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+{"text":"\\section{Introduction}\n\nThe site where r-process nuclei above A=90 have been synthesized\nremains a major unsolved problem in nucleosynthesis theory\n\\citep[e.g.,][]{Arnould07}.  Historically, many possibilities have\nbeen proposed \\citep[see][]{Meyer94}, but today, there are two\nprincipal contenders - neutron star mergers\n\\citep{Lattimer77,Freiburghaus99} and the NDW\n\\citep{Woosley94,Qian96,Hoffman97,Otsuki00,Thompson01}.  Observations\nof ultra-metal-poor stars suggest that many r-process isotopes were\nalready quite abundant at early times in the galaxy\n\\citep{Cowan95,Sneden96,Frebel07}, suggesting both a primary origin\nfor the r-process and an association with massive stars. NDWs would\nhave accompanied the first supernovae that made neutron stars and,\ndepending upon what is assumed about their birth rate and orbital\nparameters, the first merging neutron stars could also have occurred\nquite early.\n\nBoth the merging neutron star model and the NDW have problems\nthough. In the simplest version of galactic chemical evolution,\nmerging neutron stars might be capable of providing the necessary\nintegrated yield of the r-process in the sun, but they make it too\nrarely in large doses and possibly too late to be consistent with\nobservations \\citep{Argast04}.  On the other hand, making the\nr-process in NDWs requires higher entropies, shorter time-scales, or\nlower electron mole numbers, $Y_e$, than have been demonstrated in any\nrealistic, modern model for a supernova explosion \\citep[though\n  see][]{Burrows06}.\n\nPrevious papers and models for nucleosynthesis in the NDW have focused\non the production of nuclei heavier than iron using either greatly\nsimplified dynamics \\citep{Beun08,Farouqi09} or nuclear\nphysics \\citep{Qian96,Otsuki00,Arcones07,Fischer09,Huedepohl09}.\nPost processing nuclear network calculations have been performed using \nthermal histories from accurate models of the dynamics \n\\citep{Hoffman97,Thompson01,Wanajo06}, but the calculations sampled only a \nlimited set of trajectories in the ejecta. No one has yet combined the \ncomplete synthesis of a realistic NDW with that of the rest of the \nsupernova.\n\nTo address this situation, and to develop a framework for testing the\nnucleosynthesis of future explosion models, we have calculated\nnucleosynthesis using neutrino luminosity histories taken from two PNS\ncalculations found in the literature \\citep{Woosley94,Huedepohl09}.\nThis was done using a modified version of the implicit one-dimensional\nhydrodynamics code Kepler, which includes an adaptive nuclear network\nof arbitrary size.  This network allows for the production of both\nr-process nuclei during neutron-rich phases of the wind and production\nof light p-elements during proton-rich phases.  Since the results of wind\nnucleosynthesis depend sensitively on the neutrino luminosities and\ninteraction rates \\citep{Qian96,Horowitz02}, we have included accurate\nneutrino interaction rates that contain both general relativistic and\nweak magnetism corrections.  \n\nThe synthesis of all nuclei from carbon through lead is integrated\nover the history of the NDW and combined with the yield from the rest\nof the supernova, and the result is compared with a solar distribution.\nIf a nucleus produced in the NDW is greatly overproduced relative to\nthe yields of abundant elements in the rest of the supernova, there is\na problem. If it is greatly underproduced, its synthesis in the NDW is\nunimportant.  If it is co-produced, the NDW may be responsible for the\ngalactic inventory of this element.  An important outcome of this\nstudy are the yields expected from a ``plain vanilla'' model for the\nNDW. Are there any elements that are robustly produced and thus might\nbe used as diagnostics of the wind in an early generation of stars?\n\nIn \\S\\ref{wind_physics}, we discuss the general physics of neutrino\ndriven winds and analytically delineate the regions in neutrino\ntemperature space were different modes of nucleosynthesis occur.  We\nthen discuss our numerical model in \\S\\ref{computational_method}.  In\n\\S\\ref{results}, the results of the time dependent models are\npresented.  We conclude with a discussion of how the NDW might affect\ngalactic chemical evolution and consider if this allows the strontium\nabundance in low metallicity halo stars to be used as a tracer of\nsupernova fallback at low metallicity.  Additionally, we investigate\nif observed abundances in SN 1987A can put constraints on late time\nneutrino luminosities from PNSs.  Finally, we discuss some possible\nmodifications of the basic model that might improve the r-process\nproduction. These ideas will be explored more thoroughly in \na subsequent paper.\n\n\\section{General Concepts and Relevant Physics} \\label{wind_physics}\n\nAfter collapse and bounce in a core collapse supernova, a condition of\nnear hydrostatic equilibrium exists in the vicinity of the\nneutrinospheres. The temperature of the outer layers is changing on a\ntime scale determined by the Kelvin-Helmholtz time of the PNS,\n$\\tau_{KH} \\approx 10 s$ \\citep{Burrows86,Pons99}.  This is much\nlonger than the dynamical time scale of the PNS envelope, so the\nhydrostatic part of the envelope is in an approximate steady state.\nThe neutrino heating rate, which is determined by the neutrino\nluminosities from the neutrino sphere, must then balance the local\nneutrino cooling rate.  Heating and cooling are dominated by the\ncharged current processes $(\\nu_e + n) \\rightleftharpoons (e^- + p)$\nand $(\\bar \\nu_e + p) \\rightleftharpoons (e^+ + n)$ \\citep{Qian96}.\nEquating these rates, while neglecting the neutron-proton mass\ndifference and weak magnetism corrections and assuming the geometry\ncan be approximated as close to plane-parallel gives the temperature\nstructure of the neutron star atmosphere as a function of radius,\n$T_{atm} \\approx 1.01 \\, \\textrm{MeV} \\, R_{\\nu,6}^{-1\/3}\nL_{\\nu,51}^{1\/6}\\epsilon_{\\nu,MeV}^{1\/3} \\left(y_\\nu\/y\\right)^{1\/3} $,\nwhere $L_{\\nu,51}$ and $\\epsilon_{\\nu,MeV}$ are the electron neutrino\nluminosity and average neutrino energy at the neutrino sphere in units\nof $10^{51} \\, \\textrm{ergs s}^{-1}$ and MeV, respectively.  The\ngravitational redshift factor is $y = \\sqrt{1-2GM_{NS}\/r c^2}$ which, \nwhen evaluated at the neutrino sphere, $R_\\nu$, is $y_\\nu$.  Notice that\nthe only dependence on radius is carried in the redshift factor, so that\nthe atmosphere is close to isothermal.\n \nAt the radius, $r_c$, where the pressure in the envelope becomes\nradiation dominated, the material becomes unstable to outflow\n\\citep{Salpeter81}.  Since the neutrino luminosity is significantly\nlower than the neutrino Eddington luminosity, a thermally driven wind\nresults \\citep{Duncan86}.  The density at which this wind begins can\nbe found approximately by equating the radiation pressure to the\nbaryonic pressure.  This results in a critical density, $ \\rho_c\n\\approx \\sci{8.3}{7} \\, \\textrm{g} \\, \\textrm{cm}^{-3} \\,\nR_{\\nu,6}^{-1} L_{\\nu,51}^{1\/2}\\epsilon_{\\nu,MeV}\n\\left(y_\\nu\/y\\right), $ at which significant outflow begins and the\nkinetic equilibrium of weak interactions ceases to hold.  Under these\nconditions, nuclear statistical equilibrium is maintained on a time\nscale much shorter than the dynamical time scale and, for these\ntemperatures and densities, there will be no bound nuclei present.\nSince the electron fraction is set by kinetic equilibrium, the\ncomposition of the wind does not depend on any previous nuclear\nprocessing, so any nucleosynthesis from the wind will be primary.\n\nAssuming that most neutrino heating occurs near $r_c$, the entropy is\nconstant once the temperature cools to the nucleon recombination\ntemperature, $kT \\approx 0.5$ MeV.  Therefore, the final nuclear\nabundances in the wind depend mainly on the wind entropy, electron\nfraction, and the dynamical timescale at the radius where alpha \ncombination occurs \\citep{Qian96,Hoffman97}.  To\ndetermine the contribution of the wind to the nucleosynthesis of the\nentire supernova, the mass loss rate must also be known.  Estimates\nfor these quantities are given in the Appendix along with a discussion\nof the effect of general relativistic corrections.\n\nIntegrating the mass loss rate (equation \\ref{eq:mdot}) for a typical\nneutrino luminosity history implies that the wind will eject\napproximately $10^{-3} \\, M_\\odot$ of material.  This in turn means\nthat for the wind to contribute to the integrated yields of the\nsupernova for a particular isotope, that isotope needs to overproduced\nrelative to its solar mass fraction by a factor of at least $10^5$ in\nthe wind, assuming the rest of the supernova ejects $\\sim10 \\, M_\\odot$ and has\nover production factors of its most abundant metals of order 10.\n\nAt early times in the wind, the PNS is losing lepton number\n\\citep[e.g.][]{Burrows86}.  Neutrino interactions in the wind then tend to\nincrease the lepton number of the wind, and, to maintain charge\nneutrality, cause the wind to become proton rich.  Under these\nconditions, the wind may synthesize some of the light p-process\nelements via the so called $\\nu$p-process \\citep{Frohlich06,Pruet06}.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=0.7]{fig1.eps}\n\\end{center}\n\\caption{Neutrino two-color plot produced using the analytic relations\n  in the Appendix.  A neutron star with gravitational mass 1.4\n  $M_\\odot$ \\ has been assumed with a neutrinosphere radius of 10 km.\n  The total neutrino luminosity is assumed to scale as $L_{\\nu_e,tot}\n  = 10^{51} (\\langle T_{\\nu} \\rangle\/ 3.5 \\, \\textrm{MeV})^4 \\,\n  \\textrm{erg} \\textrm{s}^{-1}$.  This luminosity is split between\n  neutrinos and anti-neutrinos so as to ensure that the net\n  deleptonization rate of the PNS is zero.  The thick black line\n  corresponds to an electron fraction of $Y_e = 0.5$.  Above this\n  line, neutron-rich conditions obtain and below it the matter is\n  proton-rich.  The white region is where there no free neutrons\n  remain after charged particle reactions cease.  The N = 50 (tan)\n  region corresponds to final neutron-to-seed ratios between 0.01 and\n  15.  The ``first peak'' (yellow) region corresponds to a neutron-to-seed\n  ratio between 15 and 70, and the ``second peak'' (orange) region is where\n  the neutron-to-seed ratio is greater than 70.  The dashed lines\n  correspond to the base ten logarithm of the mass loss rate in solar\n  masses per second. }\n\\label{fig:tcp_base}\n\\end{figure*}\n\nAfter the initial deleptonization burst though, the net lepton number\ncarried by the wind will be small. Since the anti-electron\nneutrinosphere sits deeper in the PNS than the electron\nneutrinosphere, the electron neutrinos will be cooler than the\nelectron anti-neutrinos \\citep{Woosley94}.  If this asymmetry is large\nenough, the wind can become neutron rich at later times.  Combined\nwith an $\\alpha$-rich freeze out, this can give conditions favorable\nfor the r-process \\citep{Woosley92}.\n\nIn both cases, the resulting nucleosynthesis is characterized by the\nintegrated neutron to seed ratio after charged particle reactions\nfreeze out.  For proton-rich winds, alpha-particles recombine into\n$^{12}$C by the standard triple alpha reaction and then alpha-capture\nand proton-capture reactions carry the nuclear flow up to\napproximately mass 60 \\citep{Woosley92}.  The slowest reaction in this\nsequence is $^4$He($2\\alpha$,$\\gamma$)$^{12}$C, so the total number of\nseed nuclei produced is equal to the number of $^{12}$C nuclei\nproduced.  When only free protons are present, the integrated neutron density is\ndetermined by the rate of anti-neutrino capture on free protons.  An\nestimate for the neutron to seed ratio for proton-rich winds is given\nby equation \\ref{eq:nsp}.  A neutron to seed ratio of only a few is\nrequired to bypass the few long-lived waiting points which hinder\nproduction of the light p-isotopes \\citep{Pruet06}.  Still, it is\nchallenging to produce even a small neutron to seed ratio, since the\ndynamical time scale of the wind is short compared to the anti-neutrino\ncapture time scale.\n\nIn the neutron-rich case, seed nuclei are produced by a different\nreaction sequence $^4$He($\\alpha$n,$\\gamma$)$^9$Be($\\alpha$,n)$^{12}$C\n\\citep{Woosley92}.  For the conditions encountered in the wind, the\nneutron catalyzed triple-alpha reaction proceeds about ten times as\nquickly as $^4$He($2\\alpha$,$\\gamma$)$^{12}$C.  This increases the\nseed number compared with that obtained in a proton-rich wind with\nsimilar dynamical properties.  Also, since there are free neutrons,\ncapture can proceed up to the N=50 closed shell isotopes $^{88}$Sr,\n$^{89}$Y, and $^{90}$Zr.  Here, the neutron density is just determined\nby the number of free neutrons left after charged particle reactions\nfreeze out.  The neutron to seed ratio in neutron-rich winds can be\napproximated using equation \\ref{eq:nsn}.  Charged particle reactions\ncontinue up to the N=50 closed shell, at which point it becomes\nunfavorable to capture alpha particles due to small separation\nenergies and large coulomb barriers \\citep{Hoffman96}.  If neutrons are exhausted during\nthis process, the wind will mainly produce the isotopes $^{88}$Sr,\n$^{89}$Y, and $^{90}$Zr \\citep{Hoffman97}.  This happens when the\ncondition\n\\begin{equation} \n\\label{eq:n50_prdod} \n\\frac{\\bar Z}{\\bar A} \\approx 0.42-0.49 = \\frac{Y_e\n  f_\\alpha}{2Y_e(f_\\alpha-1) + 1}\n\\end{equation} \nis met.  Here, $f_\\alpha \\approx 14 Y_s\/Y_{\\alpha,i}$ is the fraction\nof the initial helium abundance that gets processed into heavy nuclei.\nNeutron to seed ratios of approximately 30 and 110 are required to\nproduce first and second peak r-process nucleosynthesis, respectively.\n\nUsing the analytic results for the wind dynamics and nucleosynthesis\ngiven in the Appendix (equations \\ref{eq:rcrit}, \\ref{eq:ent},\n\\ref{eq:mdot}, \\ref{eq:tau},\\ref{eq:ye}, \\ref{eq:nsp}, \\ref{eq:nsn},\nand using the neutrino interaction rates given in\n\\S\\ref{sec:neutrino_rates} to fix the thermodynamic state at $r_c$),\none can easily explore the neutrino temperature parameter space to\ndetermine the neutrino temperatures and fluxes that are most conducive\nto the r-process or the production of the light p-process.  Figure\n\\ref{fig:tcp_base} is a neutrino two-color plot where it is assumed\nthat the deleptonization rate is zero and that the neutrino luminosity\nscales with the temperature to the fourth power ($L_{\\nu_e,tot} =\n10^{51} (\\langle T_{\\nu} \\rangle\/ 3.5 \\, \\textrm{MeV})^4 \\,\n\\textrm{erg} \\textrm{s}^{-1}$).  The different nucleosynthetic regions\nare delineated by the final calculated neutron to seed ratio.  To give\na feeling for how a particular point in parameter space might\ncontribute to the integrated nucleosynthesis of the wind, the mass\nloss rate is also shown.\n\nFor a significant amount of material to move past the N = 50 closed\nshell during neutron-rich conditions, the anti-neutrino temperature\nmust be approximately $60$\\% higher than the neutrino temperature.\nFor second peak r-process nucleosynthesis to occur, the asymmetry must\nbe greater than $100$\\%.  Modern PNS cooling calculations do not give\nsuch large asymmetries \\citep{Pons99,Huedepohl09}.\n\nUnder proton-rich conditions, only a small region of the\nparameter space at high neutrino and low anti-neutrino temperature is\nfavorable for the $\\nu$p-process.  There will be a small amount of\nneutron production in the white region, but it is unlikely that\nsignificant production of the light p-process elements $^{74}$Se,\n$^{78}$Kr, $^{84}$Sr, and $^{92}$Mo will occur.  The region in\nneutrino temperature space where there is significant neutron\nproduction is unlikely to be reached.  This region is small due to the\nshort dynamical time scale of the wind, which reduces the time over\nwhich anti-neutrinos can capture on free neutrons.  One should note\nthat, very soon after shock formation in the supernova, a wind\nsolution may not be appropriate and material will be entrained closer\nto the PNS for a longer period of time.  This scenario would be similar\nto the the conditions used in \\cite{Pruet06}.\n  \nTherefore, based upon simple principles, it seems unlikely that the\nstandard wind scenario will produce r-process or light p-process\nisotopes in solar ratios, as is required by observations of metal poor\nhalo stars \\citep{Sneden96}.  This same conclusion has been reached by\nother authors \\citep{Hoffman97,Thompson01}, but is repeated\nhere in simple terms.  We will find that our numerical calculations\ngive similar results and that there is no significant r-process\nnucleosynthesis associated with the wind.  Still, the wind can produce \nsome isotopes that may have an observable signature.  For standard PNS \nluminosities, the wind will spend a significant amount of time in the \nregion of parameter space were N = 50 closed shell nucleosynthesis occurs.  \n\n\n\n\n\\section{Computational Method}\n\\label{computational_method}\n\nTo more accurately investigate the integrated nucleosynthesis of the\nNDW, we have updated the implicit Lagrangian hydrodynamics code Kepler\n\\citep{Weaver78,Woosley02} to carry out time-dependent simulations of\nthe wind dynamics and nucleosynthesis.  Kepler has been used\npreviously to study time-independent winds\\citep{Qian96}, but the weak\nand nuclear physics employed there was rudimentary and nucleosynthesis\nwas not tracked.  Trajectories from Kepler were used for post-processing\ncalculations of nucleosynthesis in \\cite{Hoffman97}.\n\nKepler solves the non-relativistic hydrodynamic equations in\nLagrangian coordinates assuming spherical symmetry.  First order\ngeneral relativistic corrections are included in the gravitational\nforce law (cf. \\cite{Shapiro83}).  All order $v\/c$ effects are\nneglected.  This is justified since the maximum wind speeds\nencountered are, at most, a few percent of the speed of light.  The\nmomentum equation is then \n\\begin{equation} \n\\td{v_r}{t} = -4 \\pi r^2 \\pd{P}{m} - \\frac{Gm}{r^2}\\left( 1 +\n\\frac{P}{\\rho c^2} + \\frac{4 \\pi P r^3}{m c^2} \\right) \\left(1 -\n\\frac{2 G m}{r c^2} \\right)^{-1}\n\\end{equation}\nwhere the symbols have their standard meanings.  As has been shown by\nprevious studies \\citep{Qian96,Cardall97,Otsuki00,Thompson01}, general\nrelativistic corrections to the gravitational force can have an\nappreciable effect on the entropy and dynamical time scale of the\nwind.  The equation of state includes a Boltzmann gas of nucleons and\nnuclei, an arbitrarily relativistic and degenerate ideal electron gas,\nand photons.\n\n\\subsection{Weak Interaction Physics}\n\\label{sec:neutrino_rates}\n\nEnergy deposition from electron neutrino capture on nucleons, neutrino\nannihilation of all neutrino flavors, and neutrino scattering of \nall flavors on electrons is included in the\ntotal neutrino heating rate.  Neutrino ``transport'' is calculated in the \nlight-bulb approximation.  The energy deposition rate is dominated\nby neutrino captures on nucleons.  The neutrino annihilation rates\ngiven in \\cite{Janka91} are employed.  For the scattering rates, the\nrates given in \\cite{Qian96} are used, but we include general\nrelativistic corrections.  Standard neutrino capture rates are\nemployed in the limit of infinitely heavy nucleons with first order\ncorrections.  In this limit, the cross section is \n(Y.Z. Qian, private communication)\n\\begin{equation}\n  \\sigma_{ \\nu n \\atop \\bar \\nu p} =\\frac{G_F^2 cos^2(\\theta_C)}{ \\pi\n    (\\hbar c)^4}\\left[g_V^2+3g_A^2 \\right] \\left(\\epsilon_{\\nu} \\pm\n  \\Delta \\right)^2\\left(1 \\pm W_{M, {\\nu \\atop {\\bar \\nu}}}\n  \\epsilon_{\\nu}\\right) \n\\end{equation} \nHere, $G_F$ is the Fermi coupling constant, $\\theta_C$ is the Cabibo angle,\n$g_V$ and $g_A$ are the dimensionless vector and axial-vector coupling \nconstants for nucleons, $\\Delta$ is the proton neutron mass difference, \n$\\epsilon_\\nu$ is the neutrino energy, and $W_{M, {\\nu \\atop {\\bar \\nu}}}$ \naccounts for the weak magnetism\nand recoil corrections to the neutrino-nucleon cross section when the\nbase cross section is derived in the limit of infinitely heavy\nnucleons \\citep{Horowitz02}.  This correction reduces the\nanti-neutrino cross section and increases the neutrino cross section\n(by about a total of 10\\% at the energies encountered in NDWs), which,\nfor a given incident neutrino spectrum, significantly increases the\nasymptotic electron fraction.  