracle)}}$.\nIn RNN literature, this form of training is known as teacher forcing.\nWhile the effectiveness of this scheme was not so clear in the previous study \\cite{RSAN},\nwe found it mandatory when we cope with noise and many speakers, \nboth of which are inevitable during training with meeting-like data.\n\n\\subsection{Decoding scheme: decoding with consistency check}\n\\label{sec:decode}\nReal meeting data contains a lot of unexpected sound events that are hardly observed in the training data,\nsuch as laughing sounds, a sudden change in tone of voice, coughing sound, \nand rustling sounds from e.g. papers, to name a few.\nAll of these sounds can be a cause for online RSAN to mistakenly detect a new spurious speaker and increase the number of source count.\nWhen it increases the source count by mistake because of such unexpected unseen sounds, \nit tends to wrongly split a speaker into two; the new embedding vector \nand an embedding vector already associated with the speaker.\nIt causes over-estimation errors in the source counting,\nand degrades embedding vector representation of the speaker, leading to degradation in overall performance.\n\nLet us denote the speaker embedding vector set\nat $b'$-th block as $\\{\\vect{z}_{b',i}\\}_{1 \\leq i \\leq I_{b'}}$\nwhere $I_{b'}$ corresponds to the total number of iteration in $b'$-th block.\nThen, to reduce such over-estimation error in source counting, \nwe propose to perform the following consistency check for the speaker embedding vector set, \n$\\{\\vect{z}_{b',i}\\}_{1 \\leq i \\leq I_{b'}}$, when online RSAN increases the speaker count.\nSpecifically, we propose to decode all the past blocks with the embedding vector set $\\{\\vect{z}_{b',i}\\}_{1 \\leq i \\leq I_{b'}}$.\nThen, if masks estimated with $\\vect{z}_{b',I_{b'}}$, $\\{\\hat{\\mat{M}}_{b,I_{b'}}\\}_{1 \\leq b \\leq b'}$, does not exceed $t_{\\text{res-mask}}$,\nit indicates that a speaker corresponding to $\\vect{z}_{b',I_{b'}}$ did not indeed appear \nin the past blocks and appeared for the first time at $b'$-th block. \nAnd thus, we accept the increase in the source count and keep using $\\{\\vect{z}_{b',i}\\}_{1 \\leq i \\leq I_{b'}}$ for further process.\nOtherwise, we do not accept the increase in the speaker count,\nand discard $\\{\\vect{z}_{b',i}\\}_{1 \\leq i \\leq I_{b'}}$ and replace the set of embedding vectors \nwith ones from the previous block $\\{\\vect{z}_{b'-1,i}\\}_{1 \\leq i \\leq I_{b'-1}}$ for further process.\n\n\\section{Experiments}\n\\label{sec:exp}\nIn this section, we evaluate online RSAN in comparison with state-of-the-art diarization methods,\nand shows its effectiveness. \n\n\\subsection{Experimental conditions}\n\\subsubsection{Data}\nWe generated three sets of noisy reverberant multi-speaker datasets for training;\n(dataset A) 20000 mixtures each of which is 10 seconds in length, and contains 1 or 2 speakers' speech signals, \n(dataset B) 10000 mixtures each of which is 60 seconds in length, and contains 1 to 6 speakers' speech signals,\nand (dataset C) 2372 mixtures each of which is 60 seconds in length, and contains 1 to 6 speakers' speech signals. \nTo all dataset, we added CHiME4 noise with SNR of 10 to 20~dB, and reverberation of $\\textrm{RT}_{\\textrm{60}}$ of 300 to 700~ms.\nUtterances for dataset A and B are taken from WSJ0 \\cite{WSJ0}, i.e., read speech, \nwhile those for dataset C are taken from headset recordings of real meetings recorded in our office, i.e., spontaneous speech. \nIn dataset A, each mixture was generated such that the first \\SI{5}{\\second} \ncontain one or two speakers with a probability of \\SI{50}{\\percent} each,\nwhile the second half contains zero, one or two speakers \nwith a probability of \\SI{15}{\\percent}, \\SI{55}{\\percent} and \\SI{30}{\\percent}, respectively.\nSimilarly, in dataset B, \nthe first \\SI{5}{s} of the test utterance contains zero or one speaker with a probability of \\SI{50}{\\percent} each,\nwhile the mixture in the remaining time is generated such that it contains zero, \none, two or three speakers with a probability of \\SI{5}{\\percent}, \\SI{75}{\\percent} and \\SI{15}{\\percent} and \\SI{5}{\\percent} respectively.\n\n\nEvaluation was done with two datasets; (1) simulated meeting-like data\ncomprising 1000 mixtures similar to dataset B but with unseen speakers,\nand (2) real meeting data recorded at our office with a distant microphone-array \\cite{ArakiHSCMA2017}.\nReal meeting dataset consists of 8 meetings, each of which is 15 to 20 minutes in length.\nThe number of meeting participants varies from 4 to 6, all of who are unseen during training.\nThe meeting recording contains a significant amount of babble noise (SNR of 3 to 15~dB),\nand reverberation of $\\textrm{RT}_{\\textrm{60}}$ of 500~ms.\nThe percentage of overlapped speech in these meetings is found to be $25.7$~\\% on average.\n\n\n\\subsubsection{Implementation details of online RSAN}\nNN architecture and hyper-parameters for online RSAN was same as \\cite{onlineRSAN_ICASSP2019}.\nIt consists of one fully connected layer on top of two BLSTM layers.\nMultichannel input feature was calculated based on signals observed at 2 microphones,\nand thus overall online RSAN model in this study is a 2-channel system.\nFor the evaluation based on the simulated meeting-like data, \nthe online RSAN model was first trained with the training dataset A for 300 epochs,\nand then further trained with dataset B for 50 epochs.\nThen, to cope with real meeting data, the model was further trained with dataset C for 5 epochs.\nThe block size of online RSAN was set at 10 seconds.\n$t_{\\text{res-mask}}$ was set at 0.2.\nTo obtain diarization results with online RSAN,\nwe performed power-based voice activity detection (VAD) on extracted streams \nbased on a threshold value common to one meeting.\n\n\n\\subsubsection{Methods to be compared with}\nIn the evaluation with the simulated meeting-like data, \nthe performance of online RSAN was compared with a system similar to a top-performing system \\cite{DIHARD_JHU} in DIHARD-1 challenge \\cite{DIHARD_data}.\nFor this, we used off-the-shelf implementation and model from \\cite{dihard19}.\nSince it is based on clustering of x-vectors \\cite{x-vector},\nit will be referred to as ``x-vector clustering\" hereafter.\nIt is a single-channel system.\n\nFor the real meeting data evaluation, the performance of online RSAN was compared with ``x-vector clustering\"\nand a multi-channel diarization method based on online clustering of Time-Difference-Of-Arrival (TDOA) feature \\cite{Hori_TASLP2011},\nwhich will be referred to as ``TDOA clustering\". \nThe TDOA feature was calculated based on signals observed at 8 microphones.\nDiarization performance was evaluated in terms of diarization error rate (DER) \\cite{der} including speaker overlapped segments,\nwhile the speech enhancement performance was evaluated in terms of signal-to-distortion ratio (SDR) in BSSeval \\cite{BSSeval}.\nThe sampling frequency was 8k~Hz for all the methods.\n\n\\subsection{Experiment 1: Evaluation with simulated meeting-like data}\nBefore proceeding to evaluation with real meeting data,\nwe briefly examine the performance of online RSAN and whether it is ever possible to\ncope with noisy reverberant mixtures containing many speakers.\nTable \\ref{tbl:results_sim1} shows DERs of online RSAN and x-vector clustering, averaged over 1000 mixtures.\nIt was found that online RSAN works for noisy reverberant data, and outperformed the state-of-the-art x-vector clustering.\nSDR improvement obtained with online RSAN was 10.01~dB, which we believe is reasonably high.\n\n\\begin{table}[t]\n \\centering\n \\caption{DERs for simulated meeting-like data (\\%)}\n \\label{tbl:results_sim1}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\begin{tabular}{c} x-vector clustering \\end{tabular} & \\begin{tabular}{c} Online RSAN \\end{tabular} \\\\\n\\hline\n44.39 & \\textbf{33.7} \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{figure*}[t]\n \\begin{center}\n \\includegraphics[width=130mm]{.\/fig\/specgram_ver2.jpg}\n \\end{center}\n \\caption{Spectrograms of 3 minute excerpt from real meeting data: observed signal (a), headset recordings of each speaker (b,c,d), and signals estimated by online RSAN (b',c',d')}\n \\label{fig:spectrogram}\n\\end{figure*}\n\n\\subsection{Experiment 2: Real meeting data}\nNow we evaluate the performance of online RSAN with real data.\nWe first evaluate effect of consistency-checking decoding (proposed in section \\ref{sec:decode}).\nAlso, since it may not be clear how much and what kind of training data would be necessary \nfor online RSAN to cope with real data,\nwe examined the effect of training with dataset C, i.e., spontaneous speech.\nTable \\ref{tbl:results_real1} suggests that both training with spontaneous speech \nand decoding with the consistency check is important. \nNote that, although one may think that DERs here are very high, \nit is in a similar range as DIHARD2 as you can see in \\cite{dihard19}.\n\n\\begin{table}[t]\n\\centering\n \\caption{Effect of model training with spontaneous speech, and decoding with consistency check when dealing with real meeting data}\n \\label{tbl:results_real1}\n \\begin{tabular}{l c}\n \\hline\n system & DER (\\%) \\\\ \\hline\n Online RSAN trained only with dataset A \\& B & 73.9 \\\\\n \\ \\ + dataset C & 56.3 \\\\\n \\ \\ \\ \\ +decoding with consistency checking & \\textbf{49.6} \\\\ \\hline\n \\end{tabular}\n\\end{table}\n\nNow, let us compare, with the other methods, the performance of online RSAN model trained \non dataset A, B, and C\nand decoded with the consistency check.\nWhile we use a different threshold for power-based VAD for each meeting for table \\ref{tbl:results_real1}\nto take a closer look at the difference between each setting,\nwe reran the experiments with a common threshold for all meetings \nto make a fair comparison with the other methods.\nThe threshold was determined based on the validation dataset.\nTable \\ref{tbl:results_each} shows DERs of online RSAN, \nx-vector clustering and TDOA clustering.\nWhile there are some sessions where deep learning-based algorithms, i.e. x-vector clustering and online-RSAN did not work well,\non average, even in the real meeting scenario, online RSAN can outperform these conventional approaches.\n\nFigure \\ref{fig:spectrogram} shows spectrograms of unprocessed real meeting data \n(3-minute excerpt from one of the meetings used for the evaluation), \ncorresponding headset recording of each speaker, and signals estimated by online RSAN. \nNoise signals estimated by online RSAN are omitted from the figure.\nAudio examples corresponding to the spectrograms are available on our web-page \\cite{demo_page}.