Assuming a thermal distribution, these\ncross sections result in the neutrino energy deposition rate for\nanti-electron neutrino capture \n\\begin{equation} \n\\begin{array}{rl}\n    \\dot q_{\\bar \\nu p}=&\\sci{4.2}{18} \\textrm{ergs\n      s}^{-1}\\textrm{g}^{-1}\\frac{Y_p L_{\\bar \\nu,51}}{ \\langle\n      \\mu\\rangle r_6^2} \\\\ \\times & \\biggl[-W_M^{\\bar\\nu\n        p}\\frac{\\langle \\epsilon_{\\bar \\nu}^4 \\rangle }{\\langle\n        \\epsilon_{\\bar \\nu} \\rangle } + (1 + 2W_M^{\\bar\\nu p} \\Delta)\n      \\frac{\\langle \\epsilon_{\\bar \\nu}^3 \\rangle }{\\langle \\epsilon_{\\bar\\nu}\n        \\rangle } \\\\ & - (2 \\Delta + W_M^{\\bar\\nu p} \\Delta^2\n      )\\frac{\\langle \\epsilon_{\\bar \\nu}^2 \\rangle }{\\langle \\epsilon_{\\bar \\nu}\n        \\rangle } + \\Delta^2 \\biggr]\n\\end{array}\n\\end{equation} \nand a similar expression for electron neutrino capture.  The neutrino\nenergy distributions are parameterized by assuming a Fermi-Dirac\nspectrum.  The neutrino energy averages, $\\avg{\\epsilon_\\nu^n}$, are\nevaluated using this distribution.  The neutrino energy moments and\nluminosity are evaluated in the rest frame of the fluid.  With general\nrelativistic corrections for the bending of null geodesics, the\naverage neutrino angle is given by\n\\begin{equation} \n\\langle \\mu \\rangle = \\frac{1}{2} +\n\\frac{1}{2}\\sqrt{1-\\left( \\frac{R_\\nu y_\\nu}{r y}\\right)^2}.\n\\end{equation} \nSpecial relativistic corrections are negligible in the regions where\nneutrino interactions are important.\n\nThe lepton capture rates used are calculated in the limit of\ninfinitely heavy nucleons.  This results in a positron capture energy\nloss rate\n\\begin{equation}\n\\begin{array}{rl}\n\\dot q_{e^+ n} =&\\sci{6.9}{15}\\, \\textrm{ergs g}^{-1} \\, \\textrm{s}^{-1} \\, Y_n T_{10}^6 \\\\\n\\times & \\int_0^\\infty du f_{e}(u,-\\eta)  \\left( u^5 + 3 \\delta u^4 + 3 \\delta^2 u^3 + \\delta^3 u^2  \\right)\n \\end{array}\n\\end{equation}\nhere $f_e(u,\\eta) = (\\exp(u-\\eta)+1)^{-1}$, $\\eta$ is the electron degeneracy parameter, \n$\\delta$ is the proton neutron mass difference divided by $k_bT$, and $Y_n$ is the neutron \nfraction.  A similar rate is employed for electron capture.\n\nFor the neutrino losses, we include electron and positron capture on\nnucleons and include thermal losses as tabulated in \\cite{Itoh96}.\nThe energy loss rate in the wind is dominated by the electron\ncaptures.\n\n\\subsection{Nuclear Physics}\n\nDuring a hydrodynamic time step in Kepler, the nuclear energy generation\nrate and the changing nuclear composition are calculated\nusing a modified version of the 19-isotope network described in\n\\cite{Weaver78}.  Neutrino and electron capture rates on nucleons are\ncoupled to the network, which are calculated under the same\nassumptions as the charged current energy deposition\/loss rates\ndescribed above.  Therefore, non-equilibrium evolution of the electron\nfraction is accurately tracked.\n\nAlthough this network is appropriate for calculating energy generation\nthroughout the entire wind, it is not large enough to accurately track\nthe nucleosynthesis once alpha recombination begins at $ T \\approx\n0.5$ MeV.  Therefore, for temperatures below $20 \\textrm{GK}$ an\nadaptive network is run alongside the hydrodynamics calculation.  The\ndetails of this network can be found in \\cite{Woosley04} and \n\\cite{Rauscher02}.  As a fluid\nelement passes the temperature threshold, the composition from the\n19-isotope network is mapped into the adaptive network.  Typically,\nthe network contains approximately 2000 isotopes.  Where available,\nexperimental nuclear reaction rates are employed, but the vast\nmajority of the rates employed in the network come from the\nstatistical model calculations of \n\\cite{Rauscher00}.  In general, the nuclear physics employed in these\ncalculations is the same as that used in \\cite{Rauscher02}.  The\nnucleon weak interaction rates employed in the 19-isotope network are\nalso used in the adaptive network.\n\n\n\\subsection{Problem Setup and Boundary Conditions}\n\nTo start the neutrino driven wind problem, an atmosphere of mass 0.01\n$M_\\odot$ is allowed to relax to hydrostatic equilibrium on top of a\nfixed inner boundary at the neutron stars radius.  The mass enclosed\nby the inner boundary is the neutron star's mass.  The photon\nluminosity from the neutron star is assumed to be nearly Eddington,\nbut we have found that the properties of the wind are insensitive to\nthe the luminosity boundary condition.  Once hydrostatic equilibrium\nis achieved, the neutrino flux is turned on and a thermal wind forms.\nThis wind is allowed to relax to a quasi-steady state, and then the 19\nisotope network is turned on and the wind is, once again, allowed to\nreach a quasi-steady state.  After this point, the neutrino flux is\nallowed to vary with time, and the adaptive network is turned on.  \n\nAs the calculation proceeds, the mass of the envelope being followed\ndecreases and could eventually all be blown away.  To prevent this,\nmass is added back to the innermost mass elements at a rate equal to\nthe mass loss rate in the wind.  The mass added to a fluid element at\neach time step is a small fraction of its total mass.  We find that\nmass recycling has no effect on the properties of the wind. It is\nsimply a way of treating a problem that is essentially Eulerian in a\nLagrangian code.\n\nFor most runs, a zero outer boundary pressure and temperature are \nassumed.  To investigate the effect of a wind termination shock, a\ntime dependent outer boundary condition is included in some of \nthe simulations detailed below.  The pressure \nof the radiation dominated region behind the supernova \nshock is approximately given by \\citep{Woosley02}\n\\begin{equation}\n\\label{eq:Pbound}\nP_{ps} \\approx \\frac{E_{sn}}{4 \\pi (v_{sn}  t)^3}\n\\end{equation}   \nwhere $E_{sn}$ is the explosion energy of the supernova, $v_{sn}$ is\nthe supernova shock velocity, and $t$ is the time elapsed since the \nshock was launched.  As was discussed in \\cite{Arcones07}, this \nresults in a wind termination shock at a radius where the condition\n$\\rho_w v_w^2 + P_w \\approx P_{ps}$\nobtains, where $v_w$ is the wind velocity and $\\rho_w$ is the \nwind density.  To avoid an accumulation of too many zones, \nmass elements are removed from the calculation once they \nexceed a radius of $10,000 \\, \\textrm{km}$.  This is well \noutside the sonic point and nuclear burning has \nceased by this radius in all calculations .\n \n\\section{Numerical Results}\n\\label{results}\n\nTo survey both low and intermediate mass core collapse supernovae, \nneutrino emission\nhistories were taken from two core collapse calculations, one from a\n$20 M_\\odot$ \\citep{Woosley94} supernova calculation and the other \nfrom a $8.8 M_\\odot$ \\citep{Huedepohl09}\nsupernova calculation.  Since the PNSs studied have significantly\ndifferent masses and neutrino emission characteristics, one is able to\nget a rough picture of how integrated nucleosynthesis in the NDW\nvaries with progenitor mass.\n\n\\subsection{Neutrino Driven Wind from a $20 M_\\odot$ Supernova}\n\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\includegraphics[scale=0.45]  {fig2.eps}\n\\end{center}\n \\caption{Neutrino luminosities and temperatures taken from the model\n  of \\cite{Woosley94}.  The top panel is the neutrino luminosities.  The \n  bottom panel is the average neutrino energies.  The solid line corresponds \n  to $\\nu_e$, the dashed line corresponds to $\\bar \\nu_e$, the dot-dashed \n  line corresponds to $\\nu_{\\mu,\\tau}$.}\n \\label{fig:woos94_neut}\n\\end{figure}\n\nThe first set of neutrino luminosities and temperatures are taken from\n\\cite{Woosley94}.  This calculation began with a $20 M_\\odot$ progenitor\nmeant to model the progenitor of 1987A \\citep{Woosley88}.  The\nresulting neutron star had a gravitational mass of $1.4 M_\\odot$ and\nthe neutrino sphere was taken to be at 10 km.  The neutrino\nluminosities and average energies as a function of time from this\nmodel are shown in figure \\ref{fig:woos94_neut}.  After about 4\nseconds, the neutrino energies become constant and the large\ndifference between the electron neutrino and anti-neutrino energies\nimplies that the wind will be neutron rich.  This supernova model had\nsome numerical deficiencies (Sam Dalhed, Private Communication). The\nentropy calculated for the wind in \\cite{Woosley94} ($S\/N_Ak \\approx 400$)\nwere unrealistically large due to some problems with the equation of\nstate. Here, that is not so important because the NDW is being\ncalculated separately, but this study did rely on older neutrino\ninteraction rates and did not include weak magnetism corrections (see \\S\n\\ref{sec:neutrino_rates}).  Therefore, the results obtained using\nthese neutrino histories are only suggestive of what might happen in a\nmore massive star.  If weak magnetism were taken into\naccount, the calculated electron and anti-electron neutrino\ntemperatures would probably be somewhat further apart.\n\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\includegraphics[scale=0.5]  {fig3.eps}\n\\end{center}\n\\caption{Wind structure after two seconds in the model using \nthe neutrino luminosities from \\cite{Woosley94}.  The top panel \nshows the density in units of $10^8 \\, \\textrm{g} \\, \\textrm{cm}^{-3}$ \n(solid line) and the radial velocity in units of $10^3 \\, \\textrm{km} \\, \n\\textrm{s}^{-1}$ (dot-dashed line).  The middle panel shows the net \nenergy deposition rate from weak and strong interactions in units of \n$10^{20} \\, \\textrm{erg} \\, \\textrm{g}^{-1} \\, \\textrm{s}^{-1}$ (dot-dashed line) \nand the entropy (solid line).  The bottom panel shows the temperature \nin units of $\\sci{2}{9} \\, \\textrm{K}$ (solid line), the electron \nfraction (dot-dahsed line), and the fraction of material contained in \nnuclei (dotted line). }\n \\label{fig:wnd_struct}\n\\end{figure}\n\nThe calculation was run for a total of 18 seconds.  During this time,\nthe mass loss rate decreased by almost three orders of magnitude while\na total mass of $\\sci{2}{-3} M_\\odot$ was lost in the wind.  A\nsnapshot of the wind structure two seconds after bounce is shown in\nfigure \\ref{fig:wnd_struct}.  Note that the wind velocity stays very\nsub-luminal throughout the calculation.  Therefore, the neglect of\nspecial relativistic effects is reasonable.  The secondary bump in the\nenergy deposition rate occurs at the same radius where nucleons and\nalpha-particles assemble into heavy nuclei.  This increases the\nentropy by about 10 units.  Clearly, the electron fraction is set\ninterior to were nuclei form.  The radius where nuclei form is at a\nlarge enough value that the alpha effect \\citep{Fuller95} is not\nsignificant at early times in the wind.  However, as the neutrino luminosity\ndecreases with time, nucleon recombination occurs at a smaller radius,\nand the alpha effect becomes increasingly important.\n\nThe time evolution of the wind as calculated by Kepler is shown in\nfigure \\ref{fig:woos94_wndprop}.  The increase in asymptotic entropy is mainly\ndriven by the decrease in neutrino luminosity, since the average\nneutrino energies do not vary greatly.  The analytic approximation\n(calculated using equation \\ref{eq:ent} and the neutrino interaction\nrates given in \\S\\ref{sec:neutrino_rates}) to the entropy tracks the\nentropy calculated in Kepler fairly well.  This implies that the\nvariation in the neutrino luminosity with time does not significantly alter the\ndynamics from a steady state wind. In contrast to the high entropies\nreported in \\citet{Woosley94}, the entropy here never exceeds 130.\nFor the time scales and electron fractions also obtained, such a low\nvalue of entropy is not sufficient to give a strong r-process (see\nbelow).\n\nThe electron neutrino and anti-neutrino energies do move further apart\nas a function of time though, which causes the wind to evolve from\nproton-rich conditions at early times to neutron-rich conditions\nlater.  A transition occurs from the synthesis of proton-rich isotopes\nvia the $\\nu p$-process at early times to the $\\alpha-$process mediated\nby the reaction sequence\n$\\alpha$($\\alpha$n,$\\gamma$)$^9$Be($\\alpha$,n)$^{12}$C later.  The\nslight difference between the analytic approximation and the Kepler\ncalculation of $Y_e$ is due to the alpha effect \\citep{Fuller95}.\n\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\includegraphics[scale=0.43]  {fig4.eps}\n\\end{center}\n\\caption{Properties of the neutrino driven wind from the \n\\cite{Woosley94} supernova model as a function of time. The thick \nlines correspond to the numerical results from Kepler and the thin \nlines correspond to the predictions of the analytic estimates described \nin the appendix.  The solid line is the dimensionless entropy per baryon, \nthe dashed line is the electron fraction, the dash dotted line is the \ndynamical timescale, and the dotted line is the mass loss rate.  All \nof the quantities are taken extracted from where the wind temperature \nreaches 2 GK.}\n \\label{fig:woos94_wndprop}\n\\end{figure}\n\nIntegrated production factors for the wind are shown in figure\n\\ref{fig:WWsolopf}.  The production factor for the species $i$ is\ndefined as\n\\begin{equation} \nP_i = \\frac{X_{i,w}M_{w}}{X_{i,\\odot}(M_{w}+M_{sn})},\n\\end{equation} \nwhere $X_{i,w}$ is the mass fraction of species $i$  in the wind\nafter all material has decayed to stable isotopes, $M_w$\nis the mass ejected in the wind, and $M_{sn}$ is the amount of mass\nejected by the entire supernova.  $X_{i,\\odot}$ is the mass fraction\nof isotope $i$ in the sun for which the values of \\citet{Lodders03}\nwere used. The only isotopes that are co-produced in the wind\nalone are $^{87}$Rb, $^{88}$Sr, $^{89}$Y, and $^{90}$Zr, with\nproduction factor of $^{88}$Sr about a factor of 3 higher than the\nother two N = 50 closed shell isotopes.  Before eight seconds, the\nproduction factors had been much closer.  After eight seconds though,\nthe wind is dominated by $^{88}$Sr because $Y_e \\sim 0.45$ and only\n53\\% of alpha particles are free after freeze out which puts\n$\\frac{\\bar Z}{\\bar A}\\approx 0.41 $ of the heavy nuclei just\nbelow the range given in equation \\ref{eq:n50_prdod}.  There are not\nenough free neutrons to make any significant amount of heavier nuclei,\nand this results in significant production of the stable N = 50 closed\nshell isotope with the lowest $\\frac{\\bar Z}{\\bar A}$.\n\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\includegraphics[scale=0.65]  {fig5.eps}\n\\end{center}\n\\caption{Isotopic production factors from the NDW model when  \nthe neutrino luminosities from \\cite{Woosley94} are used.  The production \nfactors are calculated assuming that $18.4 \\, M_\\odot$ of material \nwas ejected in the supernova in addition to the wind.  The top dashed line\ncorresponds to the greatest production factor in the wind, the solid line is a\nfactor of two below that, and the bottom dashed line is a factor of two below the\nsolid line.  These lines specify an approximate coproduction band for the wind alone.}\n \\label{fig:WWsolopf}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\includegraphics[scale=0.42]  {fig6.eps}\n\\end{center}\n \\caption{Neutrino two-color plot when the anti-neutrino luminosity \n is 1.2 times neutrino luminosity, and the total luminosity scales \n with average temperature to the fourth.  Similar to figure \n \\ref{fig:tcp_base}.  The red line is the neutrino temperatures \n from \\cite{Woosley94}.}\n \\label{fig:tcp_woos}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\includegraphics[scale=0.65]  {fig7.eps}\n\\end{center}\n\\caption{Isotopic production factors from the NDW model \nemploying the neutrino luminosities from \\cite{Woosley94} with \nthe anti-electron neutrino temperature reduced by $15\\%$.  The \nproduction factors are calculated assuming that $18.4 \\, M_\\odot$ \nof material was ejected in the supernova in addition to the wind.\nThe horizontal lines are similar to those in figure \\ref{fig:WWsolopf}.}\n \\label{fig:WWredpf}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\includegraphics[scale=0.65]  {fig8.eps}\n\\end{center}\n\\caption{Isotopic production factors from the NDW model \nemploying the neutrino luminosities from \\cite{Woosley94} \nwith weak magnetism corrections turned off.  The production \nfactors are calculated assuming that $18.4 \\, M_\\odot$ of material \nwas ejected in the supernova in addition to the wind.\nThe horizontal lines are similar to those in figure \\ref{fig:WWsolopf}.}\n \\label{fig:WWnowmpf}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\includegraphics[scale=0.65]  {fig9.eps}\n\\end{center}\n\\caption{Isotopic production factors from the NDW model when  \nthe neutrino luminosities from \\cite{Woosley94} are used and an external boundary \npressure is specified as described in the text, which results in a wind termination\nshock.  The production \nfactors are calculated assuming that $18.4 \\, M_\\odot$ of material \nwas ejected in the supernova in addition to the wind.  The horizontal lines are \nsimilar to those in figure \\ref{fig:WWsolopf}.}\n \\label{fig:WWpbd}\n\\end{figure}\n\n\nDuring the first four seconds, the wind is proton rich and the isotopes\n$^{69}$Ga, $^{70,72}$Ge, $^{74,76}$Se, and $^{78,80,82}$Kr \nare produced by proton captures on seed nuclei produced by the triple-alpha \nreaction and subsequent ($\\alpha$,p) reactions.  Although the mass \nloss rate is much higher when the wind \nis proton rich, the alpha-fraction freezes out at 98\\% of its initial value, \nwhich results in significantly decreased production of heavy nuclei.  \nThe difference in final alpha fraction between the neutron- and proton-rich \nphases of the wind is due mainly to the difference in speed of the reaction \nchains $\\alpha$($2\\alpha$,$\\gamma$)$^{12}$C and\n$\\alpha$($\\alpha$n,$\\gamma$)$^9$Be($\\alpha$,n)$^{12}$C, but also \nto the decreased entropy at early times.\n\nWe can compare this with the analytic predictions for nucleosynthesis\nby plotting the neutrino temperature evolution from this model on a\nneutrino ``two-color plot'' (figure \\ref{fig:tcp_woos}).  Here we have\nset $L_{\\bar \\nu_e} = 1.2 L_\\nu$ which is approximately correct at\nlate times in the calculation of \\cite{Woosley94}.  The wind never\nreaches a region in which r-process nucleosynthesis is expected, but\nspends a significant amount of time making nuclei in the N = 50 closed\nshell isotones.\n\n\\subsubsection{Variations in Neutrino Properties}\nSince the neutrino temperatures from the original model were\nuncertain, several other models were calculated. One had a reduced (by $15\\%$)\nelectron antineutrino temperature; another had the weak\nmagnetism corrections to the neutrino interaction rates turned off.  A\nsmaller antineutrino temperature is more in line with recent\ncalculations of PNS cooling \\citep{Pons99,Keil03}.  Because the\nmodel of \\cite{Woosley94} did not include weak magnetism corrections,\nour model with weak magnetism corrections turned off is more\nconsistent with the original supernova model.\n\nThe production factors for the model with a reduced electron\nantineutrino temperature are shown in figure \\ref{fig:WWredpf}.  The\nyield of $^{88}$Sr is reduced by almost a factor of ten from the base\ncase, while the production factors of $^{89}$Y and $^{90}$Zr are\nreduced by a factor of three. In this case, the wind also produces the\nproton-rich isotopes $^{74}$Se, $^{78}$Kr, and $^{84}$Sr.  The\ncoproduction line for lighter elements like oxygen in a $20 M_\\odot$\nsupernova at solar metallicity is around 18, so the wind could contribute\nto the total nucleosynthesis if the antineutrino temperature was\nreduced, but its contribution would be small.  \n\nThe yields when weak magnetism corrections are ignored are shown in\nfigure \\ref{fig:WWnowmpf}.  Without weak magnetism, the electron\nfraction drops below 0.4 at late times when the entropy is fairly\nhigh.  Equation \\ref{eq:n50_prdod} is no longer satisfied and material\nmoves past the N = 50 closed shell towards A $\\approx$ 110.  Some\nr-process isotopes are produced, such as $^{96}$Zr and $^{100}$Mo, but\nnot anywhere near solar ratios, and no material reaches the first\nr-process peak.\n\n\\begin{figure*}\n\\begin{center}\n\\leavevmode\n\\includegraphics[scale=0.75]  {fig10.eps}\n\\end{center}\n\\caption{Combined isotopic production factors of the neutrino \ndriven wind with unaltered neutrino temperatures and including weak magnetism\ncorrections added to those of a $20 M_\\odot$ stellar model from \\cite{Woosley95}.  \nThe solid black line is the coproduction line with $^{16}$O.  \nThe dashed lines are a factor of two above and below the coproduction line.  The \nneutrino driven wind is responsible for the production of $^{88}$Sr, \n$^{89}$Y, and $^{90}$Zr.