\nAs it can be seen, the headset recordings and the estimated signals look quite similar,\nwhich suggests that online RSAN extracted each speaker's voice clearly, and counted the number of speakers correctly. \nNote that, as the third speaker did not speak for the first 2 minutes, online RSAN did not output masks for that speaker in that period, \nand once that speaker starts speaking, it correctly increased the number of source extraction iteration to 4 (i.e., 3 speakers + noise) \nand started tracking that speaker.\n\n\n\\begin{table*}[]\n\\centering\n\\label{tbl:results_each}\n\\caption{DERs for noisy reverberant real meetings for each system}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n\\begin{tabular}{c} Meeting \\\\ ID \\end{tabular} & \\# of spk. & \\begin{tabular}{c} x-vector \\\\ clustering \\end{tabular} & \\begin{tabular}{c} TDOA \\\\ clustering \\end{tabular} & \\begin{tabular}{c} Online \\\\ RSAN \\end{tabular} \\\\ \\hline\n1 & 6 & 51.2 & 46.8 & \\textbf{41.4} \\\\ \\hline\n2 & 6 & 61.8 & 64.6 & \\textbf{58.4} \\\\ \\hline\n3 & 6 & 73.5 & 62.6 & \\textbf{49.0} \\\\ \\hline\n4 & 5 & 57.6 & \\textbf{23.8} & 55.7 \\\\ \\hline\n5 & 5 & 64.2 & \\textbf{47.5} & 72.5 \\\\ \\hline\n6 & 6 & 71.4 & 67.2 & \\textbf{40.3} \\\\ \\hline\n7 & 4 & 68.1 & 73.6 & \\textbf{45.5} \\\\ \\hline\n8 & 4 & 72.4 & 70.9 & \\textbf{48.8} \\\\ \\hline \\hline\n & Average & 65.0 & 57.1 & \\textbf{51.4} \\\\ \\hline\n\\end{tabular}\n\\end{table*}\n\n\\subsection{Discussion}\nAlthough online RSAN could cope with real meeting data to some extent, \nits performance was far from ``perfect\".\nWe found that most errors in the online RSAN process are due to its insufficient performance in source separation and tracking.\nIt sometimes confidently extracts or tracks two different speakers' signals \nwith one speaker embedding vector $\\{\\vect{z}_{b,i}\\}_{1 \\leq b \\leq B}$,\nprobably because their voice characteristics are similar from the system's point of view.\nThis type of error should be reduced by, for example, \nemploying more advanced NN architecture \\cite{Takahashi_Interspeech2019}, \nand increasing the number of speakers in training data like we propose in \\cite{Delcroix_ICASSP2020}.\n\n\n\n\n\\section{Conclusion}\nThis paper proposed several practical techniques required for all-neural diarization, source separation and \nsource counting model called online RSAN to cope with real meeting data.\nIt was shown that incorporation of the proposed consistency-checking decoding \nand training with spontaneous speech is effective.\nBased on the experiments with real meeting recordings, \nonline RSAN was shown to perform effective speech enhancement, and simultaneously outperform state-of-the-art diarization systems.\nOur future work includes incorporation of advanced source separation NNs into online RSAN,\nand evaluation in terms of ASR accuracy.\n\n\n\n\n\n\n\n\\bibliographystyle{.\/bibliography\/IEEEbib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{First-principles calculations}\\label{sec:DFT_details}\n\nAll first-principles calculations were performed using the projector augmented wave (PAW) method~\\cite{PhysRevB.50.17953}\nas implemented in the Vienna \\emph{Ab initio} Simulation Package (VASP)~\\cite{PhysRevB.47.558, PhysRevB.54.11169}.\nFor density functional theory (DFT) calculations, the standard PAW potentials ({\\tt Zr\\_sv} and {\\tt O}) and a plane wave cutoff of 600 eV were employed.\nThe electronic interactions were described using the Perdew-Burke-Ernzerhof functional (PBE)~\\cite{PhysRevLett.77.3865}\nand the strongly constrained appropriately normed functional (SCAN)~\\cite{PhysRevLett.115.036402}.\nA $\\Gamma$-centered $k$-point grid with a spacing of 0.31 $\\AA^{-1}$ between $k$ points\n(corresponding to a 2$\\times$2$\\times$2 $k$-point grid for a 96-atom cell) was employed.\nThe Gaussian smearing method with a smearing width of 0.05 eV was used.\nWhenever ground state structures were required, the electronic optimization was performed until the total energy difference between two iterations was less than 10$^{-6}$ eV.\nThe structures were optimized until the forces were smaller than 5 meV\/\\AA.\nThe phonon dispersions were calculated by finite displacements using the Phonopy code~\\cite{PhysRevB.78.134106}.\nFor all phonon calculations, a 96-atom supercell and a $2\\times2\\times2$ $k$-point grid were used.\nTo account for the long-range dipole-dipole force constants, the nonanalytic contribution\nto the dynamical matrix was treated using the method of Ref.~\\cite{PhysRevB.55.10355}.\nThe static dielectric tensor and atomic Born effective charges were calculated using the PBEsol functional~\\cite{PhysRevLett.100.136406}.\n\n\nFor the random phase approximation (RPA) calculations, the GW PAW potentials ({\\tt Zr\\_sv\\_GW} and {\\tt O\\_GW})\nwere used. These GW PAW potentials are constructed by using additional projectors above the vacuum level,\nand therefore, they describe well the high-energy scattering properties of the atoms and are more accurate for the polarizability-dependent RPA calculations.\nAs shown in Ref.~\\cite{PhysRevResearch.2.043361}, the RPA is capable to simultaneously describe well both the structural and electronic properties of ZrO$_2$.\nDue to the large computational cost involved in RPA calculations, a reduced plane wave cutoff of 520 eV and\na `bcc'-like generalized regular grid with 8 $k$ points in the full Brillouin zone\nwere used. The RPA energies and forces are calculated using an efficient low-scaling algorithm~\\cite{PhysRevB.90.054115,PhysRevB.94.165109,PhysRevLett.118.106403}.\nThe energy cutoff for the response function was chosen to be the same as the plane wave cutoff (520 eV)\nand the number of imaginary time\/frequency points ($N_{\\omega}$) was set to 10,\nwhich is sufficient to ensure convergence of RPA energies (see Fig.~\\ref{RPA_energy_wrt_NOMEGA16}) and forces (see Fig.~\\ref{RPA_force_wrt_NOMEGA16}).\nThe stress tensor $\\sigma_{\\alpha\\beta}$ at the RPA level was calculated via finite differences of the RPA total energies using twelve slightly distorted structures through\n\\begin{equation}\\label{eq:stress_tensor}\n\\sigma_{\\alpha\\beta}\\approx\n -\\frac{1}{\\Omega} \\frac{E(e_{\\alpha\\beta}=+\\delta)-E(e_{\\alpha\\beta}=-\\delta)}{2\\delta},\n\\end{equation}\nwhere $\\alpha$ and $\\beta$ represent the Cartesian coordinate indices, $\\Omega$ is the system volume, and $e_{\\alpha\\beta}$ is the strain tensor.\nHere, $\\delta=0.02$ was adopted. This value on the one hand ensures the convergence of the stress tensors for PBE calculations\nto within 2.5 kbar (see Fig.~\\ref{stress_vs_displacement}), and\non the other hand is sufficiently large to minimize the shell effects (plane wave $\\lG$-vectors moving in and out of the cutoff sphere) for RPA calculations. We note that the stress tensors calculated by the finite difference method is less prone to the Pulay stress than those calculated internally within VASP. The latter often suffers from basis set incompleteness errors because of the fixed plane wave basis set when the cell is distorted. This is the reason why we chose to use a relatively large plane wave cutoff of 600 eV for the DFT calculations, whose stress tensors are directly calculated by using the VASP internal routines. Even then the diagonal components of the stress tensor are corrected by the calculated Pulay stress before generating the machine-learned force fields (MLFFs). The phonon frequencies at $\\Gamma$ predicted by RPA were calculated by finite differences, with\nthe long-range dipole-dipole interactions calculated at the level of PBEsol.\n\n\n\n\n\\section{MLFF training}\\label{sec:training_details}\n\nOur MLFFs were initially trained on-the-fly during FP molecular dynamics (MD) simulations\nbased on the Bayesian linear regression~\\cite{PhysRevB.100.014105,book_Bishop}.\nFor a comprehensive description of the on-the-fly MLFF generation implemented in VASP, we refer to Refs.~\\cite{PhysRevLett.122.225701,PhysRevB.100.014105,doi:10.1063\/5.0009491}.\nA concise summary of this method can be found in Refs.~\\cite{doi:10.1021\/acs.jpclett.0c01061,Peitao_2020}.\nFor the on-the-fly training, the PBEsol functional~\\cite{PhysRevLett.100.136406} was used,\nsince it predicts accurate lattice parameters for all the three phases of ZrO$_2$~\\cite{Carla_2021} on par with SCAN, but it is cheaper.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.47\\textwidth, trim = {0 0.05cm 0.2cm 0.2cm}, clip]{FigS1_RPA_energy_wrt_NOMEGA16.eps}\n\\end{center}\n\\caption{Calculated RPA energies (in meV\/atom) of a monoclinic structure of ZrO$_2$ with 24 atoms in the cell\nas a function of the employed number of imaginary\/frequency points $N_{\\omega}$.\nNote that the RPA energy obtained using $N_{\\omega}$=16 is taken as reference.\nThe RPA energy obtained using $N_{\\omega}=10$ converges to within 0.012 meV\/atom.\n }\n\\label{RPA_energy_wrt_NOMEGA16}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.47\\textwidth, trim = {0 0.05cm 0.2cm 0.2cm}, clip]{FigS2_RPA_force_wrt_NOMEGA16.eps}\n\\end{center}\n\\caption{Calculated RPA forces (in meV\/\\AA) of each atom in a monoclinic structure of ZrO$_2$ with 24 atoms in the cell\nreferenced to the ones obtained using $N_{\\omega}=16$.\nThe results achieved by three $N_{\\omega}$ (12, 10, and 8) are shown. One can observe that the forces obtained using $N_{\\omega}=10$\nare converged to within 1 meV\/\\AA.\n }\n\\label{RPA_force_wrt_NOMEGA16}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.47\\textwidth, trim = {0 0.05cm 0.2cm 0.2cm}, clip]{FigS3_stress_vs_displacement.eps}\n\\end{center}\n\\caption{Stress tensors (in kbar) calculated by finite differences at a plane wave cutoff of 520 eV\n referenced to the ones calculated internally by VASP using a plane wave cutoff of 1600 eV\n[($\\sigma_{xx}$, $\\sigma_{yy}$, $\\sigma_{zz}$, $\\sigma_{xy}$, $\\sigma_{yz}$, $\\sigma_{zx}$)=(79.15, 132.58, 58.07, $-$5.52, $ -$22.18, 49.09) kbar]\nas a function of strain displacement. The very large plane wave cutoff of 1600 eV was chosen to minimize the Pulay stress.\nA finite temperature MD snapshot of m-ZrO$_2$ with 24 atoms in the cell was employed for this test.\nA similar behavior was observed for the tetragonal as well as cubic phase of ZrO$_2$ (not shown). }\n\\label{stress_vs_displacement}\n\\end{figure}\n\n\nThe FPMD simulations were performed in the isothermal-isobaric (NPT) ensemble at ambient pressure using a Langevin thermostat~\\cite{book_Allen_Tildesley}\ncombined with the Parrinello-Raman method~\\cite{PhysRevLett.45.1196}. The time step was set to 2.5 fs.\nFor the MLFF generation, the separable descriptors~\\cite{doi:10.1063\/5.0009491} were used.