}\n \\label{fig:comb_pf}\n\\end{figure*}\n\n\\subsubsection{Effect of a Wind Termination Shock}\nTo investigate the possible effect of a wind termination shock on nucleosynthesis,\nanother model was run with a boundary pressure and temperature \ndetermined by equation \\ref{eq:Pbound}.  An explosion energy of \n$10^{51} \\, \\textrm{erg}$ was assumed and the shock velocity was taken as\n$\\sci{2}{9} \\, \\textrm{cm s}^{-1}$.  This resulted in a wind termination\nshock that was always at a radius greater than $10^3 \\, \\textrm{km}$.  \nThe production factors from this model are shown in figure \n\\ref{fig:WWpbd}.  Similar to the simulation without a wind termination\nshock, the N=50 closed shell elements dominate the wind's nucleosynthesis.\n\n\n\nThe main difference between the case with and without a wind termination \nshock is a shift in the mass of isotopes produced during the proton-rich\nphase.  This can be seen in the increased production of Mo. \nDuring this phase, the post shock temperature varied from \n2.5 GK down to 0.8 GK and the density varied from\n$\\sci{5}{4} \\, \\textrm{g cm}^{-3}$ to $\\sci{5}{2} \\, \\textrm{g cm}^{-3}$.  \nThese conditions are very favorable for \ncontinued proton capture once the long lived waiting point isotopes \n$^{56}$Ni and $^{64}$Ge are bypassed by (n,p) reactions.  Because \nthese conditions persist for at least a second after a fluid element passes\nthrough the wind termination shock, significantly more proton captures \ncan occur on seed nuclei that have moved past mass $\\sim 64$ relative\nto the case with no termination shock.  Still, not many more neutrons \nare produced per seed nucleus relative to the base run.  Therefore, the net \nnumber of seeds that get past the long lived waiting points remains small\nand the proton-rich wind does not contribute to the integrated nucleosynthesis.     \nIt should also be noted that a different treatment of the wind's \ninteraction with the supernova shock might result in a breeze solution \nwhich may supply more favorable conditions for $\\nu$p-process \nnucleosynthesis \\citep{Wanajo06}.    \n\n\\subsubsection{Total Supernova Yields}\n\n\nIn figure \\ref{fig:comb_pf}, the production factors from a $20\nM_\\odot$ supernova model from \\cite{Woosley95} have been combined with\nthe production factors we calculated in the NDW with the unaltered\nneutrino histories of \\citep{Woosley94} with weak magnetism corrections\nincluded.  The wind could be\nresponsible for synthesizing the isotopes $^{87}$Rb, $^{88}$Sr,\n$^{89}$Y, and $^{90}$Zr.  $^{88}$Sr production is above the\nco-production band, but the rest are in agreement with the stellar\nyields.  This overproduction of $^{88}$Sr is similar to the result of\n\\cite{Hoffman97}.\n\nFor the model with a reduced anti-electron neutrino temperature\ncombined with the yields from the $20 M_\\odot$ supernova model, the\nwind contributes 28\\%, 42\\%, 35\\%, 75\\%, 75\\%, and 80\\% of the total\n$^{74}$Se, $^{78}$Kr, $^{84}$Sr, $^{88}$Sr, $^{89}$Y, and $^{90}$Zr\nabundances in the supernova, respectively.  This wind model does not result in any\nisotopes being overproduced relative to the rest of the yields of the\nsupernova.  For the case with weak magnetism turned off, the nuclei\nproduced by the wind are overproduced relative to those made in the rest of \nthe star by factor of nearly 100, hence this\nwould need to be a very rare event if this model were realistic.\n\nClearly, weak magnetism corrections and variations in the neutrino\ntemperatures have a very significant effect on nucleosynthesis in the\nwind.  Aside from the effects of an extra source of energy\n(\\ref{modifications}), the neutrino spectra are the largest current\ntheoretical uncertainty in models of the NDW.\n\n\\subsection{Neutrino Driven Wind from a $8.8 M_\\odot$ Supernova}\n\\label{eight_results}\n\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\includegraphics[scale=0.45]  {fig11.eps}\n\\end{center}\n \\caption{Neutrino luminosities and temperatures taken from the model \n of \\cite{Huedepohl09}.  The line styles are the same as in figure \n \\ref{fig:woos94_neut}.  }\n \\label{fig:jnk_neut}\n\\end{figure}\n\nThe second PNS model is a more modern one-dimensional calculation of an\nelectron-capture supernova \\citep{Huedepohl09} that started from an\n$8.8 M_\\odot$ progenitor model \\citep{Nomoto84}. This resulted in a\nPNS with a gravitational mass of $1.27 M_\\odot$ and a radius of 15 km.\nTogether the lower mass and increased radius imply a lower\ngravitational potential at the neutrinosphere.  This work employed\nneutrino interaction rates which took weak magnetism and ``in-medium''\neffects into account.  The neutrino luminosities and average energies\nas a function of time are shown in figure \\ref{fig:jnk_neut}. The\nmaximum difference between the electron and anti-electron neutrino\naverage energies is significantly less than in the model of\n\\cite{Woosley94}.  This is likely due in part to both the decreased\ngravitational potential of the PNS and the more accurate neutrino\ninteraction rates in the newer model.\n\nThe calculation was run for a total of nine seconds, at which point\nthe mass loss rate had dropped by two orders of magnitude.  The total\namount of mass ejected in the wind was $\\sci{3.8}{-4} \\, M_\\odot$.  In\nfigure \\ref{fig:jnk_wndprop}, the properties of the NDW calculated\nusing Kepler are plotted as a function of time.  Notice that the\nentropy never reaches above 100 in this model, which diminishes the\nlikelihood of significant nucleosynthesis.  For comparison, we also\ninclude the analytic estimates detailed above.  There is reasonable\nagreement between the analytic and the numerical calculations, but not\nnearly as good as in the $20 M_\\odot$ model.\n\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\includegraphics[scale=0.45]  {fig12.eps}\n\\end{center}\n\\caption{Properties of the neutrino driven wind from the \n\\cite{Huedepohl09} supernova model as a function of time.  \nThe lines have the same meaning as in figure \\ref{fig:woos94_wndprop}.}\n \\label{fig:jnk_wndprop}\n\\end{figure}\n\n\nIn contrast to the simulation run with the neutrino luminosities of\n\\citet{Woosley94}, the electron fraction continues to increase with\ntime.  The difference between the average electron neutrino energy and\nelectron anti-neutrino energy is, at most, about 3 MeV, compared to a\nmaximum of 8 MeV in the \\citet{Woosley94} calculations.  Also, the\ndifference between the average neutrino energies decreases as a\nfunction of time, compared to an increase with time in\n\\citet{Woosley94}.  Finally, the energies of all kinds of neutrinos\nare lower in the \\citet{Huedepohl09} calculation, so that the\nproton-neutron rest mass difference significantly suppresses the\nanti-neutrino capture rate relative to the neutrino capture rate.\nThese differences are presumably due to both the different neutron\nstar masses and neutrino interaction rates employed.\n\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\includegraphics[scale=0.65]  {fig13.eps}\n\\end{center}\n\\caption{Isotopic production factors from the NDW model employing \nthe neutrino luminosities from \\cite{Huedepohl09}.  The production \nfactors are calculated assuming that $7.4 \\, M_\\odot$ of material was \nejected in the supernova in addition to the wind.  \nThe horizontal lines are similar to those in figure \\ref{fig:WWsolopf}.\nNotice that none of the production factors are significantly greater \nthan one.}\n \\label{fig:jnkpf}\n\\end{figure}\n\nThe conditions in this model thus preclude {\\sl any} r-process\nnucleosynthesis, but they are potentially favorable for production of\nsome low mass p-process isotopes by the $\\nu$p-process.  The\nintegrated isotopic production factors are shown in figure\n\\ref{fig:jnkpf}.  The total ejected mass was take as 7.4 $M_\\odot$, as\n1.4 $M_\\odot$ neutron star is left behind in the calculation of\n\\cite{Huedepohl09}.  During the calculation a maximum network size of\n988 isotopes is reached.  The p-process elements $^{74}$Se and\n$^{78}$Kr are co-produced with $^{63}$Cu, $^{67}$Zn, and $^{69}$Ga,\nbut the maximum production factor for any isotope is 1 when weighted\nwith the total mass ejected in the supernova.  Therefore, in this\nsimple model, the proton-rich wind from low mass neutron stars \nwill not contribute significantly to galactic chemical evolution.\n\nThe entropies encountered when the mass loss rate is high are low\n($\\sim 50$), so that there is more production of $^{56}$Ni by\ntriple-alpha and a subsequent $\\alpha$p-process.  As the neutron\nabundance available for the $\\nu$p-process is given by\n\\begin{equation} \nY_n \\approx \\frac{\\lambda_\\nu Y_p}{\\rho N_A \\sum_i\nY_i \\avg{\\sigma v}_{i(n,p)j}}, \n\\end{equation} \nincreased seed production reduces the available neutron abundance and\ntherefore hinders production of the p-process elements $^{74}$Se,\n$^{78}$Kr, $^{84}$Sr, and $^{92}$Mo.  Additionally, at early times,\nthe dynamical time scale is short which implies a smaller integrated neutron \nto seed ratio, $\\Delta_n$ (see the appendix).        \n\nThe yields of from this model cannot be combined with the yields from \nthe rest of the supernova because they are not published.  As \\cite{Nomoto84}\nhas discussed, the mass inside the helium burning shell was close to the \nmass of the neutron star that was left after the explosion.  Therefore \nthe ejecta of the supernova is expected to have small production factors.\nThis implies that, even when the yields of the NDW are combined with the \nrest of the supernova, it is unlikely that these low mass core collapse\nsupernovae will contribute significantly to galactic chemical evolution.\n\n\\subsubsection{Effect of a Wind Termination Shock}\n\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\includegraphics[scale=0.65]  {fig14.eps}\n\\end{center}\n\\caption{Isotopic production factors from the NDW model employing \nthe neutrino luminosities from \\cite{Huedepohl09}, but including a time\ndependent external boundary pressure which results in a wind \ntermination shock.  The production \nfactors are calculated assuming that $7.4 \\, M_\\odot$ of material was \nejected in the supernova in addition to the wind.  \nThe horizontal lines are similar to those in figure \\ref{fig:WWsolopf}.\nNotice that the production factors are almost unchanged when an \nexternal boundary pressure is added.}\n \\label{fig:jnkpfpbd}\n\\end{figure}\n\n\nAs was mentioned above, it is very possible that a transonic wind solution \nmay not be appropriate this early in the supernovas evolution.  \\cite{Fischer09} \nhave found that a wind termination shock is not present in a one-dimensional\nsupernova model using the progenitor from \\cite{Nomoto84}.  Still, it is \ninteresting to consider the effect of a reverse shock on the wind nucleosynthesis.\n\nA second simulation was run with a time dependent boundary pressure given\nby equation \\ref{eq:Pbound}, with $E_{sn} = 10^{50} \\, \\textrm{erg}$ and \n$v_{sn} = \\sci{2}{9} \\, \\textrm{cm s}^{-1}$.  This results in a wind termination\nshock at a radius of approximately $\\sci{3}{8} \\, \\textrm{cm}$ throughout the \nsimulation.  Inside the wind termination shock the wind dynamics are very similar\nto those in the run with no boundary pressure.  The production factors from \nthis model are shown in figure \\ref{fig:jnkpfpbd}.  Clearly, there is almost no\ndifference in the nucleosynthesis in the runs with and without a wind termination\nshock.\n \nAfter $0.75 \\, \\textrm{s}$, the post shock temperature drops below 1 GK and \nthe wind termination shock has little effect on subsequent nucleosynthesis.\nBecause the post shock temperature is high for less than one second, the \nwind termination shock has very little effect on the integrated nucleosynthesis.  \nA larger explosion energy would likely result in a larger effect on the nucleosynthesis, \nbut there are still very few neutrons available to bypass the long lived waiting points\nand it seems unlikely that the production factors would be increased by \nmore than a factor of a few.\n\n\\section{Discussion}\n\\label{discussion}\n\n\\subsection{Comparison with SN 1987A}\n\nSince the progenitor model used in \\cite{Woosley94} was a model for SN\n1987A, it is interesting to compare our predicted abundances with\nthose observed in the ejecta of that event.  $^{88}$Sr, produced by\nthe NDW, dominates the elemental strontium yield when the results of\nthe 20 $M_\\odot$ wind model are combined with those predicted by\n\\cite{Woosley88}, who ignored the wind.  For the base NDW model, [Sr\/Fe]$=\n0.8$, if weak magnetism corrections are neglected, [Sr\/Fe]$= 1.6$; and\nif the anti-neutrino temperature is reduced in the base model by 15\\%,\n[Sr\/Fe]$=0.2$.\n\n\\cite{Mazzali92} found [Sr\/Fe]$\\approx 0.3$ in the ejecta of SN 1987A\n2-3 weeks after the explosion. This observation has a substantial\nerror bar due to the uncertainty of modelling the spectrum of the\nexpanding ejecta. Of even greater concern is comparing of our models\nfor bulk yields with supernova photospheric abundances observed on a\ngiven day when the observations are probably not even probing the\ninnermost ejecta.  Still, if one assumes that the [Sr\/Fe] ratio found\nby \\cite{Mazzali92} represents the value for all of the ejecta (for\ninstance by assuming that ``mixing'' was extremely efficient), a weak\nconstraint can be put on the neutrino fluxes and energies predicted by\nthe model of \\cite{Woosley94}. The NDW model with a reduced\nanti-neutrino temperature is much closer to the observed value of\n[Sr\/Fe] than the other two models.  This suggests the anti-neutrino\ntemperature may have been overestimated in the original model, a\nchange that would be more consistent with more modern calculations of\nneutrino spectra formation in PNS atmospheres \\citep{Keil03}. But\nobviously, a meaningful constraint will require a more complete modeling\nof the multi-dimensional explosion and time-dependent spectrum of SN\n1987A.\n\n\n\\subsection{Strontium and Yttrium in Halo Stars} \n\nSince strontium and yttrium are abundantly produced in our models, it\nmay be that the NDW has contributed to their production throughout\ncosmic history. An interesting possibility is that the abundances of\nthese elements might trace the birth rate of neutron stars at an early\ntime.  Taking a standard r-process abundance pattern from metal poor\nstars with strong r-process enhancments, \\cite{Travaglio04} find that\n8\\% and 18\\% of solar strontium and yttrium, respectively, are not\nproduced by either the ``standard'' r-process or any component of the\ns-process.  It therefore seems plausible that charged particle\nreactions in the NDW could make up this ``missing'' component.\n\nAny nucleosynthesis that happens in the NDW will be primary,\ni.e. provided that the mass function of neutron stars at birth does\nnot itself scale with metallicity, similar nucleosynthesis will occur\nfor stars of any population.  Below [Fe\/H]$\\sim -1.5$, no component of\nthe s-process contributes to the abundances of N = 50 closed shell\nisotopes \\citep{Serminato09}.  If the NDW escapes the potential well\nof the PNS, and contributes to the galactic budget of N = 50 closed\nshell isotopes, it should provide a floor to [Sr\/Fe] and [Y\/Fe].  Based\nupon the arguments of \\cite{Travaglio04}, this floor would be at\n[Sr\/Fe]$\\approx-0.18$ and [Y\/Fe]$\\approx -0.16$.  These numbers assume\nthat when the main r-process source contributes in addition to the NDW,\n[Sr\/Fe] and [Y\/Fe] approach their solar values even though the\ns-process has yet to contribute.  This is consistent with\nobservations.\n\nIn defining this floor, one must assume that the abundances in a\nparticular star sample a large number of individual supernovae.\nThis is because the production of N=50 closed shell elements likely\ndepends on the PNS mass and therefore the progenitor mass.  As we have\nfound, [Sr\/Fe]$=0.8$ in the $20 M_\\odot$ model with reduced\nanti-neutrino temperatures, but the $8.8 M_\\odot$ model produces no\nstrontium.  Observations show that below [Fe\/H]$\\sim -3$, the spreads\nin [Sr\/Fe] and [Y\/Fe] increase significantly and the mean values\nfalloff some \\citep{Francois07,Cohen08,Lai08}.  Single stars have\nvalues of [Sr\/Fe] below the predicted floor.  This could be because, \nat this metallicity, the metals in a particular\nstar come from only a handful of supernovae.\n\nAnother possible explanation of this variation is that supernova fall\nback varies with metallicity.  Since the NDW is the\ninnermost portion of the supernova ejecta, it will be the most\nsusceptible to fallback.  It has been found that the amount of\nsupernova fallback depends strongly on the metallicity of the\nprogenitor, especially going between zero and low metallicity\n\\citep{Zhang08}.  Additionally, mixing is also greatly reduced in zero\nmetallicity stars compared to solar metallicity stars due to the formers \ncompact structure \\citep{Joggerst09}. \n\nThe current understanding of supernova fallback suggests that the\nnucleosynthetic contribution of the NDW will be suppressed at very low\nmetallicity.  Of course, the ejection of iron by the supernova is also\nvery susceptible to fallback, so the effect of fallback on the\nevolution of [Sr,Y\/Fe] is complicated and may require fine tuning to\ngive the observed decrease.\nA somewhat different explanation was offered by \\citet{Qian08} who\nattributed the fall off of [Sr\/Fe] at low metallicity to the evolution\nof the ``hypernova'' rate with metallicity. For their purposes,\nhypernovae were stellar explosions that contributed iron without\nmaking much strontium. \n\nGiven the sensitivity of strontium and yttrium yields to uncertain NDW\ncharacteristics, especially neutrino fluxes and temperatures, it may\nbe some time before the complex history of these elements is even\nqualitatively understood. It is likely though that their abundances in\nhalo stars will ultimately be powerful constraints upon the evolution\nof supernova physics as a function of metallicity.\n\n\\subsection{Possible Modifications of the Basic Model}\n\\label{modifications}\n\nAs is clear from figures \\ref{fig:WWsolopf} and \\ref{fig:jnkpf}, the\nsimplest case of a non-magnetic non-rotating NDW from a neutron star\nwithout additional energy deposition does not produce r-process nuclei in\nsignificant abundances.  Are there extensions to this simple scenario\nthat {\\sl could} make the wind a site of the r-process?\n\nAs was pointed out by \\cite{Metzger07}, the combination of rotation\nand magnetic fields can decrease the dynamical time scale by magnetic\n``flinging''.  This is not particularly effective.  Adding a\nnon-thermal source of kinetic energy means that less thermal energy\nmust be put into the wind for it to escape the potential well.\nTherefore, lower entropies are achieved.  It seems unlikely that this\nmechanism, by itself, will salvage the NDW as a site for the full\nr-process.  If there were a way to make the rotation rate of the PNS\nhigh enough, it might be possible that there would be a centrifugally\ndriven outflow.  Then the electron fraction would be determined by kinetic\nequilibrium much deeper in the PNS envelope, and the material in the\noutflow would have an electron fraction much lower than that seen in\nthe wind.\n\nTo test this possibility, we ran calculations with a centrifugal force\nterm added and corotation with the PNS enforced out to $10^3$ km.\nUnfortunately, for reasonable PNS spin rates (20 ms period), \nwe found this had little effect on the nucleosynthesis.  These\ncalculations were in a regime were the electron fraction was still set\nby neutrino interactions.\n\nMany authors have discussed the possible effects of both\nmatter-enhanced\\citep{Qian95,Sigl95} and collective neutrino\n\\citep{Pastor02,Duan06} oscillations on NDW nucleosynthesis.  If\nelectron antineutrinos could undergo a collective oscillation near\nthe launch radius while the electron neutrinos did not, this would\nincrease the average energy of the antineutrinos if the $\\mu$\nand $\\tau$ neutrinos have a significantly higher temperature,\nfacilitating a reduction in the electron fraction.  For a normal mass hierarchy however,\nmatter enhanced neutrino oscillations would probably cause electron\nneutrino flavor conversion, which would {\\sl increase} the electron fraction\nand decrease the probability of significant r-process nucleosynthesis\n\\citep{Qian95}.\n\nCollective neutrino oscillations can cause antineutrino oscillations\nin the region were the electron fraction is set, and thereby decrease\nthe electron fraction where pure MSW oscillations would have predicted\nan increased electron fraction\\citep{Duan06}.  Clearly, the main\neffect of oscillations would be on the composition of the wind, not\nthe dynamics.  