\nThe cutoff radius for the three-body descriptors and the width of the Gaussian functions used for broadening the atomic\ndistributions of the three-body descriptors were set to 6 $\\AA$ and 0.4 $\\AA$, respectively. The number of radial basis\nfunctions and maximum three-body momentum quantum number of the spherical harmonics used to expand the atomic distribution\nfor the three-body descriptors were set to 15 and 4, respectively. The parameters for the two-body descriptors were\nthe same as those for the three-body descriptor.\n\nAs mentioned in the main text, two MLFFs were constructed on the fly.\nThe first MLFF was trained on a 96-atom cell, whereas the second one was trained on a smaller 24-atom cell.\nWe note that the second MLFF was not used and this training step was just used to collect a new reference dataset,\nfrom which a subset of structures were extracted using a singular value decomposition (SVD) rank compression of the kernel matrix using a CUR algorithm~\\cite{Mahoney697,PhysRevB.100.014105}\nand then recalculated by high-level QM calculations.\nFor the first MLFF, the detailed training procedures can be found in Ref.~\\cite{Carla_2021}, and in the end, 592 structures were collected in the training dataset.\nFor the second MLFF trained on smaller unit cells, the following training strategy was employed.\n($i$) We first trained the force field by heating the monoclinic ZrO$_2$ from 0 K to 1800 K using 20 000 MD steps starting from the DFT relaxed structure.\n($ii$) Then, we continued training the tetragonal phase by a heating run from 1700 K to 2600 K using 10 000 MD steps.\n($iii$) The force field was further trained by heating the cubic phase from 2500 K to 2800 K using 3333 MD steps.\nNote that in steps ($ii)$ and ($iii$), the initial structures were obtained by equilibrating the tetragonal and cubic structures\nat 1700 K and 2500 K, respectively, using the on-the-fly MLFF scheme, but the thus generated MLFFs were discarded.\n($iv$) To include the ideal tetragonal and cubic structures at 0 K, additional short heating runs from 0 K to 10 K using 100 MD steps\nwere performed starting from the DFT relaxed tetragonal and cubic structures, respectively.\nEventually, only 1275 FP calculations were performed out of 33 533 MD steps, i.e., nearly 96.2\\% of the FP calculations were bypassed.\n\n\n\\begin{table*}\n \\caption{ Zero-temperature structural parameters of all three phases of ZrO$_2$ and energy differences between\n phases predicted by different exchange-correlation functionals and MLFFs. $d_z$ represents the displacement of the oxygen (in unit of $c$) atoms along the $z$ direction with respect\n to ideal cubic position, and $\\beta$ is the angle between the lattice vectors $\\bf a$ and $\\bf c$ in the monoclinic phase.\n The values in parentheses are RPA predicted energy differences for the structures that are optimized by MLFF-RPA$^{\\Delta}$.\nThe experimental structural parameters and volumes are extrapolated to 0 K~\\cite{PhysRevB.49.11560,https:\/\/doi.org\/10.1111\/j.1151-2916.1985.tb15247.x}.}\n \\begin{ruledtabular}\n \\begin{tabular}{lccccccccc}\n & PBE & MLFF-PBE & SCAN & MLFF-SCAN & MLFF-SCAN$^{\\Delta}$ & MLFF-RPA$^{\\Delta}$& Expt. \\\\\nMonoclinic & & & & & & & \\\\\n$a$ (\\AA) & 5.194 & 5.189 & 5.150 & 5.152 & 5.155 & 5.146 & 5.151 \\\\\n$b$ (\\AA) & 5.248 & 5.251 & 5.225 & 5.223 & 5.220 & 5.219 & 5.212 \\\\\n$c$ (\\AA) & 5.381 & 5.378 & 5.326 & 5.329 & 5.335 & 5.313 & 5.317 \\\\\n$\\beta$ (deg.) & 99.66 & 99.68 & 99.35 & 99.40 & 99.40 & 99.39 & 99.23 \\\\\nVolume ($\\AA^3$\/f.u.) & 36.15 & 36.11 & 35.35 & 35.37 & 35.41 & 35.20 & 35.22 \\\\\n\\\\\nTetragonal & & & & & & & \\\\\n$a$ (\\AA) & 3.624 & 3.625 & 3.600 & 3.600 & 3.601 & 3.592 & 3.571 \\\\\n$c$ (\\AA) & 5.284 & 5.290 & 5.220 & 5.230 & 5.231 & 5.189 & 5.182 \\\\\n$c\/a$ & 1.458 & 1.459 & 1.450 & 1.453 & 1.453 & 1.445 & 1.451 \\\\\n$d_z$ & 0.057 & 0.058 & 0.052 & 0.054 & 0.054 & 0.048 & 0.047 \\\\\nVolume ($\\AA^3$\/f.u.) & 34.70 & 34.76 & 33.82 & 33.90 & 33.91 & 33.47 & 33.01 \\\\\n$\\Delta E_{t-m}$ (eV\/f.u.) & 0.110 & 0.109 & 0.074 & 0.074 & 0.075 & 0.067 (0.069) & --- \\\\\n\\\\\nCubic & & & & & & & \\\\\n$a$(\\AA) & 5.120 & 5.126 & 5.088 & 5.090 & 5.091 & 5.076 & --- \\\\\nVolume ($\\AA^3$\/f.u.) & 33.56 & 33.68 & 32.92 & 32.97 & 32.99 & 32.70 & --- \\\\\n$\\Delta E_{c-t}$ (eV\/f.u.) & 0.102 & 0.100 & 0.085 & 0.083 & 0.083 & 0.053 (0.047) & --- \\\\\n \\end{tabular}\n \\end{ruledtabular}\n \\label{tab:lattice_constants}\n\\end{table*}\n\n\nAt the end of the on-the-fly training, we determined the final regression\ncoefficients\nby using a factor of 10 for the relative weight of the energy equations with respect to the equations for the forces and stress tensors,\nand a SVD to solve the least-squares problem.\nThese were found to improve the overall accuracy of MLFFs~\\cite{Peitao_2020,Carla_2021}.\n\n\nFinally, all the structures contained in $T^{(1)}$ and $T^{(3)}$ were recalculated by PBE and SCAN,\nwhereas RPA calculations were performed only for the structures in $T^{(3)}$ (including 168 structures of 24 atoms, see the main text).\nMLFF-$\\Delta$ was obtained by machine learning the differences in energies, forces and stress tensors between high-level and low-level QM calculations\nusing the separable descriptors~\\cite{doi:10.1063\/5.0009491} with low spatial resolution (0.8 $\\AA$) and a small number of radial basis functions (8) for\nboth the radial and angular parts.\n\nThe MLFFs directly trained using PBE and SCAN are referred to as MLFF-PBE and MLFF-SCAN, respectively.\nThe MLFFs indirectly trained using SCAN and RPA via the $\\Delta$-machine learning ($\\Delta$-ML) approach are denoted as MLFF-SCAN$^\\Delta$\nand MLFF-RPA$^\\Delta$, respectively.\nWe note that although it is possible to generate the MLFF-RPA$^\\Delta$ based on the MLFF-PBE (constructed using a plane wave cutoff of 600 eV and standard PAW potentials), the RPA calculations in particular for the forces are rather demanding using such a large cutoff energy for structures of 24 atoms.\nTherefore, it is expedient to generate a second MLFF-PBE using a reduced plane wave cutoff of 520 eV and the GW PAW potentials as in the RPA calculations,\non which the MLFF-RPA$^\\Delta$ is built. We note that the changes in the cutoff energy (from 600 eV to 520 eV) and the potentials (from PAW to GW PAW) for ZrO$_2$\nhave negligible effects on the accuracy of the resulting MLFF-PBE if the stress tensors calculated by the VASP internal routines were corrected for the Pulay stress.\nFor generating MLFF-$\\Delta$ where $\\Delta=$RPA$-$PBE, both PBE and RPA calculations were performed on the structures in $T^{(3)}$\nusing the plane wave cutoff of 520 eV and the GW PAW potentials.\n\n\n\n\n\\section{MLFF validation}\\label{sec:validation_details}\n\n\n\n\nThe MLFFs including MLFF-PBE, MLFF-SCAN, and MLFF-SCAN$^\\Delta$\nhave been validated on a test dataset containing 120 structures of 96 atoms\n(40 monoclinic structures at $T$=1000 K, 40 tetragonal structures at $T$=2000 K, and 40 cubic structures at $T$=3000 K).\nHowever, for MLFF-RPA$^\\Delta$,\ndue to the high computational cost for the RPA calculations, a reduced test dataset containing 60 structures of 24 atoms (20 monoclinic structures\nat $T$=1000 K, 20 tetragonal structures at $T$=2000 K, and 20 cubic structures at $T$=3000 K) was used.\nAll the structures in the test datasets were generated from MD simulations using the NPT ensemble and MLFF-SCAN.\n\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth, clip]{FigS4_phonon_MLSCAN_SCAN.eps}\n\\end{center}\n\\caption{Phonon dispersion relations of (a) monoclinic, (b) tetragonal, and (c) cubic ZrO$_2$ at 0 K predicted by SCAN (grey lines), MLFF-SCAN (blue lines),\nand MLFF-SCAN$^{\\Delta}$ (red lines). Almost no difference is observed between MLFF-SCAN$^{\\Delta}$ and MLFF-SCAN for all three phases,\nindicating their comparable accuracies.\n}\n\\label{fig:phonon_MLSCAN_SCAN}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth, clip]{FigS5_phonon_MLRPA_SCAN.eps}\n\\end{center}\n\\caption{Phonon dispersion relations of (a) monoclinic, (b) tetragonal, and (c) cubic ZrO$_2$ at 0 K predicted by MLFF-RPA$^\\Delta$ (red lines).\nThe results from SCAN (grey lines) are also given for comparison, showing very similar phonon dispersions with MLFF-RPA$^\\Delta$ for all three phases.\n}\n\\label{fig:phonon_MLRPA_SCAN}\n\\end{figure*}\n\n\nAs shown in Table I of the main text, all generated MLFFs are very accurate with small validation errors.\nThe errors for MLFF-SCAN are slightly larger than those for MLFF-PBE.\nThis is likely due to the poor numerical performance of the SCAN functional~\\cite{doi:10.1063\/1.5094646,doi:10.1021\/acs.jpclett.0c02405}.\nThe MLFF-SCAN$^\\Delta$ derived by the $\\Delta$-ML approach\nexhibits almost comparable accuracy as MLFF-SCAN that was directly trained by SCAN.\nThe good accuracies of the obtained MLFF-PBE and MLFF-SCAN\n have also been showcased in their good predictions of structural parameters,\nenergy differences between different phases (Table~\\ref{tab:lattice_constants}), and phonon dispersion relations\n(Fig.~\\ref{fig:phonon_MLSCAN_SCAN}) as compared to their respective DFT counterparts.\nAs expected, PBE overestimates the lattice constants and the energy differences between the phases.\nSCAN improves upon PBE because of the improved treatment of intermediate Van der Waals interactions.\nOur DFT results are overall in good agreement with Ref.~\\cite{PhysRevResearch.2.043361}.\nFrom Table~\\ref{tab:lattice_constants} one can also observe that MLFF-SCAN$^\\Delta$\nperforms almost equally well as MLFF-SCAN in the prediction of structural parameters and energy differences between the phases,\nconsistent with the error analysis shown in Table I of the main text..\nThis is also true in predicting phonon dispersion relations, as demonstrated in Fig.~\\ref{fig:phonon_MLSCAN_SCAN}.\nAlmost no difference is observed between MLFF-SCAN$^{\\Delta}$ and MLFF-SCAN for all three phases,\nvalidating the feasibility of the $\\Delta$-ML approach.\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth, clip]{FigS6_Str_plot_ZrO2.eps}\n\\end{center}\n\\caption{Crystal structures of (a) cubic, (b) tetragonal, and (c) monoclinic ZrO$_2$. Large and small spheres denote Zr and O atoms, respectively.\nThe red and purple colors in the tetragonal structure (b) are used to distinguish O atoms that are displaced\nupwards (red arrows) and downwards (purple arrows) as compared to the cubic structure, and in the monoclinic structure (c) they indicate the two nonequivalent O atoms.\n Structural models were generated using VESTA~\\cite{Momma:db5098}.