As can be seen in the neutrino two color plots,\noscillations would have to change the effective temperature of the\nanti-neutrinos by a very large amount to move from a region where N=50\nclose shell nucleosynthesis occurs to a region where the second\nr-process peak can be produced.\n\nThese effects are based upon the assumption that $\\mu$- and\n$\\tau$-neutrinos are significantly more energetic than the electron \nneutrinos.  In the calculation\nof \\cite{Woosley94}, this is the case, as can be seen in figure\n\\ref{fig:woos94_neut}.  Interestingly, the $\\mu$ and $\\tau$\ntemperatures are almost the same as the electron anti-neutrino\ntemperature in the \\cite{Huedepohl09} calculation, which can be seen\nin figure \\ref{fig:jnk_neut}.  It is not clear wether this difference\nobtains because of the difference in the PNS masses or the\nsignificantly different neutrino physics employed in the calculations.\nA detailed study of neutrino transport in static backgrounds showed\nthat the inclusion of all relevant neutrino interactions brings the\naverage energies of the $\\mu$- and $\\tau$- neutrinos closer to the\ntemperature of the anti-electron neutrinos \\citep{Keil03}.  Therefore\nit is uncertain wether or not neutrino oscillations could effect\nnucleosynthesis significantly.  Clearly, the uncertainties here are\nnot in the wind itself but in the formation of the spectra in the PNS\nand the details of neutrino transport with neutrino oscillations.\n\nFinally, it has been suggested \\citep{Qian96,Nagataki05} that adding a\nsecondary source of volumetric energy deposition can significantly\nincrease the entropy of the wind, which results in a more alpha-rich\nfreeze out and conditions that would be more favorable for r-process\nnucleosynthesis.  The addition of energy to the wind also decreases\nthe dynamical timescale.  Since the important quantity to consider for\nthe r-process is $s^3\/\\tau_d$ \\citep{Hoffman97}, \nboth effects increase the chance of having a significant neutron to\nseed ratio after freeze out.  If the NDW model is to be salvaged, this\nseems to us the minimal necessary extension.  Of course, the physical\nprocess contributing this extra energy is very uncertain.  One\npossibility is that oscillations of the PNS power sound waves\nwhich produce shocks and deposit energy in the wind, similar to the supernova\nmechanism of \\cite{Burrows06}, but smaller in magnitude.  We will\nexplore this possibility in some detain in a subsequent paper.\n\nFrom a chemical evolution standpoint, it is important to consider what\neffect neutron star mergers will have on the evolution of the\nr-process abundances.  It seems unavoidable that r-process nuclei will\nbe produced in the tidal tails ejected during these mergers\n\\citep{Freiburghaus99} and the amount of material ejected in these\nevents is approximately enough to account for the galactic inventory\nof r-process elements given the expected merger rate\n\\citep{Lattimer76,Rosswog99}.\n\nNeutron star mergers have been largely discounted because inferred\nmerger rates are small at low metallicity due to the long in spiral time\nand therefore they cannot account for the r-process enrichment seen in\nlow metallicity halo stars \\citep{Argast04}.  Of course, the inferred\nmerger rates are very uncertain as are models for the early evolution\nof the Milky Way, so that both the conclusion that neutron star\nmergers can account for the r-process inventory of the galaxy and that\nthey are not consistent with producing the r-process at low\nmetallicity are very uncertain.  Clearly, there is significant room\nfor more work in this area.\n\nTherefore, it seems reasonable that the galactic r-process abundances\ncould be accounted for by a combination of mergers and winds with an\nextra source of energy, either from an acoustically and\/or magnetically\nactive PNS.  Since not every supernova will have the requisite\nconditions for an r-process, there will be significant variation in\nthe yields from single supernovae.  This, along with the contribution\nfrom neutron star mergers, will give significant variation in the\n[r-process\/$\\alpha$-element] values found in single stars at low\nmetallicity but averaged over many stars these should track one\nanother, which is consistent with observations \\citep{Sneden08}.\n\n\\section{Conclusions}\n\nWe have performed calculations of the dynamics and nucleosynthesis in\ntime dependent neutrino driven winds.  This was done for two sets of\nneutrino spectra calculated in one-dimensional supernova models taken\nfrom the literature.  The nucleosynthesis in these models was compared\nwith supernova yields to determine if these models were consistent\nwith observations.  Additionally, we compared the results of these\nnumerical models to analytic models of the neutrino driven wind and\nfound good agreement.\n\nSimilar to most of the work on the NDW after \\cite{Woosley94}, we find\nthat it is unlikely that the r-process occurs in the neutrino driven\nwind unless there is something that causes significant deviation from\na purely neutrino driven wind.  Additionally, in the simplest case,\nthere is little production of p-process elements at early times in the\nwind.  In our calculation that used spectra from a more massive\nneutron star, the wind only produces the N=50 closed shell elements\n$^{87}$Rb, $^{88}$Sr, $^{89}$Y, and $^{90}$Zr.\n\nThis result is sensitive to small changes in the neutrino interaction rates \n(i.e. the inclusion of weak magnetism) and changes to the neutrino temperature \nof order 10\\%.  Comparing our models with the over abundance of strontium \nseen in SN 1987A suggests that the difference between the electron and \nanti-electron neutrino temperatures in the model of \\cite{Woosley94} \nmay have been to large.  We also find that the effect of a wind termination\nshock on the wind nucleosynthesis is small.\n\nUsing neutrino spectra from an $8.8 M_\\odot$ supernova that drives a\nwind which is proton rich throughout its duration \\citep{Huedepohl09}, we\nfind that no significant $\\nu$p-process occurs and the wind does not\ncontribute to the yields of the supernova. The neutrino spectra from\nthis model are probably more accurate than the spectra from the model\nof \\cite{Woosley94}.  We also investigated\nthe effect of an outer boundary pressure which resulted in a wind \ntermination shock.  This had a negligible effect on the nucleosynthesis.\n\nHowever, one also expects that the nucleosynthesis in the NDW will\nvary considerably from event to event, especially with the mass and\npossibly the rotation rate of the PNS. The winds from more massive PNS\nhave greater entropy and might, in general, be expected to produce\nheavier elements and more of them. The neutrino spectral histories of\nPNS as a function of mass have yet to be determined over a wide \nrange of parameter space.  Currently, the neutrino luminosities and \ntemperatures are the largest uncertainties in models of the NDW.\n\n\n\\begin{acknowledgements}\n\nWe would like to thank Alex Heger, David Lai, Enrico Ramirez-Ruiz,\nSanjay Reddy, and Yong-Zhong Qian for useful discussions about issues\nrelating to this work.  L. R. was supported by an NNSA\/DOE Stewardship\nScience Graduate Fellowship (DE-FC52-08NA28752) and the University of\nCalifornia Office of the President (09-IR-07-117968-WOOS).  S. W.  was\nsupported by the US NSF (AST-0909129), the University of California\nOffice of the President (09-IR-07-117968-WOOS), and the DOE SciDAC\nProgram (DEFC-02-06ER41438).  R. H. was supported by the DOE SciDAC\nProgram (DEFC-02-06ER41438) and under the auspices of the Department of \nEnergy at Lawrence Livermore National Laboratory under contract \nDE-AC52-07NA27344.\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Results and Discussion}\n\nWe first illustrate the method in some detail for the case $\\epsilon_{BB}=1.0$, and then present the general results. We also proceed to rigorously characterize the motifs and identify them in the lattice structures.\n\nIn order to name the different phases we searched the Material Project Database\\cite{Jain2013} to find a prototype isostructural phase and name the GA calculated lattice accordingly. If no match is found, then we name the phase according to the following convention:\n\\begin{equation}\\label{Eq:naming_convention}\n{\\mbox{A}_m \\mbox{B}_n}^{\\mbox{space group}}_{\\mbox{identifier}} .\n\\end{equation}\nHere $m$ and $n$ are the number of A and B particles within the unit cell. The space group is determined using the FINDSYM package\\cite{Stokes2005}, with the tolerance for lattice and atomic positions set to $0.05$. The identifier is necessary as multiple phases with the same stoichiometry and space group, differing only in Wyckoff number and positions, are found.\n\n\n\n\\begin{figure}[t]\n\\includegraphics[width=0.48\\textwidth]{fig1.png}\n\\caption{Structures searched by GA in $\\epsilon_{BB}=1.0$ and $\\gamma = 0.8$. (a) Formation energies ($E_{form}$) of structures searched by GA as a function of stoichiometry ($x$). Each point corresponds to a structure. The color of points are assigned by the type of motifs in the corresponding structure. The black solid line is the convex hull of the system, while the black dash line is the threshold for metastable structures. (b) Structure of the FCC motif (c) Structure of the MgZn$_2$ motif, which is Frank-Kasper Z$_{16}$.}\\label{Fig:epsilon_1}\n\\end{figure}\n\n\n\\textbf{The case $\\boldsymbol{\\epsilon_{BB}=1.0}$.} Here we consider $\\epsilon_{BB} = \\epsilon_{AA} = 1.0$, while $0.3 \\le \\gamma \\le 0.9$. We first compute the energy of the ground state for the pure A and B states, which previous calculations\\cite{Stillinger2001,Travesset2014} have shown to be the hcp phase. Here, however, because of the finite cut-off of LJ potentials, the fcc phase has lower energy. The identification of equilibrium phases proceeds by comparing their energy against phase separation into pure $A$ and $B$. Then, out this list of putative binary phases that are stable against phase separation, the energies are compared to establish the resulting true phase diagram equilibrium. This is how the phase diagram Fig.~\\ref{Fig:epsilon_1} is built, where there is only one stable BNSL, the MgZn$_2$ Frank-Kasper phase at $\\gamma=0.8$. We should note that maximum of the packing fraction for this phase occurs for $\\gamma_c=\\sqrt{2\/3}=0.8165$\\cite{Travesset2017a}, which is very close.\n\nSince it is common that structures that are metastable at 0 K can be observed in experiments at finite temperatures, we also considered metastable phases defined to be those within $0.1\\epsilon$\/particle in energy above the convex hull. As shown in Fig.~\\ref{Fig:epsilon_1}, there are a number of metastable phases at $x=0.333$, which are minor variations of MgZn$_2$ as we analyze further below in the context of motifs.\n\n\\textbf{General $\\epsilon_{BB}$.} On physical grounds, it is expected that the smaller the particle the weaker the interaction, hence we consider $\\epsilon_{BB} \\le 1$. In Fig.~\\ref{Fig:epsilon_gen_energy}, we provide a typical calculation for fixed $\\gamma=0.6$ as a function of both $\\epsilon_{BB}$ and $x$. As expected, see  Fig.~\\ref{Fig:epsilon_1}, the phase diagram is trivial for $\\epsilon_{BB}=1$. However, three phases TiCu$_3$, AlB$_2$ and CrB at $x=0.25, 0.333, 0.5$ are found for $\\epsilon_{BB}= 0.8$.\n\nBy repeating the calculations shown in Fig.~\\ref{Fig:epsilon_gen_energy} for the other values of $\\epsilon_{BB}$ at a fixed $\\gamma=0.6$ (see Table~\\ref{tab:my_label}), we constructed the phase diagram shown in Fig.~\\ref{Fig:epsilon_gen}. In Fig.~\\ref{Fig:epsilon_gen} we note the appearance of seven additional phases for $\\epsilon_{BB}< 0.6$ that could not be matched to any prototype: Detailed description for these and all other equilibrium phases are collected in Supporting Information Table S1. A database for all the structures is included in Supporting Information. \n\n\n\n\nSimilarly, the phase diagrams for all other values of $\\gamma$ are also presented in Supporting Information Fig. S2. Common to all these phase diagrams is the appearance of many diffusionless (martensitic), usually incongruent transformations, as a function of the  energy parameter $\\epsilon_{BB}\/\\epsilon_{AA}$. In Supporting Information Fig. S3, we have also included phase diagrams for all values of $\\epsilon_{BB}\/\\epsilon_{AA}$ in $x$ and $\\gamma$.\n\n\\textbf{Motifs.} We define motifs as the polyhedron consisted of a center particle and its first-shell neighbors. The motifs are generated according to the analysis of bond length table from neighboring particles to the center (see details in Supporting Information Fig. S6). In this study, we only include motifs with the larger A-particles as the center. We will name motifs according to\n\\begin{equation}\\label{Eq:motif_name}\n    \\mbox{Motif}-\\mbox{CN}-\\mbox{Identifier} \\ ,\n\\end{equation}\nwhere CN is the Coordination (the number of particles) and identifier discriminates among motifs with the same coordination number.\n\n\\onecolumngrid\n\n\n\n\\begin{figure}[b]\n    \\includegraphics[width=1.0\\textwidth]{fig2.pdf}\n    \\caption{Two examples of GA results for $\\gamma = 0.6$. In each figure, the solid line is the convex hull, while the dashed line is the threshold for metastable structures, see the discussion above. (a) Structures searched by GA as a function of x when $\\epsilon_{BB} = 1.0$: There are no stable binary structures between x=0 and x=1. (b) Structures searched by GA as a function of x for $\\epsilon_{BB}=0.8$. There are three stable structures which appear at $x = 0.25$ (TiCu$_3$),  $x = 0.333 $ (AlB$_2$) and $x = 0.5$ (CrB).}\\label{Fig:epsilon_gen_energy}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[]\n\\includegraphics[width=1.0\\textwidth]{fig3.pdf}\n\\caption{Phase diagram in $x$ and $\\epsilon_{BB}\/\\epsilon_{AA}$ for $\\gamma = 0.6$.}\\label{Fig:epsilon_gen}\n\\end{figure}\n\n\\twocolumngrid\n\n\nWe identified 187 equilibrium and 102,822 metastable structures. Out of the 187 equilibrium structures, we removed redundancies by a cluster alignment algorithm\\cite{Fang2010,Sun2016a}leading to only 53 equilibrium structures. Out of these 53 structures we identified 42 motifs, which are listed in the order of increasing CN in Supporting Information Fig. S4. 416,391 motifs can be found in the 102,822 metastable structures. Among them, a vast majority (312,891) of the motifs of the metastable structures also exist in the equilibrium phases. In Tab~\\ref{tab:motif}, we list the name, CN and the percentage fraction of the ten most frequent motifs present in meta-stable structures. Note that these ten already account for more than 95\\% of the 312,891 motifs. The details about how to identify the motif from a crystal and how to identify if a crystal has the motif inside have been included in the Supporting Information.\n\n\n\n\n\\begin{table}[h]\n    \\centering\n    \\caption{Ten most frequent Motifs in metastable structures}\n    \\begin{tabular}{c|c|c}\n    \\hline\n    \\hline\n        Motif & CN & Frequency \\\\ \\hline\n        FCC & 12 & 31.4\\% \\\\ \\hline\n        HCP & 12 & 18.9\\% \\\\ \\hline\n        Octahedron (Motif-6-4) & 6 & 9.8\\% \\\\ \\hline\n        Half Hexagonal Prism 1 (Motif-6-2) & 6 & 9.1\\% \\\\ \\hline\n        Triangular Prism (Motif-6-3) & 6 & 6.4\\% \\\\ \\hline\n        Half Hexagonal Prism 2 (Motif-6-1) & 6 & 5.3\\% \\\\ \\hline\n        BCC & 8 & 5.0\\% \\\\ \\hline\n        Hexagonal Prism (Motif-12-3) & 12 & 4.8\\% \\\\ \\hline\n        Half Truncated Cube (Motif-12-1) & 12 & 2.3\\% \\\\ \\hline\n        MoB (Motif-13-1) & 13 & 2.2\\% \\\\ \\hline\n        Total & & 95\\% \\\\ \\hline\n        \\hline\n    \\end{tabular}\n    \\label{tab:motif}\n\\end{table}\n\nAs an illustrative example, we consider the case of $\\epsilon_{BB}=1$ and $\\gamma=0.8$, where in Fig.~\\ref{Fig:epsilon_1} we have shown the two relevant motifs are the FCC and the Frank-Kasper Z$_{16}$.\nBy coloring each structure according to the motif, we can confirm that the metastable phases (all in red) have motifs which are small variations of the Frank-Kasper Z$_{16}$ and that the vast majority of the structures found in other searches have motifs which are variations of either FCC or Frank-Kasper Z$_{16}$.\n\n\\onecolumngrid\n\n\\begin{figure}[]\n\\includegraphics[width=0.7\\textwidth]{fig4.png}\n\\caption{(a) Map of MgZn$_2$ in $\\gamma$ and $\\epsilon_{BB}$. The red regime indicates that the structure of MgZn$_2$ is thermodynamically stable while in blue regime it is metastable. (b) Map for the Z$_{16}$ motif with red stable, blue metastable. The red regime is where the stable structure has Z$_{16}$ motif inside. Note that the motif has a wider range of both stability and metastability, as it also appears in other Laves phase, such as the MgCu$_2$ and MgNi$_2$.}\\label{Fig:mgzn2_motif_map}\n\\end{figure}\n\n\\twocolumngrid\n\n\\onecolumngrid\n\n\\begin{figure}[]\n    \\includegraphics[width=0.8\\textwidth]{fig5.png}\n    \\caption{Map for first four frequent motifs in $\\gamma$ and $\\epsilon_{BB}$ excludes the general motif FCC and HCP. (a) Octahedron (b) Half hexagonal prism 1. (c) Triangular Prism (d) Half hexagonal prism 2. Red indicates stable structures and blue indicates metastable.\n    }\\label{Fig:req_motif}\n\\end{figure}\n\n\\twocolumngrid\n\nIn Fig.~\\ref{Fig:mgzn2_motif_map} we show the domain of stability and metastability for the MgZn$_2$ phase and the Z$_{16}$ motif. The GA searches were performed on a mesh of $\\gamma$ and $\\epsilon_{BB}$ with an increment of 0.1. Here, to improve the resolution of the stability range, we examined the stability of all GA-found structures and motifs on a finer mesh in the $\\gamma$-$\\epsilon_{BB}$ plane with an increment of 0.02. Rather interestingly, the stability range of the Z$_{16}$ motif is larger than that of the MgZn$_2$ phase, indicating this motif is not unique to MgZn$_2$, but shared by other Laves and Frank-Kasper phases. Similar plots for the four more frequent motifs are shown in Fig.~\\ref{Fig:req_motif}.\n\n\n\n\\begin{table}[h]\n    \\centering\n    \\caption{Comparison between packing phases\\cite{Hopkins2012} and our study for $ 0.3 \\le \\gamma  \\le 1$. (SG=Space group), The $\\ast$ indicates there are small distortions in the LJ phase, compared with the packing phase. Motifs in the LJ column indicates that they are not stable in the GA result, but they have the motif inside in the corresponding $\\gamma$ regime.}\n    \\begin{tabular}{c c c c | c c}\n    \\hline\n    \\hline\n     Phase   & $\\gamma$-range        &  Ref  &  SG &  LJ & Distortion \\\\ \\hline\n     A$_3$B  & $[0.618, 0.660]$ & \\cite{OToole2011} &  59  & TiCu$_3$  &  \\\\\n     AlB$_2$  & $[0.528, 0.620]$  &       & 191 & AlB$_2$ & \\\\\n     AuTe$_2$ & $[0.488, 0.528]$ & \\cite{Filion2009} & 12 & Motif-6-2 &   \\\\\n     (2-2)$^{\\ast}$ & $[0.480, 0.497]$ &\\cite{Marshall2010} & 11 & Motif-6-1 & \\\\\n     (4-2) & $[0.488, 0.483]$ &\\cite{Hopkins2012} & 191 & Motif-12-3 & \\\\\n     (5-2) & $[0.480, 0.483]$ &\\cite{Hopkins2012} & 44 & & \\\\\n     (7-3) & $[0.468, 0.480]$ &\\cite{Hopkins2012} & 71 & Motif-12-3 & \\\\\n     HgBr$_2$ & $[0.443, 0.468]$ &\\cite{Filion2009} & 36 & Motif-6-4 & \\\\\n     (6-6) & $[0.414, 0.457]$ &\\cite{Hopkins2012} & 11 & Motif-6-4 &  \\\\\n     XY & $[0.275, 0.414]$ &\\cite{Hopkins2012} &  & & \\\\\n     (6,1)$_4$ & $[0.352, 0.321]$ &\\cite{Hopkins2012} & 69 & ${\\mbox{A}_2\\mbox{B}_{12}}^{(139)}_{(1)}$ & $\\ast$ \\\\\n     (6,1)$_6$ & $[0.321, 0.304]$ &\\cite{Hopkins2012} & 139 & ${\\mbox{A}_2\\mbox{B}_{12}}^{(139)}_{(1)}$ & $\\ast$ \\\\\n     (6,1)$_8$ & $[0.302, 0.292]$ &\\cite{Hopkins2012} & 139 & ${\\mbox{A}_2\\mbox{B}_{12}}^{(139)}_{(1)}$ & $\\ast$ \\\\\n     \\hline\\hline\n     \\\\\n    \\end{tabular}\n\n    \\label{tab:pg_lj_comp}\n\\end{table}\n\nQuite generally, the motifs are far more sensitive to $\\gamma$ than they are to $\\epsilon_{BB}\/\\epsilon_{AA}$, confirming that the particle size is more important than the actual intensity of the interactions. It is consistent with all calculations that stable structures with the same values of $\\gamma$ tend to share motifs. As found for MgZn$_2$ and Z$_{16}$, the regions for stability and metastability is wider than the corresponding structures, thus indicating that motifs define very general families of structures, like Laves phases. A classification of motifs by Renormalized Angle Sequences (RAS)\\cite{Lv2017,Lv2018} has been included in Supporting Information. \n\nThis study has identified 53 equilibrium lattices and 42 motifs (with the larger particle A as reference). We now discuss the relevance of these results for packing models\\cite{Filion2009,Hopkins2012}, their connection to the motifs reported in Quasi Frank-Kasper phases\\cite{Travesset2017} and their implications for binary superlattices.