\n }\n\\label{fig:Str_plot_ZrO2}\n\\end{figure*}\n\nThe MLFF-RPA$^\\Delta$ is validated on a reduced test dataset of small unit cells, i.e., 60 structures of 24 atoms,\nbecause of the large computational cost of the RPA calculations. The validation errors in energies, forces and stress\ntensors calculated by MLFF-RPA$^\\Delta$ are given in Table II of the main text. For comparison, the validation errors\nfor MLFF-PBE and MLFF-SCAN are also shown.\nFirst, one can observe that MLFF-RPA$^\\Delta$ shows comparable validation errors as MLFF-PBE and MLFF-SCAN,\nimplying their comparably good accuracy. The slightly larger validation errors calculated by MLFF-RPA$^\\Delta$\narise from the relatively noisy nature of RPA. Second, one may notice that the validation errors of energies and\nstress tensors on small unit cells of 24 atoms for all the MLFFs (not just limited to MLFF-RPA ) are larger than those\non larger unit cells of 96 atoms, whereas the RMSEs for the forces remain almost unchanged (compare Table II and Table I in the main text).\nThis difference in fact originates from the definition of RMSE and can be understood from the error propagation with respect to the system size.\nSpecifically, according to basic statistics, the variance of the difference between the DFT and MLFF\npredicted energy Var$(E^{\\rm DFT}-E^{\\rm MLFF})$ is expected to be proportional to the system size $N$,\ni.e., Var$(E^{\\rm DFT}-E^{\\rm MLFF})\\sim N$, if the errors in the predicted local energies are statistically independent.\nThe RMSE of the energy per atom (in meV\/atom) are calculated as\n\\begin{equation}\\label{eq:RMSE_E}\n{\\rm RMSE}(E)= \\sqrt{\\frac{1}{M} \\sum_i^M \\Big[(E^{\\rm DFT}_i - E^{\\rm MLFF}_i)\/N_i\\Big]^2},\n\\end{equation}\nwhere $M$ is the number of structures and $N_i$ is the number of atoms in the structure $i$.\nFrom this definition, one can readily show that\n\\begin{equation}\\label{eq:RMSE_E2}\n{\\rm RMSE}(E)= \\sqrt{\\frac{{\\rm Var}(E^{\\rm DFT}-E^{\\rm MLFF})}{N^2}} \\sim \\frac{1}{\\sqrt{N}},\n\\end{equation}\nThis means that if the system becomes $N$ times larger, then the RMSE of the energy per atom becomes $1\/\\sqrt{N}$ times smaller.\nThis explains why all the MLFFs predict a larger RMSE for the energy per atom for the 24-atom cells than for the 96-atom cells.\nThe same error analysis holds for the stress tensor, since by definition it is calculated as the derivative\nof the energy with respect to the strain and then divided by the system volume.\nNevertheless, this error propagation rule does not apply for the RMSE for the forces,\nsince the force is an intensive property that is independent of system size.\n\n\n\n\\section{Phase transitions of zirconia within the quasi-harmonic approximation}\\label{sec:phase_transition_details}\n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth, clip]{FigS7_QHA_free_energy.eps}\n\\end{center}\n\\caption{The free energy difference (per formula unit) between the tetragonal and\nmonoclinic phase as a function of temperature predicted by the MLFFs within the quasi-harmonic approximation (QHA)\nusing the quantum Bose-Einstein (BE) statistics. We note that the results obtained from the BE statistics\nand classical Maxwell-Boltzmann statistics are almost identical for temperatures above 200 K (not shown),\nindicating that the quantum effect can be neglected at high temperatures.\n }\n\\label{fig:QHA}\n\\end{figure}\n\n\nThe primitive cells of the cubic, tetragonal, and monoclinic structures of\nzirconia contain one, two, and four ZrO$_2$ formula units respectively.\nA unit cell of 12 atoms can thus be used to accommodate all three phases (see Fig.~\\ref{fig:Str_plot_ZrO2}).\n\n\nAs shown in the phonon dispersions (Fig.~\\ref{fig:phonon_MLSCAN_SCAN}),\nboth monoclinic and tetragonal phases are dynamically stable at 0 K. In addition, no soft phonons are observed for both phases\nwhen the volume is changed within 5\\%. This allows us to estimate the $T_c$ using the quasi-harmonic approximation (QHA).\nFig.~\\ref{fig:QHA} shows the free energy difference between the tetragonal and monoclinic phase\nas a function of temperature predicted by the MLFFs within the QHA using the quantum Bose-Einstein statistics.\nAccordingly, MLFF-PBE predicts a value of 1511 K for $T_c$.\nMLFF-SCAN$^\\Delta$ and MLFF-SCAN yield close values of 1148 K and 1164 K, respectively,\nwhich are very close to the one obtained by the MLFF-PBEsol (1178 K) in Ref.~\\cite{Carla_2021}.\nIn addition, for nearly the entire temperature range, MLFF-SCAN$^\\Delta$ predicts very close free energies as compared to MLFF-SCAN.\nThis is expected, since both MLFF-SCAN$^\\Delta$ and MLFF-SCAN overall show similar validation errors\nas well as lattice parameters, energy differences between the phases, and phonon dispersions.\nAs compared to MLFF-SCAN, MLFF-RPA$^\\Delta$ predicts a slightly smaller value of $T_c$ (about 1117 K).\nIn general, we find that within the QHA the predicted $T_c$ is correlated to the calculated energy differences between the two phases at 0 K (see Table~\\ref{tab:lattice_constants}).\n\n\n\n\n\\section{SVD rank compression}\\label{sec:SVD_CUR}\n\nIn order to select few most representative structures for machine-learning the differences from a large pool of dataset,\nwe employed SVD rank compression of the kernel matrix based on the leverage-score CUR algorithm~\\cite{Mahoney697,PhysRevB.100.014105}.\nIn the following, the CUR algorithm is briefly introduced.\n\nWe denote $\\mathbf{K}$ as a kernel matrix calculated in the feature space. It is a squared matrix whose elements $K_{ij}$ measure the similarity between two local reference configurations $i$ and $j$. The formulation of the CUR algorithm starts from the diagonalization of the matrix $\\mathbf{K}$:\n\\begin{align}\n\\mathbf{U}^{\\textit{T}}\\mathbf{K}\\mathbf{U}&=\\mathbf{L}=\\mathrm{diag}\\left(l_{\\mathrm{1}},...,l_{N}\\right), \\label{equation_diag}\n\\end{align}\nwhere $N$ is the dimension of the matrix $\\mathbf{K}$ and $\\mathbf{U}$ is the eigenvector matrix defined as\n\\begin{align}\n\\mathbf{U}&=\\left(\\mathbf{u}_{\\mathrm{1}},...,\\mathbf{u}_{N}\\right), \\label{equation_U}\\\\\n\\mathbf{u}_{j}&=\\left(u_{1j},...,u_{Nj}\\right)^{\\textit{T}}, \\label{equation_u}\n\\end{align}\nEq.~(\\ref{equation_diag}) can be rewritten as\n\\begin{align}\n\\mathbf{k}_{j}&=\\sum\\limits_{\\xi = 1}^{N} \\left(u_{j \\xi} l_{\\xi}\\right) \\mathbf{u}_{\\xi}, \\label{equation_kj}\n\\end{align}\nwhere $\\mathbf{k}_{j}$ denotes the $j$th column vector of the matrix $\\mathbf{K}$.\nIn our implementation, we have adopted a modified version of the original CUR algorithm~\\cite{Mahoney697}\nsuch that the columns of $\\mathbf{K}$ that are strongly correlated with the $N_{low}$ eigenvectors $\\mathbf{u}_{\\xi}$ with the smallest eigenvalues $l_\\xi$ are disregarded.\nThis is achieved by defining the leverage scoring for each column of $\\mathbf{K}$\n\\begin{align}\n\\omega_{j}&=\\frac{1}{N_{low}} \\sum\\limits_{\\xi=1}^{N} \\gamma_{\\xi j}, \\label{equation_ls1}\\\\\n\\gamma_{\\xi j}&= \\begin{cases} u_{j\\xi}^{2}, & \\mbox{if } l_{\\xi}\/l_{\\rm max}< \\epsilon \\\\ 0, & \\mbox{otherwise} \\end{cases} \\label{equation_ls2}\n\\end{align}\nwhere $l_{\\rm max}$ is the maximum eigenvalue of $\\mathbf{K}$ and $\\epsilon$ is a parameter used to define the degree of rank compression.\nBy defining Eq.~\\eqref{equation_ls1}, the $N_{low}$ columns of $\\mathbf{K}$ and the local reference configurations corresponding to those columns\nwith largest leverage scorings are disregarded. The remaining ones are deemed to be the most important ones.\nOur final representative structures are then those structures that contribute to these most important local reference configurations.\n\nUsing the SVD rank compression introduced above, we selected a subset of structures ($T^{(3)}$) from the $T^{(2)}$ dataset (including 1275 structures of 24 atoms).\nWe employed pair descriptors with low spatial resolution (0.8 $\\AA$) and a small number of radial basis functions (8) to construct the kernel.\nThe actual number of SVD compressed structures depends on the parameter $\\epsilon$ in Eq.~\\eqref{equation_ls2}.\nFor instance, the resulting subset $T^{(3)}$ contains 168 structures, when $\\epsilon=10^{-10}$.\nThis number will be reduced when $\\epsilon$ is increased. As shown in Table~\\ref{tab:error_MLFF-DFT_SM}, in addition to\nthe MLFF-SCAN$^\\Delta$ that is discussed in the main text, we generated another two new MLFFs (denoted as MLFF-SCAN$^\\Delta$-2 and MLFF-SCAN$^\\Delta$-3)\nvia the $\\Delta$-ML approach. These two MLFFs used just 102 and 72 structures for generating the MLFF-$\\Delta$.\nOne can observe that even 72 structures are sufficient to machine-learn the differences and the resulting MLFF-SCAN$^\\Delta$-3\nis still accurate, showing only slightly larger validation errors in energies as compared to MLFF-SCAN$^\\Delta$.\nMoreover, we find that MLFF-SCAN$^\\Delta$-3 predicts very similar lattice parameters, energy differences between the phases (not shown),\nas well as the phonon dispersion relations (see Fig.~\\ref{fig:phonon_two_ML_SCAN_delta}) as compared to MLFF-SCAN$^\\Delta$.\nThe same observations also apply when generating MLFF-RPA$^\\Delta$, so that in practice, similar results as in the main text could be obtained\nwith just half the number of RPA calculations.\n\n\n\\begin{table*\n\\caption {The validation root-mean-square errors (RMSE) in energies per atom (meV\/atom), forces (eV\/\\AA) and stress tensors (kbar)\nfor MLFF-SCAN, MLFF-SCAN$^\\Delta$ and MLFF-SCAN$^\\Delta$-$k$ ($k=$2 and 3).\nThe latter three MLFFs are obtained by the $\\Delta$-ML approach and\ndiffer in the level of SVD rank compression when constructing the $T^{(3)}$ dataset.\nHere, the parameter $\\epsilon$ defines the degree of rank compression [see Eq.~\\eqref{equation_ls2}] and $N_{\\rm str}$\nrepresents the number of SVD compressed structures used to machine-learn the differences.\nThe test dataset includes 120 structures of 96 atoms.\n}\n\\begin{ruledtabular}\n\\begin{tabular}{lccccc}\n & Energy & Force & Stress & $\\epsilon$ & $N_{\\rm str}$ in $T^{(3)}$ \\\\\n \\hline\nMLFF-SCAN$^\\Delta$ & 2.37 & 0.139 & 2.30 & 1E-10 & 168 \\\\\nMLFF-SCAN$^\\Delta$-2 & 2.45 & 0.139 & 2.28 & 1E-08 & 102 \\\\\nMLFF-SCAN$^\\Delta$-3 & 2.46 & 0.139 & 2.29 & 1E-07 & 72 \\\\\nMLFF-SCAN & 2.49 & 0.139 & 2.38 & & \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\label{tab:error_MLFF-DFT_SM}\n\\end{table*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth, clip]{FigS8_phonon_two_MLSCAN-delta.eps}\n\\end{center}\n\\caption{Phonon dispersion relations of (a) monoclinic, (b) tetragonal, and (c) cubic ZrO$_2$ at 0 K predicted by MLFF-SCAN$^\\Delta$ (blue lines)\nand MLFF-SCAN$^\\Delta$-3 (red lines).\n}\n\\label{fig:phonon_two_ML_SCAN_delta}\n\\end{figure*}\n\n\n\n\\newpage\n\\bibliographystyle{apsrev4-1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}