\n\n\n\\textbf{Packing Phase Diagram.} We consider the study of Hopkins \\textit{et al.}\\cite{Hopkins2012} as the reference phase diagram for packing problems, although it only includes unit cells containing up to 12 particles. Consistently with this study we concentrate on the range $0.3\\le \\gamma \\le 1$, also because for smaller $\\gamma$ there are many phases with narrow stability ranges that are less relevant in actual experimental systems.\n\n\n\nFrom Table~\\ref{tab:pg_lj_comp}, the packing of binary phase diagram contains 13 phases for the $0.3 \\le \\gamma \\le 1$ range. For large $\\gamma > 0.528$ only two phases exist; AlB$_2$ and A$_3$B, which are both found in binary LJ systems  (if allowing for small differences in A$_3$B). For $0.488 < \\gamma < 0.528$, however, the AuTe$_2$ phase is reported; We did not find such phase, but we do report the Motif-6-2 as stable for the same range of $\\gamma$, see Supporting Information, which is present in the equilibrium phases at $\\gamma=0.5$ ${{\\mbox A}_4{\\mbox B}_6}^{(166)}_{(9)}$, BaCu and TePt. Some other phases, which are reported as packing phases \\cite{Hopkins2012} but not stable in the GA search, are also identified to have the motif in the corresponding $\\gamma$ regime. This indicates that these packing phases may be meta-stable in our calculation. For smaller $\\gamma$, there is also overlap if allowing for small distortions. \n\nOther phases that have large packing fractions, such as CrB and S74e\/h(KHg$_2$ in our notation)\\cite{Filion2009}, that are metastable in the packing phase diagram become equilibrium, thus showing that the LJ system augments the number of stable phases as compared with packing models.\n\n\\textbf{Motifs and Quasi Frank Kasper Phases.} In Ref.~\\cite{Travesset2017} it was shown that all experimental BNSLs could be described as disclinations of the $\\{3,3,5\\}$ polytope, thus generalizing well known four Frank-Kasper motifs Z$_{12}$,Z$_{14}$,Z$_{15}$, Z$_{16}$\\cite{Frank1958,Frank1959} to include other motifs. \n\n\\onecolumngrid\n\n\\begin{table}[]\n    \\centering\n    \\caption{Motifs in Quasi Frank Kasper phases\\cite{Travesset2017} compared to the ones described in this work.} \n    \\begin{tabular}{c | c c c c c c}\n    \\hline\\hline\n    QFK\\cite{Travesset2017}  & $\\mbox{Z}_6$ & $\\mbox{Z}_{12}^{\\prime\\prime}$ & $\\mbox{Z}_{14}^{\\prime}$  & $\\mbox{Z}_{16}$ & $\\mbox{Z}_{18}^{\\prime\\prime}$ & $\\mbox{Z}_{24}$ \\\\\n    \\hline\n    This work & Motif-6-4 & Motif-12-2 & Motif-14-1 & Motif-16-2 & Motif-18-3 & Motif-24-1 or  \\\\\n              &        &         &         &         & & Motif-24-3 \\\\\n    \\hline\n    \\hline\n\n    \\end{tabular}\n    \\label{tab:qfk_lj_comp}\n\\end{table}\n\n\\twocolumngrid\n\nIn Table~\\ref{tab:qfk_lj_comp} we show the equivalence between Quasi Frank Kasper motifs and the ones obtained in this work, which only include those with the A-particle as reference. It should be pointed that the motifs are not completely the same, as in Ref.~\\cite{Travesset2017} the motifs were defined by the Voronoi cell and its corresponding neighbors, which is a slightly different definition than the one used in this paper. \n\n\\textbf{Experimental Results.} The list of experimentally reported BNSLs is taken from Ref.~\\cite{Travesset2017a}, where we have excluded two dimensional superlattices and those where nanocrystals cannot be approximated as spherical, see Ref.~\\cite{Boles2016}. The comparison between the results obtained in this paper and experimental BNSLs is provided in Table~\\ref{tab:exp_lj_comp}.\n\n\\begin{table}[]\n    \\centering\n    \\caption{Experimentally determined structures. NA: Phase not available in this study. NF: Phase not found in this study. The DDQC\/AT is a quasicrystal phase. The bccAB$_6$ phase is also known as C$_{60}$K$_6$ and is denoted as ${\\mbox{A}\\mbox{B}_6}^{(229)}_{(1)}$ in this paper.} \n    \\begin{tabular}{c c | c c c}\n    \\hline \\hline\n    \\multicolumn{2}{c|}{Experiment} &  \\multicolumn{3}{c}{Binary LJ}\\\\ \\hline\n    BNSL & $\\gamma$-range &     & $\\gamma$-range & $\\varepsilon_{BB}$-range  \\\\\n    NaCl &  $[0.41,0.60]$  &    &  $[0.2,0.5]$    & $[0.1,0.8]$     \\\\\n    CsCl &  $[0.71,0.90]$  &    &   NF   &      \\\\\n    AuCu &  $[0.58,0.71]$  &    &    NF  &      \\\\\n    DDQC\/AT&  $[0.41,0.43]$  &    &  NA   &      \\\\\n    AlB$_2$ &  $[0.45,0.70]$  &    & $[0.4,0.7]$    & $[0.1,0.9]$     \\\\\n    MgZn$_2$ & $[0.60,0.81]$  &    & $[0.7,1.0]$     & $[0.1,1.0]$  \\\\\n    AuCu$_3$ & $[0.40,0.60]$  &   &   NF   &      \\\\\n    Li$_3$Bi & $[0.53,0.56]$  &   &   NF   &      \\\\\n    Fe$_4$C  & $[0.55,0.65]$  &    &   NF   &     \\\\\n    CaCu$_5$ & $[0.60,0.80]$  &    &  $[0.6,0.8]$    & $[0.1,0.9]$    \\\\\n    CaB$_6$  & $[0.43,0.47]$  &    & $[0.3,0.5]$    & $[0.1,0.8]$    \\\\\n    bccAB$_6$  & $[0.45,0.50]$  &    & $[0.4,0.6]$      & $[0.1,0.5]$     \\\\\n    cubAB$_{13}$  & $[0.55,0.60]$  &   &  NF    &     \\\\\n    NaZn$_{13}$  & $[0.47,0.70]$  &    & $[0.6]$  & $[0.1,0.6]$     \\\\\n    \\hline \\hline\n    \\end{tabular}\n    \\label{tab:exp_lj_comp}\n\\end{table}\n\nSeven of the experimentally reported BNSLs, namely NaCl, AlB$_2$, MgZn$_2$, CaCu$_5$, CaB$_6$, bccAB$_6$ and NaZn$_{13}$ are found as equilibrium phases in the LJ system essentially for the same range of $\\gamma$. The fact that in our results the stability is roughly independent of $\\varepsilon_{BB}$ in certain regions provides some support for the idea that microscopic details of the potential are unimportant in this region (``universality''). Further making this point is that the same phases are stable for soft repulsive potentials in the same $\\gamma$-range \\cite{Travessetpnas2015,HorstTravesset2016,LaCour2019}. \n\nWe now analyze the phases reported in experiments that are not equilibrium in our study. One of them is beyond the scope of our calculation; DDQC\/AT, which is a quasicrystal. The Li$_3$Bi and also the AuCu$_3$ are stabilized by large deformations of the ligands, i.e. vortices\\cite{Travesset2017}, and therefore are not possible to obtain from a quasi HS approximation. The Fe$_4$C phase was observed in 2006\\cite{Shevchenko2006}, and since then, it has not been reported in any further study, which may suggest is metastable, and furthermore, it can only be stabilized by vortices\\cite{Travesset2017a}. The CsCl phase has a very narrow range of stability around $\\gamma_c = \\sqrt{3}-1=0.732$\\cite{Travesset2017a}, which is likely missed by the discretization of $\\gamma$ values in our study.\nFinally, AuCu occurs when there is ligand loss\\cite{Travesset2017,Boles2019} and is stabilized through a different mechanism involving the non-spherical shape of the nanocrystal. We therefore conclude that the binary LJ model successfully predicts those experimentally reported phases that can be described as quasi-hard spheres. This is in contrast to packing models, where MgZn$_{2}$ or CaCu$_5$ phases, widely reported in experiments are not equilibrium phases (maximum of the packing fractions). See Fig.~\\ref{Fig:summary} for a visual summary of this discussion.\n\n\\begin{figure}[]\n    \\includegraphics[width=0.48\\textwidth]{fig6.pdf}\n    \\caption{Summary of the main results of the paper: The experimental phases are classified according to: Hard sphere, OTM\/hard sphere (exist when NCs are modeled as hard spheres but are stabilized by vortices)\\cite{Travesset2017}, pure OTM(only stable with vortices), and other (observed in special cases, such as ligand detachment\\cite{Boles2019}). See also Table~\\ref{tab:exp_lj_comp}. Consistent with the LJ assumptions, only the hard sphere phases are found in our work. The Experiment Pred includes those strong candidates to be found experimentally, as discussed below.}\\label{Fig:summary}\n\\end{figure}\n\n\n\n\\section{Conclusions}\n\nBy the use of Genetic Algorithm (GA), we have been able to predict stable structures under different sizes of particles and strengths of interaction ($\\gamma \\in$  [0.3 to 0.9], $\\epsilon_{BB} \\in$ [0.1 to 1.0]). We report 53 stable phases, which cover a significant part of currently reported structures.  Besides that, we also predict 35 stable structures which are not in Material Project database. We find that the type of stable structures strongly depends on $\\gamma$, but weakly on $\\epsilon_{BB} < 1$, providing evidence that the stability of the lattices has a weak dependence on the potential details (universality). By comparing our results with other theoretical and experimental works, it is shown that regardless of potential details, the same $\\gamma$ regime has the same stable structure, which reinforce that the stable structure has a weak dependence on the potential details.\n\n\nThere are two aspects about the limitations of the hard sphere description: The first is that it does \\textit{not} provide a free energy: the observed phases are not the ones with maximum packing fraction\\cite{Hopkins2012}, but rather, ones where the packing fractions is maximum for the particular structure. This is where the Binary LJ becomes important: the stable phases are the ones that minimize the free energy (modeled as the LJ potential). The second limitation is that it does not model large deformations of the ligand shell: these cases go beyond the LJ model and is evident from Fig.~\\ref{Fig:summary}, showing that these phases are absent.\n\nThe crystalline motifs are employed to describe the large amount of metastable structures. We find that metastable structures mostly can be described from the motifs present in equilibrium structures, thus suggesting the possibility of building superlattices by patching all motifs that can tile the 3D space, as similarly done in the more restricted case of Frank-Kasper phases\\cite{DutourSikiric2010}. It also raises the possibility of motifs being present within the liquid\\cite{Damasceno2012} as a way to anticipate the emergent crystalline structure.\n\nComparing with available experimental results, see Table~\\ref{tab:exp_lj_comp} and Fig.~\\ref{Fig:summary}, the binary LJ model captures all the equilibrium phases where nanocrystals can be faithfully described as quasi hard spheres: NaCl, AlB$_2$, MgZn$_2$, CaCu$_5$, CaB$_6$, bccAB$_6$ and NaZn$_{13}$. The other phases reported in experiments either require the presence of vortices, as predicted by the OTM\\cite{Travesset2017,Travesset2017a}, or are stable over a very narrow range of $\\gamma$ values, likely missed by the necessary discrete number considered in our study.\n\nPacking phase diagram models reported 14 equilibrium phases in the interval $\\gamma \\in [0.3,1)$, see Table~\\ref{tab:pg_lj_comp}, while our study reports 53, thus showing that binary LJ have a more complex phase diagram. Rather interestingly, phases such as MgZn$_2$ or CaCu$_5$, which are very common in experiments, are absent in the packing phase diagram; Although very useful in identifying at which $\\gamma$ values a phase is likely to appear, packing models give very poor predictions on which, among all possible phases, will actually be observed.\n\nThe two guiding principles for stability of BNSLs in experiments are high packing fraction (or low Lennard-Jones Energy) and tendency towards icosahedral order, as reflected in the motifs\\cite{Travesset2017a,Coropceanu2019}. Therefore, we expect that those equilibrium Lennard-Jones phases with Quasi Frank-Kasper motifs, \nfor example, the BNSLs ${\\mbox{A}_{2}\\mbox{B}_{4}}^{(227)}_{(1)}$ and\n${\\mbox{A}_{2}\\mbox{B}_{12}}^{(139)}_{(1)}$\n(Motif-16-2), or $\\mbox{Zr}_{2}\\mbox{Cu}^{(139)}_{(1)}$ (Motif-14-1), will be excellent candidates to search for BNSLs,  see Fig.~\\ref{Fig:summary}. Definitely, these ideas will be developed further in the near future, where the 53 stable lattices will be studied with more realistic nanocrystal models described at the atomic level. \n\nIn this work we focused on spherically symmetric potentials with additive interactions, as described by relations like\n\\begin{equation}\n\\epsilon_{AB}=\\frac{1}{2}(\\epsilon_{AA}+\\epsilon_{BB}) \\ .\n\\end{equation}\nIt is of interest to consider more general models, where these restrictions are lifted. This, however, will be the subject of another study.\n\n\\section{Methods}\nThe crystal structure searches with GA were only constrained by stoichiometry, without any assumption on the Bravais lattice type, symmetry, atom basis or unit cell dimensions (up to a maximum of particles per unit cell). During the GA search, energy was used as the only criteria for optimizing the candidate pool. At each GA generation, 64 structures are generated from the parent structure pool \\textit{via} the mating procedure described in Ref.~\\cite{Deaven1995,Oganov2006,Ji2010}. The mating process was based on real-space \"cut-and-paste\" operations that was first introduced to optimize cluster structures~\\cite{Deaven1995}. This process was extended to predict low-energy crystal structures by Oganov~\\cite{Oganov2006} and reviewed in Ref.~\\cite{Ji2010}. Here, we follow the same procedure that was described in detail in Ref.~\\cite{Ji2010} and was implemented in the Adaptive Genetic Algorithm (AGA) software.\n\nWith a given set of LJ parameters, we performed three GA searches independently, with each GA search running for 1000 generations. The maximum number of particles per unit cell used in each search was 20, and thus, phases with large unit cells, the most relevant being NaZn$_{13}$, could not be included. Therefore, we include NaZn$_{13}$ into our calculation manually. All energy calculations and structure minimizations were performed by the LAMMPS code \\cite{Plimpton1995} with some cross checks using HOOMD-Blue\\cite{AndersonMe2008a} with FIRE minimization\\cite{Bitzek2006}. The database of binary lattices in HOODLT\\cite{Travesset2014} was also used.\n\n\n\\section{Supporting Information}\nSupporting information contains: \nList and maps of structures searched by genetic algorithm;\nphase diagrams of equilibrium structures; equilibrium motif database; maps of motifs; algorithms for motif identification and renormalized angle sequence\n\n\\section{acknowledgement}\n\nA.T acknowledges discussions with I. Coropceanu and D. Talapin. We also thank Prof. Torquato for facilitating the data of his group packing studies. Work at Ames Laboratory was supported by the US Department of Energy, Basic Energy Sciences, Materials Science and Engineering Division, under Contract No. DE-AC02-07CH11358, including a grant of computer time at the National Energy Research Supercomputing Center (NERSC) in Berkeley, CA. The Laboratory Directed Research and Development (LDRD) program of Ames Laboratory supported the use of GPU-accelerated computing. Y. S. was partially supported by National Science Foundation award EAR-1918134 and EAR-1918126.\n\n\n\\bibliographystyle{apsrev4-1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\\setcounter{page}{2}\n\n\nAn interesting application of gauge\/gravity duality to condensed matter physics arises in the study of \nmomentum relaxation. This is so mainly because the resulting zero frequency conductivities are finite, \nallowing us to study transport in a more realistic way.\nTo this end, one must find gravitational solutions which break translational \ninvariance along the boundary directions due to the presence of one or more spatially dependent sources. \nGenerically this involves numerically solving the Einstein equations which give an elliptic PDE problem in this context.\nSome studies focus on configurations which describe a lattice in the dual field theory, for instance when the chemical potential is a periodic function \nof a spatial direction on the boundary.\nGravitational solutions of this kind have been successfully constructed \n\\cite{Horowitz:2012ky, Horowitz:2012gs, Horowitz:2013jaa, Donos:2014yya, Rangamani:2015hka}\nand they indeed reproduce the expected\nlow frequency dynamics, i.e.\\ the zero frequency delta functions in the conductivities are resolved into finite width\nDrude peaks.   \n\nA considerable technical simplification arises if a certain global symmetry is present, say, in the matter sector \nof the bulk theory. Then, translational invariance can be broken along that symmetry direction while \npreserving homogeneity of the geometry, which in turn \nimplies that the construction of such solutions only requires solving ODEs. Examples of this kind include\n\\cite{Donos:2012js, Donos:2013eha}, which have been shown to yield finite DC conductivities as well as to possess a rich \nstructure which displays transitions between different metallic and \ninsulating regimes.\\footnote{A simplification of this type arises in holography with massive gravity in the bulk \\cite{Vegh:2013sk}.}\n\n\nWe can  simplify the problem even further by arranging the bulk matter content in such a way that the \nblack branes of interest are not only homogeneous but also isotropic. This fact was exploited in \\cite{Andrade:2013gsa} \n(see also \\cite{Taylor:2014tka}), which considered a particular configuration of a set of massless scalars, termed linear axions since they \nare shift-symmetric, that allows for an analytical black brane \nsolution at non-zero chemical potential in an arbitrary number of dimensions. As expected, the DC \nelectric and thermal conductivities are finite \\cite{Andrade:2013gsa, Gouteraux:2014hca,Davison:2014lua,Donos:2014cya}, and they can be evaluated \nanalytically. However, the conductivities at non-zero frequency need to be computed numerically since this \ninvolves solving coupled fluctuation equations around the background solution. Interestingly, it has been noted that \ndeviations from Drude physics are present when the strength of momentum relaxation is large \n\\cite{Kim:2014bza, Davison:2014lua, Davison:2015bea}. \n\n\nRecently, it has been noted that the equations of General Relativity simplify considerably in the limit in \nwhich the number of spacetime dimensions, $D$, is taken to be large, which provides an efficient tool to \napproximate finite $D$ results, as a perturbative calculation in $1\/D$  \\cite{Emparan:2013moa}. \nThe main ingredient of the construction is the fact that when the number of spacetime dimensions is large, \nthe gravitational potential becomes very steep near the horizon which yields to a natural separation of the \ndynamics that localise near the horizon and the ones that probe regions far away from it. \nIn particular, this implies that the quasi-normal mode (QNM) spectrum splits into near-horizon {\\it decoupled} modes,\nand {\\it coupled} modes that are delocalised \\cite{Emparan:2014cia, Emparan:2014aba, Emparan:2015rva}. \nAs explained in these references, while the coupled modes are quite generic, i.e.\\ \nshared by many black holes, the decoupled modes are sensitive to the particularities of different \nsolutions. \n\nIn this paper we initiate the study of holographic inhomogeneities and the resulting momentum relaxation using large $D$ techniques.\nOur first goal is to study the decoupled quasi-normal modes which control the characteristic decay rate of the  \nelectric and thermal conductivities in the linear axion model at non-zero chemical potential. \nWe find that momentum conservation is restored unless we scale the strength of the axions with $D$, i.e.\\ the decoupled QNM frequency vanishes to leading order. \nTherefore we scale the axion strength appropriately and obtain a QNM capturing the momentum decay rate at leading order in large $D$. We calculate corrections to this in $1\/D$.\nWe see that in certain regimes the QNM frequencies are well-described using the large $D$ approximation. \nSecond, we will compute the AC thermal conductivity for zero chemical potential, which at leading order is exactly of Drude form. These results are consistent with the corresponding QNM calculation. To our knowledge, this is the first \nanalytical realisation of Drude behaviour outside of the hydrodynamic regime in the context of holography.\nWith these results at hand, we will comment on the signature of the transition from coherent to incoherent regimes, \ni.e.\\ the breakdown of Drude physics, in the large $D$ approximation.   \n\nThis paper is organised as follows. In section \\ref{sec:model} we review the axion model, its transport properties and the appropriate master fields for the conductivity calculation. In section \\ref{sec:QNM} we compute the large $D$ decoupled QNMs which control momentum relaxation, giving analytical expressions for their frequencies. In section \\ref{sec:conductivity} we compute the AC thermal conductivity as an expansion in $1\/D$. We conclude in section \\ref{sec:conclusions}. \n\n\n\\section{Momentum relaxation in arbitrary D}\n\\label{sec:model}\n\n\nIn this section we review the holographic model of momentum relaxation proposed in~\\cite{Andrade:2013gsa}. \nWe will discuss the main properties of the background solution and describe the computation of the two-point \nfunctions using a gauge invariant master field formalism. \n\n\n\\subsection{Linear axion background}\n\\label{sec:axion}\n\n\nThe holographic model of momentum relaxation in $D = n+3$ bulk dimensions which we consider throughout this \npaper is given by the action \\cite{Andrade:2013gsa}\n\\begin{equation}\\label{S0}\n\tS_0 = \\int d^{n+3} x \\sqrt{- g} \\left( R + (n+1)(n+2) \\ell^{-2} - \\frac{1}{4} F^2 - \\frac{1}{2} \\sum_{I=1}^{n+1} (\\partial \\psi_I)^2  \\right)\n\\end{equation}\n\\noindent where $F = dA$ is the field strength of a $U(1)$ gauge field, $\\psi_I$ are $(n+1)$ massless scalar fields \nand $\\ell$ is the AdS radius which we set to one henceforth. \n\nThis model admits the following analytical black brane solution\\footnote{This solution was previously derived in \\cite{Bardoux:2012aw} \nin a different context.}\n\\begin{equation}\\label{NL ansatz}\n\tds^2 = - f(r) dt^2 + \\frac{dr^2}{f(r)} + r^2 \\delta_{a b} dx^a dx^b, \\qquad A = A_t(r) dt, \\qquad \\psi_I = \\delta_{I a} x^a\n\\end{equation}\n\\noindent where $a$ labels the $(n+1)$ boundary spatial directions $x^a$ and\n\\begin{align}\n\tf(r) &= r^2 - \\frac{\\alpha^2}{2 n} - \\frac{m_0}{r^n} + \\frac{n \\mu^2}{2(n+1)} \\frac{r_0^{2n}}{r^{2n}}\\label{axion soln} \\\\\n\tA_t(r) &= \\mu \\left(1  - \\frac{r_0^n}{r^n} \\right)\\label{axion gauge field}\n\\end{align}\nHere $r_0$ is the horizon of the brane, $\\mu$ is the chemical potential in the dual theory and $m_0$ is related to the total energy\nof the solution. \nThe Hawking temperature is given by \n\\begin{equation}\n\tT = \\frac{f'(r_0)}{4 \\pi} = \\frac{1}{4 \\pi} \\left( (n+2) r_0 - \\frac{\\alpha^2}{2 r_0} - \\frac{n^2 \\mu^2}{2 (n+1) r_0} \\right).\n\\end{equation}\nNote that, despite the fact that the geometry is isotropic and homogeneous, the solution manifestly breaks translational\ninvariance due to the explicit dependence of $\\psi_I$ on $x^a$. This feature is reflected in the its thermoelectric DC conductivities, \nwith the $\\delta$-function at zero frequency present in the Reissner-Nordstr\\\"om solution removed due to the breaking of translational invariance. \n\n\\subsection{Transport}\n\\label{sec:Trans}\nConductivities can be computed in terms of two-point functions, which are given in AdS\/CFT by studying linear fluctuations\naround the black holes under consideration.  Here we are interested in the electric and thermal conductivities at zero spatial \nmomentum, which can be obtained in terms of the retarded two-point functions\n\\begin{equation}\\label{2pt fns}\n\tG_{JJ} (\\omega) = \\langle J^1 J^1 \\rangle (\\omega), \\quad G_{QJ} (\\omega) = \\langle Q^1J^1 \\rangle (\\omega) \\quad G_{QQ} (\\omega) = \\langle Q^1Q^1 \\rangle (\\omega)\n\\end{equation}\n\\noindent where $Q^i = T^{ti} - \\mu J^i$ and $T^{ij}$ and $J^i$ are the the stress tensor and $U(1)$ current of the field theory, respectively. \nHere we have chosen to compute the conductivities along the axis $x^1$. Because the black holes of interest are isotropic, this does not \nresult in loss of generality. \nWe can then express the electric conductivity $\\sigma(\\omega)$, the thermo-electric conductivity $\\beta(\\omega)$ and the thermal conductivity $\\kappa(\\omega)$ \nin terms of the two-point functions \\eqref{2pt fns} by means of the Kubo formulae:\n\\begin{align}\n\\nonumber\n\t\\sigma(\\omega) &= \\frac{i}{\\omega} (G_{JJ} (\\omega) - G_{JJ} (0))  , \\\\\n\\nonumber\n\t\\beta(\\omega)   &= \\frac{i}{\\omega T} (G_{QJ} (\\omega) - G_{QJ} (0))  , \\\\ \n\\label{conductivityDefs}\n\t\\kappa(\\omega) &= \\frac{i}{\\omega T} ( G_{QQ} (\\omega) - G_{QQ} (0) )  \n\\end{align}\n\nAnalytical traction may be gained in the DC limit, where these conductivities can be computed. \nAs shown in \\cite{Andrade:2013gsa} the DC electrical conductivity is given by\n\\begin{equation}\n\t\\sigma(0) = r_0^{n-1} \\left(1 + n^2\\frac{\\mu^2}{\\alpha^2} \\right)\n\\end{equation}\nwhilst the thermal and thermo-electric conductivities for general $n$ are given in \\cite{Donos:2014cya}\n\\begin{equation}\n\t\\kappa(0) =  r_0^{n+1}\\frac{(4\\pi)^2 T}{\\alpha^2}, \\qquad \\beta(0) =  r_0^{n}\\frac{4\\pi \\mu}{\\alpha^2}. \\label{kappaDC} \n\\end{equation}\n\nSeparately, the conductivities may be approximated analytically for small $\\alpha$ by the Drude formula. \nFor instance, at $n=1$ the thermal conductivity is given by \n\\begin{equation}\\label{drude}\n\t\\kappa(\\omega) = \\frac{\\kappa(0)}{1 - i \\omega \\tau}, \\qquad \\omega \\ll T\n\\end{equation}\nwhere $\\tau$ is the characteristic time of momentum relaxation, set by $\\alpha$. Since there is only \none characteristic time scale, we say that transport is {\\it coherent} in this regime.\\footnote{See \\cite{Hartnoll:2014lpa} for a discussion on this terminology.}\nIncreasing $\\alpha$, the deviations from \\eqref{drude} become large, driving the system into an {\\it incoherent} \nphase. This transition was first observed in this holographic system by a numerical analysis in $n=1$ \\cite{Kim:2014bza}, \nand later on also noticed in the presence of a charged scalar condensate in \\cite{Andrade:2014xca}. A closely related coherent\/incoherent\ntransition has been reported for the thermal conductivity at zero chemical potential for $n=1$ in \\cite{Davison:2014lua}, which \nfocussed on an explanation in terms of QNM: the system behaves coherently when there is an isolated, long-lived, \npurely dissipative excitation in the spectrum. Moreover, this analysis was extended in perturbation theory to include \nchemical potential \\cite{Davison:2015bea}, with qualitatively similar results. \n \n\n\\subsection{Master fields}\n\\label{sec:MF}\nA general approach to computing the conductivities in the background \\eqref{NL ansatz}-\\eqref{axion gauge field} utilises a minimal, consistent set of perturbations,\n\\begin{equation}\\label{linear perts}\n  \\delta A = e^{- i \\omega t} a(r) dx^1, \\qquad \t\\delta (ds^2) = 2 e^{- i \\omega t} r^2 h(r) dt dx^1 , \n  \\qquad \\delta \\psi_1 = e^{- i \\omega t} \\alpha^{-1} \\chi(r).\n\\end{equation}\nThe linearised equations of motion which govern the perturbations \\eqref{linear perts} can be written as\n\\begin{align}\n\ta'' + \\left[ \\frac{f'}{f} + \\frac{(n-1)}{r} \\right] a' + \\frac{\\omega^2}{f^2} a + \\frac{\\mu n}{f} \\frac{r_0^n}{r^{n-1}} h' &= 0 \\\\\n\t\\chi'' + \\left[ \\frac{f'}{f} + \\frac{(n+1)}{r} \\right] \\chi' + \\frac{\\omega^2}{f^2} \\chi - \\frac{i \\omega \\alpha^2}{f^2} h &=0 \\\\\n\t\\frac{i \\omega r^2}{f} h' + \\frac{i \\omega n \\mu}{f} \\frac{r_0^n}{r^{n+1}} a - \\chi' &=0 \n\\end{align}\nwhere primes denote derivatives with respect to $r$. For odd $n$, the near boundary expansions for the physical fields are given by \n\\begin{align}\n\\label{UV phys1}\n\th &= h^{(0)} + \\ldots + \\frac{h^{(n+2)}}{r^{n+2}}  \t  + \\ldots\\\\\n\ta &= a^{(0)} + \\ldots + \\frac{a^{(n)}}{r^n}   + \\ldots \\\\\n\\label{UV phys3}\n\t\\chi &= \\chi^{(0)} + \\frac{\\chi^{(1)}}{r} + \\frac{\\chi^{(2)}}{r^2}  + \\ldots\n\\end{align}\nFor even $n$, the expansions \\eqref{UV phys1}-\\eqref{UV phys3} contain logarithms, as a result of the Weyl anomaly \npresent in even boundary dimensions \\cite{Henningson:1998gx}. These terms will play no role in the following, so we \nshall omit them. \nThe terms $\\chi^{(1)} $, $\\chi^{(2)} $ are fixed by the equations of motion as\n\\begin{equation}\\label{chi 1 2}\n\t\\chi^{(1)} = 0 , \\qquad \\chi^{(2)}  = \\frac{\\omega( \\omega \\chi^{(0)} - i \\alpha^2 h^{(0)})}{2 n}\n\\end{equation}\nThe gauge invariant sources for the electric and thermal conductivity are $a^{(0)}$ and \n$ s^{(0)} =\\omega \\chi^{(0)} - i \\alpha^2 h^{(0)} $, respectively (see e.g.\\ \\cite{Donos:2013eha}).\n\nAs shown in \\cite{Andrade:2013gsa}, the perturbation equations can be decoupled in terms of two gauge invariant master \nfields $\\Phi_\\pm$, given by \n\\begin{equation}\\label{MF def}\n   f  r \\chi'  = \\frac{\\omega}{\\mu} ( \\tilde c_+ \\Phi_+ + \\tilde c_- \\Phi_-  ),  \\qquad\n\ta = - i (\\Phi_+ + \\Phi_-)\n\\end{equation}\n\\noindent where\n\\begin{equation}\\label{cpm}\n\t\\tilde c_\\pm = \\frac{1}{2 r_0^n} \\left\\{ (n+2) m_0 \\pm [ (n+2)^2 m_0^2 + 4 r_0^{2n} \\mu^2 \\alpha^2  ]^{1\/2} \\right \\}.\n\\end{equation}\n\n\\noindent The master fields are governed by the equations  \n\\begin{equation}\\label{MF eqs pm}\n\tr^{3-n} ( f r^{n-1} \\Phi_\\pm' )' + \\left(  \\frac{r^2 \\omega^2}{f} - \\frac{n^2 \\mu^2 r_0^{2n}}{r^{2n}} + \n\tn \\tilde c_\\pm \\frac{r_0^n}{r^n} \\right) \\Phi_\\pm  = 0.\n\\end{equation}\n\\noindent As shown in \\cite{Son:2002sd}, in order to obtain the retarded correlators the fluctuations must satisfy ingoing boundary conditions \nat the black hole horizon. These can be implemented by simply imposing the ingoing condition on the master field \n\\cite{Berti:2009kk}, which amounts to\n\\begin{equation}\\label{ingoing bc}\n\t\\Phi_\\pm(r) = (r - r_0)^{- i \\omega\/(4 \\pi T)} ( \\Phi_\\pm^H + \\ldots  ), \\qquad {\\rm near } \\; r = r_0\n\\end{equation}\n\\noindent where $\\Phi_\\pm^H$ are arbitrary constants and the ellipsis denotes regular subleading terms. \n\nThe near boundary asymptotics of the master fields are given by \n\\begin{equation}\\label{UV MF}\n\t\\Phi_\\pm = \\Phi^{(0)}_\\pm + \\ldots + \\frac{1}{r^{n}} \\Phi^{(n)}_\\pm + \\ldots\n\\end{equation}\nFrom \\eqref{UV phys1}-\\eqref{UV phys3} and \\eqref{MF def}, we learn that the asymptotic data in \\eqref{UV MF} \nis related to the physical asymptotic data as\n\\begin{align}\n\t\\Phi^{(0)}_\\pm &= \\pm \\frac{1}{\\omega(\\tilde c_- - \\tilde  c_+)} ( 2 \\mu \\chi^{(2)} + i \\omega \\tilde c_- a^{(0)}  ) \\\\\n\t\\Phi^{(n)}_\\pm &= \\pm \\frac{i}{\\omega^2(\\tilde c_- - \\tilde  c_+)} ( - \\alpha^2 (n+2) \\mu h^{(n+2)} + \\omega^2 \\tilde c_- a^{(n)}  ) \n\\end{align}\n\\noindent where $\\chi^{(2)}$ is related to the gauge invariant source for the stress tensor by \\eqref{chi 1 2}.\nIn order to compute the two-point functions at $\\mu\\neq 0$, a detailed computation of the on-shell action is needed due to the \nnon-trivial interplay between the physical sources and vevs in $\\Phi_\\pm$\\footnote{This computation was carried out\nfor $n=1$ in \\cite{Davison:2015bea}.}. However,  it is easy to see that in order to obtain the poles in such correlators \nit suffices to solve for the spectra of $\\Phi_\\pm$ with Dirichlet boundary conditions $\\Phi^{(0)}_\\pm =0$. \n\n\n\n\n\\subsubsection{The neutral case}\n\\label{neutral master}\n\nFor $\\mu = 0$, all the gauge invariant information is contained in the thermal conductivity. To compute it, the \nrelevant fluctuations are \\eqref{linear perts} with $a(r) = 0$. The physical boundary data satisfies \\eqref{chi 1 2} \nand the gauge invariant source for the stress tensor is again $ s^{(0)}$. \nVia simple manipulations of the equations of motion, we can derive the master field equation\n\\begin{equation}\\label{MF eq neutral}\n\tr^{3-n} ( f r^{n-1} \\Phi' )' + \\left(  \\frac{r^2 \\omega^2}{f}  + n (n+2) \\frac{m_0}{r^n} \\right) \\Phi = 0 \n\\end{equation}\n\\noindent where the master field $\\Phi$ is given by\n\\begin{equation}\\label{Phi def}\n\t\\Phi =   \\frac{f r \\chi'}{i\\omega}\n\\end{equation}\nNote that \\eqref{MF eq neutral} is the $\\mu \\to 0$ limit of the equation for $\\Phi_+$ \\eqref{MF eqs pm} with $m_0 \\geq 0$.\nEquation \\eqref{MF eq neutral} has been previously derived\nfor $n=1$ in \\cite{Davison:2014lua}.\nAs in the $\\mu \\neq 0$ case, the UV asymptotics for $\\Phi$ can be written as\n\\begin{equation}\\label{UV Psi}\n\t\\Phi = \\Phi^{(0)} + \\ldots + \\frac{\\Phi^{(n)}}{r^{n}}  + \\ldots\n\\end{equation}\n\\noindent where once again we are not writing down the terms involving $ \\log r $ which are present for \neven $n$. The independent coefficients in \\eqref{UV Psi} are related to the boundary data \\eqref{UV phys1} and \\eqref{UV phys3}\nby\n\\begin{equation}\\label{Phi UV data}\n\t\\Phi^{(0)} = \\frac{ i\\omega \\chi^{(0)} + \\alpha^2 h^{(0)}}{n}, \\qquad \n\t\\Phi^{(n)} = \\frac{ (n+2) \\alpha^2}{\\omega^2}   h^{(n+2)}\n\\end{equation}\n\nUp to an overall $\\omega$-independent factor, $\\xi$, which we will fix later using the DC results, the two-point function $G_{QQ}$ can be written as \n\\begin{equation}\\label{G2 neutral}\n\tG_{QQ} = \\xi\\frac{\\Phi^{(n)}}{\\Phi^{(0)}} \n\\end{equation}\nHere we have chosen a renormalization scheme in which all local contributions to \\eqref{G2 neutral} are removed by \ncounterterms \\cite{deHaro:2000vlm}. \n\n\n\\section{QNM frequencies}\n\\label{sec:QNM}\n\n\nFinding analytical solutions to the master field equations \\eqref{MF eqs pm} and \\eqref{MF eq neutral} for general \nvalues of the parameters seems out of reach. Closely following \\cite{Emparan:2013moa, Emparan:2014cia, Emparan:2014aba, \nEmparan:2015rva},  we obtain perturbative \nsolutions using $1\/n$ as the expansion parameter. \nIn this section we will find expressions for the decoupled QNM for $\\mu \\neq 0$ \nto order $n^{-1}$ and for $\\mu = 0 $ to order $n^{-3}$, finding good agreement with numerical calculations at finite $n$\nin a certain region of parameter space.  \nIn  section~\\ref{sec:conductivity}  we will carry out the computation of the AC thermal conductivity to  order $n^{-2}$, obtaining a result consistent with our \nQNM calculation. \n\n\nAs explained in \\cite{Emparan:2014cia, Emparan:2014aba, Emparan:2015rva}, the spectrum of QNM in the large $n$ limit splits into decoupled \nmodes, which are normalisable in the near horizon geometry, and non-decoupled modes,  which are not. The latter \nare shared by many black holes so we do not expect to obtain information about the conductivities in this set of \nmodes, since, in particular, they are part of the spectra of black holes which are translationally invariant along \nthe boundary directions.\nWe focus on the decoupled modes and find that they indeed correspond to `Drude poles', i.e.\\ they are the purely \nimaginary modes which control the relaxation time of the system. As stated in \\cite{Emparan:2014cia, Emparan:2014aba, Emparan:2015rva}, \na necessary condition for the existence of decoupled QNMs is the presence of negative minima in the effective potential $V_{\\pm}$ defined by recasting the master field \nequation as\n\\begin{equation}\n\t\\left( \\frac{d^2}{d r_*^2} + \\omega^2 - V_{\\pm} \\right) \\Psi_\\pm = 0\n\\end{equation}\n\\noindent where $d r_* = dr \/f(r)$ is the tortoise coordinate. This form can be achieved by letting $ \\Phi_\\pm(r) = r^{(1-n)\/2} \\Psi_\\pm(r)$ \nin the master field equations \\eqref{MF eqs pm} and \\eqref{MF eq neutral}. By examining $V_{-}$, we conclude that there are \nno decoupled QNMs for $\\Phi_-$.  \n\nWhen taking the $n \\to \\infty$ limit, it is important to assign the scaling with $n$ of different physical \nquantities. Our goal is to capture the effects of momentum relaxation, so we will rescale quantities as appropriate so that $\\alpha$ appears at infinite $n$. More concretely, we will \ntake the $n \\to \\infty$ limit holding $r_0$, $\\mu$ and $\\hat \\alpha$ fixed, where\n\\begin{equation}\n  \t\\hat \\alpha = \\frac{\\alpha}{\\sqrt{n}}.\n\\end{equation}  \nThis scaling mirrors the scaling of momenta required in \\cite{Emparan:2015rva}.\nIt is convenient to define the radial variable $\\rho$ by\n\\begin{equation}\n \t\\rho = \\left( \\frac{r}{r_0} \\right)^n.\n\\end{equation} \nHere we work with $r_0=1$. We will keep $\\rho$ finite as $n \\to \\infty$, performing expansions of\n\\begin{equation}\nr = \\rho^{1\/n} = 1 + \\frac{1}{n}\\log{\\rho} + \\ldots\n\\end{equation} \nwhich takes us into the horizon region.\nIn order to obtain the perturbative solution we are after, we postulate the following expansions for the fields and the \nQNM frequency $\\omega = \\omega_\\pm$,\n\\begin{equation}\\label{n expansions}\n\t\\Phi_\\pm(\\rho) = \\sum_{i = 0} \\frac{\\Phi_{\\pm,i}(\\rho)}{n^i}, \\qquad \\omega_\\pm = \\sum_{i = 0} \\frac{\\omega_{\\pm,i}}{n^i}\n\\end{equation}\nOur boundary conditions are normalisability in the near horizon, i.e.\\ \n\\begin{equation}\n\t\\Phi_{\\pm,i}(\\rho) \\to 0 , \\qquad {\\rm at} \\, \\, \\rho \\to \\infty \n\\end{equation}\n\\noindent and ingoing boundary conditions at the horizon. These can be written as boundary conditions for the $\\Phi_{\\pm,i}$\nin \\eqref{n expansions} by expanding \\eqref{ingoing bc} in powers of $1\/n$. \nThe remainder of the computation of the decoupled QNM proceeds in close parallel to the one described in \n\\cite{Emparan:2015rva}, and we shall simply quote our results.\n\nFor $\\Phi_+$, we find decoupled QNMs with frequencies\n\\begin{align}\n\\omega_{+} &= -i \\hat\\alpha ^2\\left\\{\\frac{ \\left(2-\\hat\\alpha ^2\\right)}{2-\\hat\\alpha ^2+\\mu ^2}  - \\frac{1}{n}\\left[\\frac{2 \\hat\\alpha^2}{\\left(2-\\hat\\alpha ^2+\\mu ^2\\right) } \\,  \\log \\left(\\frac{2-\\hat\\alpha ^2}{2-\\hat\\alpha ^2-\\mu ^2}\\right) \\right.\\right. \\nonumber\\\\\n &\\phantom{=\\ }\\left.\\left. +\\frac{2 \\left(2 - \\hat\\alpha^2\\right)^3 + \\left(12 - \n    8 \\hat\\alpha^2 + \\hat\\alpha^4\\right) \\mu^2 + \\left(2 - 3 \\hat\\alpha^2\\right) \\mu^4 }{\\left(2-\\hat\\alpha ^2+\\mu ^2\\right)^3 }  \\right]+O\\left(n^{-2}\\right)\\right\\}. \\label{eq:QNMcharged}\n\\end{align}\nWe find that it is impossible to satisfy the  boundary conditions for $\\Phi_-$, so we conclude that there are no decoupled QNMs for this field, as argued above. \nIn the $\\mu=0$ case we are able to obtain two higher orders in the expansion for the $\\Phi$ frequency:\\footnote{Interestingly, these QNM frequencies can be obtained directly from the QNMs of black branes in AdS without momentum relaxation \\cite{Emparan:2015rva} by mapping the momenta $\\hat{q}^2 = \\hat\\alpha^2\/2$ and the spatial metric curvature parameter $K=-\\hat\\alpha^2\/2$.}\n\\begin{align}\n\\omega &= -i \\hat{\\alpha}^2  \\left\\{1 - \\frac{2}{n} - \\frac{2\\left(12+(\\pi^2-6)\\hat{\\alpha}^2\\right)}{3\\left(\\hat{\\alpha}^2-2\\right)n^2} \\right. \\nonumber\\\\\n &\\phantom{=\\ }\\left.+\\frac{8\\left[-12+\\hat{\\alpha}^2\\left((\\hat{\\alpha}^2 -4)(\\pi^2-3)-3(\\hat{\\alpha}^2 +2)\\zeta(3)\\right)\\right]}{3\\left(\\hat{\\alpha}^2-2\\right)^2 n^3}+O\\left(n^{-4}\\right)\\right\\} \\label{eq:QNMneutral}.\n\\end{align} \n\nA notable feature of the frequencies \\eqref{eq:QNMcharged} and \\eqref{eq:QNMneutral} is a breakdown of the expansion when $\\hat\\alpha^2+\\mu^2 =2$. In fact this behaviour could have been predicted by examining a large~$n$ expansion of the DC thermal conductivity, \\eqref{kappaDC},\n\\begin{equation}\n\\kappa(0) = \\kappa(0)|_{n\\to\\infty} \\left(1 + \\frac{4 + \\mu^2}{(2-\\hat\\alpha^2 - \\mu^2)n}+ O\\left(n^{-2}\\right)\\right).\n\\end{equation}\nThis breakdown can be traced back to a change in the way that the temperature scales with $n$ at large $n$:\n\\begin{equation}\nT = \\frac{(2 - \\hat\\alpha^2 -\\mu^2)n}{8 \\pi} + O\\left(n^{0}\\right).\n\\end{equation}\nConsequently, in order to examine the point $\\hat\\alpha^2+\\mu^2 =2$ we must repeat our large $n$ analysis there. For $\\mu=0$ and $\\hat\\alpha^2 =2$ the master field equation \\eqref{MF eq neutral} can be solved exactly for any $n$. This generalises the analysis performed at $n=1$ in \\cite{Davison:2014lua}. The additional divergence in the $\\mu\\neq 0$ case \\eqref{eq:QNMcharged} at $2+\\mu^2 = \\hat\\alpha^2$ coincides with the change of $n$ scaling of the mass parameter of the background solution, and occurs at a higher value of $\\hat\\alpha^2$ than the divergence discussed above. \n\nMoving on, we would like to compare these large $n$ analytical expressions \\eqref{eq:QNMcharged} and \\eqref{eq:QNMneutral} with finite $n$ numerics. For clarity we focus on $\\mu=0$, for which the comparison is presented in figure~\\ref{QNMplot} for values $n=1,11$ and $101$.  The $n=1$ case was previously analysed numerically in \\cite{Davison:2014lua} wherein it was noted that a pole collision occurred as $\\alpha$ was dialled. In this figure~\\ref{QNMplot}, we demonstrate that there are numerous such pole collisions at $n=1$, indicated by each extremum of the curve. For $n>1$ we find only one pole collision. Interestingly, the oscillations are centred on the critical value $\\hat\\alpha = \\sqrt{2}$ discussed above, and the locations where the crossings occur coincide with the existence of analytic regular, normalisable modes whose frequencies have integer imaginary part as discussed in appendix \\ref{appendixCritical}. We discuss these collisions in the context of a transition from coherent to incoherent behaviour in the conclusions, section \\ref{sec:conclusions}.\n\nFinally, we note that there is excellent agreement at sufficiently large finite $n$ between the numerical results and the large $n$ expansion, which interestingly includes the breakdown near $\\hat\\alpha = \\sqrt{2}$.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{qnmTripA}\\\\\n\\includegraphics[width=0.45\\textwidth]{qnmTripB}\\\\\n\\begin{picture}(0.1,0.1)(0,0)\n\\put(-160,240){\\makebox(0,0){$n=1$}}\n\\put(45,240){\\makebox(0,0){$n=11$}}\n\\put(-55,120){\\makebox(0,0){$n=101$}}\n\\put(-215,225){\\makebox(0,0){$-\\Im(\\omega)$}}\n\\put(-115,100){\\makebox(0,0){$-\\Im(\\omega)$}}\n\\put(0,10){\\makebox(0,0){$\\hat\\alpha$}}\n\\put(-105,135){\\makebox(0,0){$\\hat\\alpha$}}\n\\put(105,135){\\makebox(0,0){$\\hat\\alpha$}}\n\\end{picture}\n\\caption{Purely imaginary quasi-normal mode frequencies of the linear axion black brane at $\\mu=0$ as a function of $\\hat\\alpha = \\alpha\/\\sqrt{n}$ computed numerically for various $n$ as labelled (solid curves). The dashed lines show the large $n$ analytical counterpart \\eqref{eq:QNMneutral} computed to order $n^{-2}$. The vertical dotted line is the `critical' value $\\hat\\alpha = \\sqrt{2}$, where the master field equation can be solved analytically for any $n$, giving the integer crossing frequencies. The $n=1$ case shows several pole collisions,  indicated by each turn over of the curve, whilst for $n>1$ we see only one collision for this range. Units are given by $r_0=1$.\\label{QNMplot}}\n\\end{center}\n\\end{figure}\n\n\n\\section{AC conductivity}\n\\label{sec:conductivity}\n\nIn this section we compute the AC thermal conductivity for the neutral theory to order $n^{-2}$, and compare the resulting expressions with numerics. Specifically we look at the frequency range which captures the decoupled mode describing the essential momentum relaxation physics, i.e.\\ we take $\\omega = O\\left(n^0\\right)$.\n\nThe computation begins with the approach outlined in \\cite{Emparan:2013moa}. The basic structure of the calculation is a matched asymptotic expansion made possible by the new small scale $r_0\/n$, corresponding to the localisation of gradients near the horizon. This new scale allows us to separate the bulk geometry into a near and far zone, defined as follows:\n\\begin{eqnarray}\n\\text{near zone:}\\quad & r-r_0\\ll r_0, & \\quad \\log \\rho \\ll n\\\\\n\\text{far zone:}\\quad & r-r_0\\gg \\frac{r_0}{n}, &\\quad \\log \\rho \\gg 1.\n\\end{eqnarray}\nAs we have previously, we shall take $r_0=1$. Note that these zones overlap: in particular, the overlap zone is described by $\\log \\rho \\gg 1$ in the near zone, and $\\log \\rho \\ll n$ in the far zone. Thus the calculation proceeds by solving in both zones and matching at the overlap; the near zone will allow us to imprint the ingoing horizon boundary conditions on the solution, whilst the far zone will enable us to read off the normalisable and non-normalisable data and allow the computation of the two-point function. \n\n\\subsection{Near zone}\nThe near zone is reached by taking the large $n$ limit whilst working at fixed $\\rho$. The calculation proceeds similarly to the QNM calculation and so we will be brief. A key difference is that we do not wish to impose normalisablity, and so the frequencies are not quantised. As before, we expand $\\Phi(\\rho) = \\sum_{i = 0} \\frac{\\Phi_i(\\rho)}{n^i}$, but we do not expand $\\omega$. After imposing the ingoing boundary conditions we obtain, \n\\begin{eqnarray}\n\\Phi_{0} &=& \\frac{a_0}{\\rho}\\\\\n\\Phi_{1} &=& \\frac{2 i a_0 \\omega}{(\\hat\\alpha^2-2)}\\frac{\\log{(\\rho-1)}}{\\rho} - \\frac{2 a_0(\\hat\\alpha^2 - i \\omega)}{\\hat\\alpha^2 -2}\\frac{\\rho-1}{\\rho}\n\\end{eqnarray}\ntogether with explicit expressions for $\\Phi_{2}$ and $\\Phi_{3}$ which we have omitted here. $a_0$ is an unconstrained  integration constant. Finally, in the overlap region, we have\n\\begin{eqnarray}\n\\Phi_{0} &=& \\frac{a_0}{\\rho}\\\\\n\\Phi_{1} &=& -2a_0\\frac{(\\hat\\alpha^2-i\\omega)}{\\hat\\alpha^2 -2} \\left(1-\\frac{1}{\\rho}\\right) + 2 a_0 \\frac{i \\omega}{\\hat\\alpha^2 -2}\\frac{\\log\\rho}{\\rho} - a_0 \\frac{2 i \\omega}{\\hat\\alpha^2-2}\\frac{1}{\\rho^2}.\n\\end{eqnarray}\nwhere again we have evaluated the overlap expressions for $\\Phi_{2}$ and $\\Phi_{3}$ but we omit them here in the interest of keeping the presentation concise.\n\n\\subsection{Far zone and matching}\nAt leading order the far zone equations can be obtained by removing any terms which decay exponentially fast with $n$ \\cite{Emparan:2013moa}. More generally, an expansion can be formed by counting powers of $r^{-n}$ in the equations of motion after inserting\n\\begin{equation}\n\\Phi = \\phi + r^{-n} \\psi + \\ldots.\n\\end{equation}\nLet us introduce a counting parameter $\\lambda$ for this purpose, i.e. we count $\\phi$ as order $\\lambda^0$. In order to obtain the conductivity we need the coefficient of $r^{-n}$, and so we need to go to order $\\lambda^1$.\nAt order $\\lambda^0$ the master field equation becomes\n\\begin{equation}\n{\\cal D}\\phi = 0,\\qquad {\\cal D} \\equiv \\partial_r^2 - \\frac{(n-1)\\hat{\\alpha}^2 - 2(n+1) r^2}{2r^3 - \\hat{\\alpha}^2 r} \\partial_r + \\frac{4 \\omega^2}{(\\alpha^2-2r^2)^2}.  \\label{lambda0}\n\\end{equation}\nThis equation can be solved explicitly in terms of Gauss hypergeometric functions,\n\\begin{eqnarray}\n\\phi &=&  \\left(1-\\frac{\\hat{\\alpha}^2}{2 r^2}\\right)^{-\\frac{i \\omega}{\\sqrt{2}\\hat{\\alpha}}} \\Bigg(A\\; _2F_1\\left(-\\frac{i \\omega}{\\sqrt{2}\\hat{\\alpha}}, 1- \\frac{n}{2} - \\frac{i \\omega}{\\sqrt{2}\\hat{\\alpha}}, 1-\\frac{n}{2}, \\frac{\\hat{\\alpha}^2}{2r^2}\\right)\\nonumber\\\\\n&& + r^{-n} B\\; _2F_1\\left(1-\\frac{i \\omega}{\\sqrt{2}\\hat{\\alpha}}, \\frac{n}{2} - \\frac{i \\omega}{\\sqrt{2}\\hat{\\alpha}}, 1+\\frac{n}{2}, \\frac{\\hat{\\alpha}^2}{2r^2}\\right) \\Bigg)\\label{phifar}\n\\end{eqnarray}\nwhere $A$ and $B$ are integration constants; $A$ will contribute to the non-normalisable part of $\\Phi$ at infinity, $\\Phi^{(0)}$, whilst $B$ will contribute to the normalisable part, $\\Phi^{(n)}$. To find this in the overlap region we need to use expressions for the Gauss hypergeometric functions for large parameters. Expanding $\\phi$ in powers of $n$ and similarly for the integration constants $A$ and $B$, we find, \n\\begin{eqnarray}\n\\phi_0 &=& A_0 - \\frac{2B_0}{(\\hat\\alpha^2 - 2)\\rho}\\label{far0overlap}\\\\\n\\phi_1 &=& A_1 - \\frac{A_0\\omega^2}{\\hat\\alpha^2-2} + \\frac{1}{\\rho} \\left(-\\frac{2B_1}{\\hat\\alpha^2-2} -2 B_0\\frac{2\\hat\\alpha^2 + \\omega^2}{(\\hat\\alpha^2-2)^2} -4 B_0\\frac{\\hat\\alpha^2}{(\\hat\\alpha^2-2)^2} \\log\\rho\\right)\\label{far0overlapB}\n\\end{eqnarray}\ntogether with similar expressions for $\\phi_2$ and $\\phi_3$. Subscripts denote the power of $1\/n$ for which it is a coefficient.\n\nAt next order in $\\lambda$, the equation for $\\psi$  is sourced by $\\phi$:\n\\begin{eqnarray}\nr^n{\\cal D} \\left(\\frac{\\psi}{r^n} \\right) &=& {\\cal S} \\label{DS}\n\\end{eqnarray}\nwhere the operator ${\\cal D}$ is defined in \\eqref{lambda0} and where\n\\begin{align}\n{\\cal S} &= r^{-1} \\frac{(2-\\hat\\alpha^2) \\left(n \\hat\\alpha^2  - 2(2+n) r^2\\right)}{(\\hat\\alpha^2-2r^2)^2} \\phi' \\nonumber\\\\\n&\\phantom{=\\ }+ r^{-2} \\frac{2-\\hat\\alpha^2}{2r^2-\\hat\\alpha^2}\\left(-n(2+n) - \\frac{8 \\omega^2 r^2}{(2r^2-\\hat\\alpha^2)^2}\\right)\\phi.\n\\end{align}\nUnlike for $\\phi$ we have not directly integrated this equation. However, we can do so order-by-order in a large $n$ expansion at fixed $r$ provided we include the correct non-perturbative contributions. To the order of $n$ considered these turn out to be,\n\\begin{eqnarray}\n\\psi &=& \\left(\\psi_{B,0}(r) + \\frac{\\psi_{B,1}(r)}{n} + \\frac{\\psi_{B,2}(r)}{n^2} + \\frac{\\psi_{B,3}(r)}{n^3}  +O\\left(n^{-4}\\right) \\right)\\nonumber\\\\\n&&+ r^{-n}\\left(\\psi_{C,0}(r) + \\frac{\\psi_{C,1}(r)}{n} + \\frac{\\psi_{C,2}(r)}{n^2} + \\frac{\\psi_{C,3}(r)}{n^3} + O\\left(n^{-4}\\right) \\right). \\label{nonpert}\n\\end{eqnarray}\nWe can solve for each $\\psi_{B,i}$ and $\\psi_{C,i}$ provided $A_0=0$, which as we shall see shortly is consistent with the required value from the matching calculation.  Each term is required for the matching calculation to work and is straightforward to obtain. This method is more efficient than solving \\eqref{DS} at arbitrary $n$ and then expanding, as in \\cite{Emparan:2013moa}. Applied to $\\phi$ above, this method gives the same result as the expansion of \\eqref{phifar}.\n\nExpressing \\eqref{nonpert} in the overlap region and combining with \\eqref{far0overlap} and \\eqref{far0overlapB} gives us $\\Phi$ in the overlap zone, which can be matched with the expression coming from the near zone calculation. This fixes the coefficients appearing in \\eqref{far0overlap},\\eqref{far0overlapB} together with additional integration constants which arise in each of the $\\psi_{B,i}$. For example, \n\\begin{align}\nA_0 &= 0,\\\\\nA_1 &= \\frac{2 a_0(\\hat\\alpha^2-i\\omega)}{2-\\hat\\alpha^2}\\\\\nA_2 &= -\\frac{2 a_0(-i \\omega^3 + \\hat\\alpha^2(4-2i\\omega+\\omega^2))}{(2-\\hat\\alpha^2)^2}\\\\\n\\nonumber\nA_3 &= \\frac{a_0}{3\\left(\\hat{\\alpha }^2-2\\right)^3} \n\\bigg\\{ \\hat{\\alpha }^4 \\left(-6 \\omega ^2-4 i \\pi ^2 \\omega \\right) \\\\\n   &-\\hat{\\alpha }^2 \\left[ 3 \\left(\\omega ^4-6 i \\omega\n   ^3+8 \\omega ^2+32\\right)+4 \\pi ^2 \\omega  (\\omega +2 i)\\right] + i \\omega ^2 \\left(3 \\omega ^3-4 \\pi ^2 \\omega +48 i\\right) \\bigg\\}\n\\end{align}\nThe $B_i$ coefficients are given in relation to the coefficients appearing in $\\psi_{B,i}$. These coefficients determine $\\Phi^{(0)}$ and $\\Phi^{(n)}$ to order $n^{-3}$. Note that since $\\psi$ does not contribute to $\\Phi^{(0)}$, and $A_0=0$, the non-normalisable data $\\Phi^{(0)}$ vanishes to leading order in $n$ and so the Green's function will grow with $n$.\n\n\\subsection{Results}\nCombining the asymptotic results for $\\phi$ and $\\psi$ discussed above brings us to the main result of this section --- the thermal conductivity \\eqref{conductivityDefs} to order $n^{-2}$: \n\\begin{eqnarray}\n\\nonumber\n&& \\kappa(\\omega) =  2\\pi \\frac{2-\\hat\\alpha^2}{\\hat\\alpha^2-i \\omega} \\\\\n\\nonumber\n&&+\\frac{4 \\pi  \\left(\\hat\\alpha^4 (2+\\omega  (\\omega -i))-i \\hat\\alpha^2 \\omega ^3+2 i \\omega  \\left(\\hat\\alpha^2-i\n   \\omega \\right)^2 \\log \\left(2-\\hat\\alpha^2\\right)-2 i \\omega  \\log (2) \\left(\\hat\\alpha^2-i \\omega \\right)^2\\right)}{n\n   \\left(\\hat\\alpha^3-i \\hat\\alpha \\omega \\right)^2}\\nonumber\\\\\n\\nonumber\n&&-\\frac{4 \\pi \\omega }{3 n^2 \\hat{\\alpha }^4 \\left(\\hat{\\alpha }^2-2\\right)^2 \\left(\\omega +i \\hat{\\alpha }^2\\right)^3}\\bigg\\{\\\\\n\\nonumber\n&& +\\hat{\\alpha }^6 \\bigg[ \\hat{\\alpha }^6 (-(6 \\log (2-\\hat{\\alpha }^2 )+\\pi ^2+6-6 \\log (2) )) \\\\\n\\nonumber\n&& +4 \\hat{\\alpha }^2 (6 \\log (2-\\hat{\\alpha }^2 )+\\pi ^2+12-6 \\log (2) )-48 \\bigg]\\\\\n\\nonumber\n&& +i \\hat{\\alpha }^6 \\bigg[  (\\hat{\\alpha }^2-2 ) (3 (\\hat{\\alpha }^2+\\pi ^2+4 ) \\hat{\\alpha }^2+2 (\\pi ^2-6)) \\\\\n\\nonumber\n&&+18 (\\hat{\\alpha }^4-4 ) \\log (2-\\hat{\\alpha }^2 ) -18 (\\hat{\\alpha }^4-4 ) \\log (2)  \\bigg] \\omega\\\\\n\\nonumber\n&& +\\hat{\\alpha}^4 \\bigg[ -12 \\hat{\\alpha }^6 (1+\\log (2))+2 \\hat{\\alpha }^4 (\\pi ^2+15 (1+\\log (2))) \\\\\n\\nonumber\n&& -4 \\hat{\\alpha }^2 (3+\\pi ^2+\\log (4096))\n +6 (\\hat{\\alpha }^2-2) (2 \\hat{\\alpha }^4-\\hat{\\alpha }^2+6) \\log (2-\\hat{\\alpha }^2 )+72 \\log (2) \\bigg] \\omega ^2\\\\\n\\nonumber\n&& -3 i \\hat{\\alpha }^2 \\bigg[ \\hat{\\alpha }^8-3 \\hat{\\alpha }^6 (5+\\log (16))+\\hat{\\alpha }^4 (28+46 \\log (2))-4 \\hat{\\alpha }^2\n (1+\\log (4096)) \\\\\n \\nonumber\n&& +2 (\\hat{\\alpha }^2-2) (6 \\hat{\\alpha }^4-11 \\hat{\\alpha }^2+2) \\log (2-\\hat{\\alpha }^2)+\\log (256) \\bigg] \\omega ^3\\\\\n\\nonumber\n&& +6 \\hat{\\alpha }^2 \\left(\\hat{\\alpha }^2-2\\right) \\bigg[-\\hat{\\alpha }^4+6 \\hat{\\alpha }^2 (1+\\log (2))-6 \\left(\\hat{\\alpha }^2-2\\right) \\log \\left(2-\\hat{\\alpha }^2\\right)-12 \\log (2)\\bigg] \\omega ^4\\\\\n\\nonumber\n&& +3 i \\bigg[ \\hat{\\alpha }^6-2 \\hat{\\alpha }^4 (3+\\log (4))+8 \\hat{\\alpha }^2 (1+\\log (4)) \\\\\n&& +4(\\hat{\\alpha }^2-2)^2  \\log (2-\\hat{\\alpha }^2 )-16 \\log (2) \\bigg ] \\omega ^5 \\bigg\\} +O\\left(n^{-3}\\right). \\label{conductivityfinal}\n\\end{eqnarray}\nWe have fixed the overall normalisation given by $\\xi$ in \\eqref{G2 neutral} by comparing with the DC value \\eqref{kappaDC}:\n\\begin{equation}\n\\kappa(0) = 2\\pi\\left(\\frac{2}{\\hat\\alpha^2}-1\\right) \\left(1+ \\frac{4}{(2-\\hat\\alpha^2)n}\\right). \\label{kappaDCNetural} \n\\end{equation}\nWe find  $\\xi = -\\hat\\alpha^2+ O(n)^{-3}$ . Interestingly, at $\\mu=0$, the expansion for $\\kappa(0)$ truncates at order $1\/n$.\n\n \nWe note that at leading order the result takes Drude form.  A comparison with numerical integration for finite $n=1,3, 11,101$ is given in figure \\ref{kappaplot}, truncating at orders $n^0$, $n^{-1}$ and $n^{-2}$.\n\nLet us start with the $n^0$ approximation (black dash in figure \\ref{kappaplot}).  Even at this leading order the broad features of the conductivity are well captured by the large $n$ expansion. We note that the agreement at larger frequencies is excellent. The width of the peak, which is related to the relaxation timescale, is also very good, in agreement with the approximation of the QNMs by the large $n$ expansion. The only notable discrepancy is the height of the peak. This can be easily understood: $\\kappa(0)$ receives a $1\/n$ correction \\eqref{kappaDCNetural} and no further corrections, thus the DC limit is not expected to agree at this order. By $n=101$ the agreement is good everywhere.\n\nAt order $n^{-1}$ (blue dots in figure \\ref{kappaplot}) the DC limit now agrees, as anticipated. Overall the approximation is good even at $n=1$, but now we see even for modest values of $n$ the analytical result is in excellent agreement with the numerical result, e.g.\\ at $n=11$. \n\nAt order $n^{-2}$ (red dash in figure \\ref{kappaplot}) the $n=11$ and $n=101$ results are not visibly affected. However for the lower values of $n=1,3$ the agreement becomes worse than at order $n^{-1}$. This is similar to the large $D$ approximation applied to the Gregory-Laflamme instability \\cite{Emparan:2015rva}, where it was argued that the series is asymptotic due to the existence of non-perturbative contributions.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{kappaTripA}\\\\\n\\includegraphics[width=0.9\\textwidth]{kappaTripB}\\\\\n\\begin{picture}(0.1,0.1)(0,0)\n\\put(-40,250){\\makebox(0,0){$n=1$}}\n\\put(155,250){\\makebox(0,0){$n=3$}}\n\\put(-40,120){\\makebox(0,0){$n=11$}}\n\\put(155,120){\\makebox(0,0){$n=101$}}\n\\put(-210,230){\\makebox(0,0){$\\Re(\\kappa)$}}\n\\put(-210,100){\\makebox(0,0){$\\Re(\\kappa)$}}\n\\put(-105,10){\\makebox(0,0){$\\omega$}}\n\\put(105,10){\\makebox(0,0){$\\omega$}}\n\\end{picture}\n\\caption{The AC thermal conductivity at $\\mu=0$ computed analytically to orders $n^0$ (black dashed), $n^{-1}$ (blue dotted) and $n^{-2}$ (red dashed) as given in \\eqref{conductivityfinal}, compared to the finite $n$ numerical result (solid) for $\\hat\\alpha = 1\/2$.  Units are given by $r_0=1$.\\label{kappaplot}}\n\\end{center}\n\\end{figure}\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\n\nWe have studied the linear axion model defined via \\eqref{S0} when the number of spacetime dimensions\nis large, focusing on the quasi-normal modes governing momentum relaxation and the AC thermal conductivity. We have kept the horizon radius $r_0$ and the chemical potential $\\mu$ fixed in this limit. We \nfound that the influence of the momentum relaxation parameter, $\\alpha$, vanished at large $n$, but could be restored by scaling $\\alpha$ \nsuch that $\\hat\\alpha \\equiv \\alpha\/ \\sqrt{n}$ is held fixed. This places the physics of momentum relaxation at leading order in the large $D$ expansion.\n\nAn important technical point which simplifies our analysis is the existence of \nmaster field equations in arbitrary dimensions \\eqref{MF eqs pm}, \\eqref{MF eq neutral}. These decoupled wave equations contain \nall the gauge invariant information required to compute the two-point functions of interest, reducing our problem \nto calculations closely related to those already carried out in the context of General Relativity in large $D$. \n\nWe obtain analytical expressions for the QNM which control the electric and heat transport as a power series in $1\/D$. \nIn the language of \\cite{Emparan:2014aba}, these are of the decoupled kind, meaning that they are normalisable \nin the near horizon geometry. \nFor $\\mu \\neq 0$, we have computed the QNM which controls the electric conductivity for   \nup to order $n^{-1}$, while for $\\mu = 0$ we obtain the thermal QNM to order $n^{-3}$. Furthermore, \nwe calculate the AC thermal conductivity for $\\mu = 0$ to order $n^{-2}$. At leading order it takes Drude form, \nand at order $n^{-1}$ it provides a good approximation even for small values of $n$, illustrating the practicality of this technique for such systems.\n\nInterestingly, our perturbative series for the QNM breaks down due to the growth of the \ncoefficients as $\\hat\\alpha \\to  \\hat \\alpha_{c} \\equiv  \\sqrt{2 r_0^2-\\mu^2}$. \nNumerically, we observe that the structure of lowest lying QNM \nchanges significantly as we approach $\\hat \\alpha_{c}$. For very small values of $\\hat \\alpha$ there exists an isolated, \npurely dissipative excitation which governs transport, i.e.\\ the system is in a coherent regime. Increasing $\\alpha$ towards \n$\\hat \\alpha_{c}$, the characteristic time scale of this mode decreases and it mixes with the rest of the QNM in the spectrum, so \nthat we enter  an incoherent phase.\\footnote{In this regime there can be numerous pole collisions at finite $n$, including a collision between the Drude mode and a higher lying excitation.}\nWe thus interpret the breakdown of the perturbative expansion as a large $D$ signature \nof the coherent\/incoherent transition. It would be interesting to revisit our analysis with transverse wavevector $k \\neq 0$ and investigate the interplay of these QNMs with diffusion.\n\nMore generally, it is interesting to observe that in the case where we do not scale the sources of the axions with $D$, momentum conservation is restored at infinite $D$. This suggests that the large $D$ expansion may be used to improve analytical control over more generic setups incorporating inhomogeneity. We leave this possibility for future work.\n\n\\acknowledgments\n\nWe are pleased to thank Marco Caldarelli, Richard Davison, Roberto Emparan, Blaise Gout\\'eraux and Kostas Skenderis for valuable comments. \nT.A.\\ is supported by the European Research Council under the European Union's Seventh Framework Programme\n(ERC Grant agreement 307955). He also thanks the Institute of Physics at University of Amsterdam for their hospitality \nduring the completion of this work. \nS.A.G.\\ is supported by National Science Foundation grant PHY-13-13986.\nB.W.\\ is supported by European Research Council grant ERC-2014-StG639022-NewNGR.  He also thanks the Department of Physics at the University of Oxford for their hospitality during the completion of this work.\nWe are grateful to Centro de Ciencias Pedro Pascual, Benasque where this work was initiated.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nOne important topic in many branches of biology is to understand evolutionary events and forces leading to current biological systems, such as a group of species or strains of a virus. To this end, evolutionary relationships among the biological system under investigation are typically represented by \na phylogenetic tree, that is, a binary tree whose leaves are labelled by the taxon units in the system.  As these events and forces, such as rates of speciation and expansion, are often not directly observable~\\cite{mooers2007some,heath2008taxon},  one popular approach is to compare empirical shape indices computed from trees inferred from real datasets with those predicted by a null tree growth model\n~\\cite{blum2006random,hagen2015age}. Furthermore, topological tree shapes are also closely related to several fundamental statistics in population genetics~\\cite{ferretti2017decomposing,arbisser2018joint} and certain important parameters in \nthe dynamics of virus evolution and propagation~\\cite{colijn2014phylogenetic}. \n\n\n\nOne important family of tree shapes are balance indices, such as Colless' index, Sackin's index and the number of subtrees (see, e.g.~\\cite{fischer2021tree} and the references therein).  Various properties concerning  these  statistics have been established in the past decades on the following two fundamental random phylogenetic tree models: \nthe Yule model (aka the Yule-Harding-Kingman (YHK) model)~\\cite{rosenberg06a,disanto2013exact,Janson2014} and the uniform model (aka the proportional to distinguishable arrangements (PDA) model)~\\cite{McKenzie2000,chang2010limit, WuChoi16,CTW19}.\nHowever, for phylogenetic trees inferred from real datasets, the Yule or uniform model may not always be a good fit~\\cite{blum2006random}, and several general classes of random trees have been proposed  for modelling and analysing the observed data,\ntwo popular ones being Ford's alpha model~\\cite{Ford2006} and Aldous' beta model~\\cite{aldous96a}.  \n\n\nIn this paper, we confine ourselves to  Ford's alpha model, a one-parameter family  of random tree growth models introduced by Daniel J. Ford in his PhD thesis \\cite{Ford2006}. \n More precisely, under the Ford model with a fixed parameter $0\\le \\alpha \\le 1$, a random tree of a given number of leaves is generated such that at any step in which a tree $T_n$ with $n$ leaves has been constructed from previous steps, a new leaf attaches to  an internal edge of $T_n$ with probability $\\frac{\\alpha}{n-\\alpha}$ and  to  a leaf edge in $T_n$ with probability $\\frac{1-\\alpha}{n-\\alpha}$. The resulting random tree model will be referred to as the Ford model (with parameter $\\alpha$) in this paper, which is also known as the alpha tree model~(see, e.g.~\\cite{coronado2019balance}). Note that the Ford model is a family of random tree models which includes the Yule model with $\\alpha=0$,   the uniform model with $\\alpha=1\/2$, and the Comb model with $\\alpha=1$.  \n\nThe tree shape indices studied in this paper are the number of cherries and that of pitchforks. Here a cherry is a subtree with precisely two leaves and a pitchfork a subtree with three leaves. \nThe asymptotic properties of the number of cherries was first studied by McKenzie and Steel  \\cite{McKenzie2000}, who showed that the number of cherries is asymptotically normal for the Yule and the uniform models as the number of leaves tends to infinity. Later, similar properties of the number of cherries are  extended to the Ford model~\\cite[Theorem 57]{Ford2006} and to the Crump-Mode-Jagers branching process~\\cite{plazzotta2016asymptotic}. \nFor the number of pitchforks, Rosenberg~\\cite{rosenberg06a} obtained its  mean and variance and Chang and Fuchs~\\cite{chang2010limit} proved that  the number of pitchforks is also  asymptotically normal for the Yule and the uniform models. For the joint distributions, Holmgren and Janson showed that~\\cite{Janson2014} the joint distribution is asymptotically normal for the Yule model. This was recently extended by us to the uniform model based on a uniform version of the extended urn models in which negative entries are permitted for their replacement matrices~\\cite{Paper1}. \n\n\nIn this paper,  we establish the strong law of large numbers and the central limit theorem \nfor the joint distribution of cherries and pitchforks under the Ford model (Theorem~\\ref{Thm:Convg-ChPh}) \nby considering an associated nonuniform urn model (Theorem~\\ref{thm:urn:edge}). These results are presented in Section~\\ref{sec:limiting}, following Section~\\ref{sec:preliminary} in which we  collect  background concerning the Ford model and  limiting theorems on uniform urn models.\nFurthermore, we derive a recurrence  formula for computing the exact joint distribution under the Ford model~(Theorem~\\ref{jointpmf}) in Section~\\ref{sec:exact}, generalizing the results in~\\cite{WuChoi16,CTW19} for the Yule and the uniform model. This recurrence formula enables us to obtain exact expressions for the mean and variance of the number of cherries and that of pitchforks and their covariance under the Ford model. This, in-particular, generalises the exact expressions of mean and variance for the number of cherries and that of pitchforks for the Yule and the uniform models~\\cite{McKenzie2000,rosenberg06a,chang2010limit}  and the number of cherries for the Ford model~\\cite[Theorem 60]{Ford2006}. \nAs an application, in Section~\\ref{sec:expansion} we obtain higher order expansions of the first and second moments of the joint distributions. \n\n\n\n\n\\section{Ford Model and Urn Model}\n\\label{sec:preliminary}\nIn this section, we first introduce  the Ford model, which is a one-parameter family of random phylogenetic tree models. Next we present a nonuniform version of the extended urn models  associated with the Ford tree model. Finally, we recall certain conditions on the related uniform version of the extended urn model under which the strong law of large numbers and the central limit theorem are obtained.\n\n\\subsection{Ford model}\n\nA rooted binary tree is a finite connected simple graph without cycles that contains a unique vertex of degree 1 designated as the root and all the remaining vertices are of degree 3 (interior vertices) or 1 (leaves). A phylogenetic tree with $n$ leaves is a rooted binary tree whose leaves are bijectively labelled by the elements in $\\{1,\\dots,n\\}$. Edges incident with leaves are referred to as pendant edges. \n\nUnder the Ford model with parameter $0\\le \\alpha \\le 1$,  a random phylogenetic tree $T_n$ with $n$ leaves is constructed recursively by adding one leaf at a  time as follows. Fix a random permutation $(x_1,\\dots,x_n)$ of $\\{1,\\dots,n\\}$.  The initial tree $T_2$  contains precisely two leaves (e.g. one cherry) which are labelled as $x_1$ and $x_2$. For the recursive step, given a tree $T_m$ with $m$ leaves constructed so far, choose a random edge in $T_m$ according to the distribution that assigns weight $1-\\alpha$ to each pendant edge (i.e., those incident with a leaf)  and weight $\\alpha$ to each of the other edges. The new leaf labelled $x_{m+1}$ bifurcates the selected edge and joins in the middle. \nEvery single addition of a leaf in the tree results into a replacement of the selected edge with two new edges.\nFinally, we let $A_n$ and $C_n$ denote the numbers of pitchforks and cherries in tree $T_n$, respectively.\n\n \\begin{figure}[ht]\n\t\\begin{center}\n\t\t{\\includegraphics[width=0.9\\textwidth]{1_Fig_example.pdf}}\n\t\\end{center}\n\t\\caption{A sample path of the Ford model and the associated trajectory under the urn model. (i) A sample path of the Ford model evolving from $T_2$ with two leaves to $T_6$ with six leaves. The labels of the leaves are omitted for simplicity. The type of an edge is indicated by the circled number next to it. For $2\\le i \\le 5$, the edge selected in $T_i$ to generate $T_{i+1}$ is highlighted in bold and the associated edge type is indicated in the circled number above the arrow. (ii) The associated urn model with six colours, derived from the types of pendants edges in the trees. \n\t\tIn vector form, $U_0=(0,2,0,0,1,0), U_1=(2,0,1,0,0,2), U_2=(0,4,,0,0,2,1), U_3=(2,2,1,0,1,3)$, and $U_4=(2,2,1,1,1,4)$. \t\n\t}\n\t\\label{fig:example}\n\\end{figure}\n\n\n\n\\subsection{An urn model associated with trees }\n\\label{subsect:urn:tree}\n\nConsider an urn containing balls of $d$ different colours where colours are denoted by integers $\\{1,2,\\dots, d\\}$. Let $U_n=(U_{n,1},\\dots, U_{n,d})$ be the configuration vector of length $d$ such that the $i$-th element of $U_n$ is the number of balls of colour $i$ at time $n$.  Let $U_0$ be the initial vector of colour configuration, then at every time $n\\geq 1$, a ball is selected uniformly at random from the urn and if the colour of the selected ball is $i$ then the ball is replaced along with $R_{i,j}$ many balls of colour $j$,  for every $1 \\leq j \\leq d$. The dynamics of the urn configuration depends on its initial configuration $U_0$ and the $d \\times d$ replacement matrix $R = (R_{i,j})_{1\\leq i,j\\leq d}$.\n\n\nWe  study the limiting properties of the numbers of cherries and pitchforks via an equivalent urn process. Towards this, we use six different colours and assign one colour to each type of edges of the tree in the following scheme introduced in~\\cite{Paper1}:  colour $1$ for all pendant edges of a cherry in a pitchfork; colour $2$ for pendant edges of a cherry not contained in a pitchfork; colour $3$ for pendant edges in a pitchfork but not in any cherry; colour $4$ for pendant edges in neither a cherry nor a pitchfork; colour $5$ for internal edges adjacent to a cherry but not in a pitchfork (i.e., those adjacent to colour $2$ edges), and colour $6$ for all other (necessarily internal) edges (including the one incident with the root). See Fig.~\\ref{fig:example} for an illustration of the scheme. \n\n\nConsider an urn with colour configuration at time $n$ as\n$U_n = (U_{n,1},\\dots, U_{n,6})$, where $U_{n,i}$ denotes the number of edges of colour $i$ in the tree at time $n$, which has precisely $n+2$ leaves. Then $U_0 = (0,2,0,0,1,0)$, since at the initial time step $(n=0)$ there is one internal edge and one essential cherry in a rooted tree; see $T_2$ in Fig.~\\ref{fig:example}. Based on the colouring scheme of the edges, at any time $n\\geq 0$, we have\n\\begin{equation}\\label{urn-pf-ch}\n(A_{n+2}, C_{n+2}) = \\frac{1}{2} \\left(U_{n,1}, U_{n,1}+U_{n,2}\\right),\n\\end{equation}\nwhere $A_{n+2}$ and $C_{n+2}$ are the numbers of pitchforks and cherries in $T_{n+2}$, respectively.\nUnder the alpha tree model, the dynamics of the corresponding urn process evolves according to the following replacement matrix\n\\[ R = \\begin{bmatrix}\n0&0&0&1&0&1\\\\\n2&-2&1&0&-1&2\\\\\n-2&4&-1&0&2&-1\\\\\n0&2&0&-1&1&0\\\\\n2&-2&1&0&-1&2\\\\\n0&0&0&1&0&1\n\\end{bmatrix}. \\]\nLet $e_i$, $1\\le i \\le 6$, denote a $6$-vector in which the $i$-th component is $1$ and $0$ elsewhere; and   $\\chi_n$   the random vector taking value $e_i$ if, at time $n$, speciation happens at an edge with type  $i$.\nThus,  we have the following recursion\n\\[U_n= U_{n-1} +\\chi_n R, \\qquad n\\geq 1, \\]\nwhere\n\\begin{equation}\\label{SelecProb}\nP(\\chi_n =e_i|\\mbox{${\\mathcal F}$}_{n-1}) \\propto \\begin{cases}\n(1-\\alpha) U_{n-1,i}, &\\text{ for } i \\in\\{1,2,3,4\\}, \\\\[1ex]\n\\alpha \\,U_{n-1,i}, & \\text{ for } i\\in \\{5,6\\}.\n\\end{cases}\n\\end{equation}\nObserve that the process $(U_n)_{n \\ge 0}$, which  describes the dynamics of the numbers of cherries and pitchfork,\nis a {\\em nonuniform urn model} since the balls are not selected uniformly at random from the urn, which is different from the classical {\\em uniform} urn models in which  the balls are selected uniformly at random from the urn (see, e.g.~\\cite[Chapter 7]{hofri2019algorithmics}).\n\n\\subsection{Limiting theorems on uniform urn models}\n\\label{limit:uniform:urn}\nIn this subsection, we recall the strong laws of large numbers and the central limit theorems on a version of uniform urn models developed in~\\cite{Paper1}, which will be related to the nonuniform urn process in Subsection~\\ref{subsect:urn:tree} later using the urn coupling idea  in \\cite{Kaur2018}. \n\n\nFor the classical uniform urn models, it has been shown (see \\cite{BaiHu2005}) that the random process $U_n\/n$ converges almost surely to the left eigenvector of $R$ corresponding to the maximal eigenvalue and the asymptotic normality holds with a known limiting variance matrix under certain assumptions on $R$.  Standard assumptions made in the urn model theory are  that the replacement matrix is  irreducible with a constant row sum and all the off-diagonal elements are non-negative~(see, e.g.~\\cite{Hosam2009}).  In  \\cite{Paper1}, we extend this to the case when off-diagonal elements of a replacement matrix can be negative satisfying  the following set of assumptions {\\bf (A1)--(A4)}, which was slightly rephrased from~\\cite{Paper1}. Let $\\text{diag}(a_1, \\dots , a_d)$ denote the diagonal matrix whose diagonal elements are $a_1,\\dots , a_d$. \n\n\\noindent\n{\\bf (A1):} {\\em Tenable:} It is always possible to draw balls and follow the replacement rule.\n\\\\\n{\\bf (A2):} {\\em Small:} \nAll eigenvalues of $R$ are real; the maximal eigenvalue $\\lambda_1$, called the {\\em principal eigenvalue} is positive with $\\lambda_1>2\\lambda$ holds for all other eigenvalues $\\lambda$ of $R$. \\\\\n{\\bf (A3):} {\\em Strictly balanced:}  \tThe column vector $\\mathbf{u}_1=(1,1,\\dots,1)^\\top$,  is a right eigenvector \nof $R$ corresponding to $\\lambda_1$;   and it has a principal left eigenvector $\\bf{v}_1$ (i.e., the \nleft eigenvectors corresponding to $\\lambda_1$) that is also a  probability  vector.\n\\\\\n{\\bf (A4):} {\\em Diagonalisable:} There exists an invertible matrix $V$ with real entries whose first row is $\\bf{v}_1$ such that the first column of $V^{-1}$ is $\\mathbf{u}_1$ and \n\\begin{equation}\n\\label{eq:R:diagonal}\nVRV^{-1} = \\text{diag}(\\lambda_1, \\lambda_2,\\dots, \\lambda_d) =: \\Lambda,\n\\end{equation}\nwhere $\\lambda_1> \\lambda_2 \\ge \\dots \\ge \\lambda_d$ are  eigenvalues of $R$. \n\n\n\\medskip\n\nLet $\\mathcal{N}(\\mathbf{0}, \\Sigma)$ be the multivariate normal distribution with mean vector $\\mathbf{0}=(0,\\dots,0)$ and covariance matrix $\\Sigma$. Then we have the following result from~\\cite[Theorems 1 \\& 2]{Paper1}, which can also be alternatively derived from~\\cite[Theorems 3.21 \\& 3.22 and Remark 4.2]{Janson2004}. \n\n\\begin{theorem}\n\t\\label{thm1&2:paper1}\n\tUnder assumptions {\\em \\bf{(A1)--(A4)}}, we have\n\t\\begin{equation}\n\t\\label{eq:asconv:urn}\n\t(n\\pev)^{-1} U_n \\ \\stackrel{\\mbox{a.s.}}{\\longrightarrow} \\  \\mathbf{v}_1\n\t~\\quad~\\mbox{and}~\\quad~ \n\tn^{-1\/2} (U_n - n\\pev \\mathbf{v}_1) \\xrightarrow{~d~}  \\mbox{${\\mathcal N}$}(\\mathbf{0}, \\Sigma),\n\t\\end{equation}\n\twhere $\\pev$ is the principal eigenvalue and $\\mathbf{v}_1$ is the principal left eigenvector of $R$, and \n\t\\begin{equation}\n\t\\Sigma = \\sum_{i,j=2}^d \\frac{\\pev \\lambda_i \\lambda _j {\\mathbf u}_i^\\top \\mbox{\\em diag}(\\mathbf{v}_1) {\\mathbf u}_j }{\\pev-\\lambda_i -\\lambda_j} \\mathbf{v}_i^\\top \\mathbf{v}_j,\n\t\\end{equation}\n\twhere  ${\\mathbf v}_j$ is the $j$-th row of $V$  and ${\\mathbf u}_j$ the $j$-th column of $V^{-1}$ for $2\\le j \\le d$.\n\\end{theorem}\n\n\n\n\\section{Limit Theorems for the Joint Distribution}\n\\label{sec:limiting}\n\nIn this section, we present the strong laws of large numbers and the central limit theorems on the joint distribution of the number of cherries and that of pitchforks under the Ford model.\n\n\n\\subsection{Main convergence results}\n\n\nFor later use, we consider the following polynomials in $\\alpha$: \n\\begin{equation}\n\\label{eq:main:cov:poly}\n\\begin{matrix*}[l]\n\\gp_1 = 8\\alpha^3-32\\alpha^2+45\\alpha-23, & \\quad\\quad\\quad \\gp_4 = 8\\alpha^3-40\\alpha^2+37\\alpha+13, \\\\\n\\gp_2 = 40\\alpha^3-164\\alpha^2+221\\alpha-97, &  \\quad\\quad\\quad  \\gp_5 = 40\\alpha^3-112\\alpha^2-31\\alpha+181, \\\\\n\\gp_3 = 56\\alpha^3-248\\alpha^2+367\\alpha-181, &  \\quad\\quad\\quad  \\gp_6 = 8\\alpha^3+4\\alpha^2-71\\alpha+71; \n\\end{matrix*}\n\\end{equation}\nand for simplicity of notation, we do not indicate the $\\phi_i$'s as  functions of $\\alpha$. Moreover, it can be verified directly that $\\phi_1, \\phi_2, \\phi_3 <0$ and $\\phi_4, \\phi_5, \\phi_6 > 0$ for $\\alpha \\in (0, 1).$\nThen, we have the following result on the joint asymptotic properties of the urn model process associated with the $\\alpha$-tree model.\n\n\n\\begin{theorem}\n\t\\label{thm:urn:edge}\n\tSuppose $(U_n)_{n\\geq 0}$  is the urn process associated with the Ford model with parameter $\\alpha\\in(0,1)$. \n\tThen, \n\t\\begin{equation}\n\t\\label{eq:conv:urn}\n\t\\frac{U_n}{n} \\ \\stackrel{\\mbox{a.s.}}{\\longrightarrow} \\  \n\t\\mathbf{v}\t\n\t~\\quad~\\mbox{and}~\\quad~\n\t\\frac{ U_n- n\\mathbf{v} }{\\sqrt{n}} \\xrightarrow{~d~}  \\mathcal{N}\\left (\\mathbf{0},\\Sigma \\right),\n\t\\end{equation}\n\tas $n \\to \\infty$, where \n\t\\begin{equation}\\label{main:Leftev1}\n\t{\\mathbf v}= \\frac{1}{2(3-2\\alpha)} \\left(2(1-\\alpha), \\,2(1-\\alpha),\\, (1-\\alpha),\\, 1+\\alpha,\\, 1-\\alpha, \\,5-3\\alpha \\right)\n\t\\end{equation}\nand with the polynomials $\\gp_1,\\dots,\\gp_6$ defined in~\\eqref{eq:main:cov:poly},\n\\begin{equation}\n\\Sigma=\\frac{1-\\alpha}{4(3-2\\alpha)^2(5-4\\alpha)(7-4\\alpha)} \n\\begin{bmatrix*}[r]\n-12\\gp_1 & 4\\gp_2 & -6\\gp_1  & -2\\gp_4 & 2 \\gp_2 & -2 \\gp_2 \\\\\n4\\gp_2 & -4\\gp_3 & 2\\gp_2  & -2\\gp_6 & -2 \\gp_3 & 2 \\gp_3   \\\\\n-6\\gp_1 & 2\\gp_2  & -3\\gp_1  & -\\gp_4 &  \\gp_2 & - \\gp_2  \\\\\n-2\\gp_4  &  -2\\gp_6  &  -\\gp_4 & \\gp_5 & - \\gp_6 &  \\gp_6  \\\\\n2 \\gp_2   &  -2 \\gp_3  &  \\gp_2& - \\gp_6 & - \\gp_3 &  \\gp_3  \\\\\n-2 \\gp_2    &  2 \\gp_3   &  - \\gp_2 & \\gp_6  &   \\gp_3 & -\\gp_3  \n\\end{bmatrix*}.\n\\end{equation}\n\\end{theorem}\nThe proof of Theorem~\\ref{thm:urn:edge} is given at  the end of this section.\n\n\n\\begin{remark} \n\\label{rem3.1} For later use, here we present the limiting results on the urn model using a scaling factor relating to the time $n$ (which is motivated by noting that the number of leaves in the tree  at time $n$ is $n+2$). However, the results can be readily rephrased using the proportion of color balls in the urn process.\n\n\\end{remark}\n\n\\begin{remark} \n\tUsing the approach outlined in~\\cite{Paper1},  Theorem \\ref{thm:urn:edge} \tcontinues to hold for  the unrooted $\\alpha$-tree models. \n\\end{remark}\n\nWith Theorem~\\ref{thm:urn:edge}, we are ready to present one of our main results in this paper \nconcerning limit theorems on the joint distribution of the number of cherries $C_n$ and the number of pitchforks $A_n$\nunder the Ford model. \n\n\n\\begin{theorem} \\label{Thm:Convg-ChPh}\nUnder the Ford model with parameter $\\alpha \\in [0,1]$, we have\n\t\\[\\frac{1}{n} (A_n,C_n) \\ \\stackrel{\\mbox{a.s.}}{\\longrightarrow} \\  (\\nu, \\mu) := \\frac{1-\\alpha}{2(3-2\\alpha)} (1,2),\\]\n\tand\n\t\\[ \\frac{(A_n, C_n) -n (\\nu, \\mu) }{\\sqrt{n}} \\xrightarrow{~d~}  \\mathcal{N}\\big((0,0), S \\big), \\]\n\twhere \\begin{equation}\n\tS= \\begin{bmatrix} \\tau^2 & \\rho \\\\ \\rho & \\si^2 \\\\ \\end{bmatrix} =\n\t\\frac{1-\\alpha}{(3- 2\\alpha)^2 (5-4\\alpha) } \\begin{bmatrix}\n\t\\frac{-24 \\alpha^3 +96\\alpha^2 -135\\alpha +69 }{4(7-4\\alpha)} & \\frac{-(2-\\alpha) (1-2\\alpha)}{2} \\\\[1ex]\n\t\\frac{-(2-\\alpha) (1-2\\alpha)}{2} & 2-\\alpha\n\t\\end{bmatrix}.\n\t\\end{equation}\n\\end{theorem}\n\\medskip\n \\begin{remark} \nWe consider special cases of $\\alpha$-tree model, which are commonly studied in phylogenetics. The first two have been established in ~\\cite{Paper1}.\n\n\\begin{enumerate}\n\t\\item The uniform model corresponds to  $\\alpha=1\/2$, where all edges, internal or leaf, are selected with equal weight and the limit results hold with\n\t\\[(\\nu, \\mu) = \\frac{1}{8}(1, 2)\n\t\\quad \\text{and } \\quad   \\begin{bmatrix} \\tau^2 & \\rho \\\\ \\rho & \\si^2 \\\\ \\end{bmatrix}   =\n\t\\frac{1}{64}\\begin{bmatrix} 3&0\\\\0&4\\end{bmatrix}. \\]\n\t\\item The Yule model corresponds to  $\\alpha =0$, where only leaf edges are selected with equal weight and the limit results hold with\n\t\\[ (\\nu, \\mu) = \\frac{1}{6} (1,2)\n\t\\quad \\text{and } \\quad  \\begin{bmatrix}\n\t\\tau^2 & \\rho \\\\\n\t\\rho & \\si^2 \\\\\n\t\\end{bmatrix}  =  \\frac{1}{45}  \\begin{bmatrix} 69\/28  &-1  \\\\  -1 & 2 \\end{bmatrix}.  \\]\n\t\n\\item The  Comb model corresponds to $\\alpha =1$,  a degenerate case. It is easy to see that  $(\\nu, \\mu) = (0,0) $ and $\\tau^2= \\rho = \\si^2  = 0$.\n\t\n\\end{enumerate}\n \\end{remark}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{Thm:Convg-ChPh}]\n\tFirst note that the case $\\alpha=1$ reduces to a degenerate case of Comb model and therefore we only consider $\\alpha\\in[0,1)$. The limiting results for the case $\\alpha=0$ has been obtained in \\cite{Paper1}, which agree with the above results when $\\alpha=0$. Thus, it is enough to prove the result for $\\alpha\\in (0,1)$.\n\t\n\t By~\\eqref{urn-pf-ch}, we have \t \n\t$ (A_n, C_n) = U_n Q $\n\twith\n\t\\begin{equation}\\label{Def-T}\n\tQ^\\top = \\frac{1}{2}\\begin{bmatrix} 1 &0&0&0&0&0\\\\\n\t1&1&0&0&0&0\n\t\\end{bmatrix}.\n\t\\end{equation}\t\t\nSince\n\t\\begin{equation}\n\t\\frac{U_n }{n} \\ \\stackrel{\\mbox{a.s.}}{\\longrightarrow} \\   {\\mathbf v}  = \\frac{1}{2(3-2\\alpha)} \\big(2(1-\\alpha), 2(1-\\alpha), 1-\\alpha, 1+\\alpha, 1-\\alpha, 5-3\\alpha \\big),\n\t\\end{equation}\n\tusing the relation from equation \\eqref{urn-pf-ch} we get\n\t\\[\\frac{1}{n} (A_n,C_n) =\\left(\\frac{U_n}{n} \\right) Q \\ \\stackrel{\\mbox{a.s.}}{\\longrightarrow} \\  {\\mathbf v}\\,Q = \\frac{1-\\alpha}{2(3-2\\alpha)} (1,2). \\]\n\tThis concludes the proof of the almost sure convergence. We now prove the central limit theorem and obtain the expression for the limiting variance matrix.\n\t\n\t\n\tDenoting covariance matrix $\\Sigma$ by $(\\sigma_{i,j})$ for $1\\le i,j\\le 6$, we   consider the matrix \n\t\\begin{align*}\n\tS&= Q^\\top  \\Sigma Q\n\t=\\frac{1}{4} \\begin{bmatrix}\n\t\\sigma_{1,1} & \\sigma_{1,1}+\\sigma_{1,2}\\\\\n\t\\sigma_{1,1}+\\sigma_{2,1}& \\sigma_{1,1}+\\sigma_{2,1}+\\sigma_{1,2}+\\sigma_{2,2}\n\t\\end{bmatrix} \\\\[1ex]\n\t&=\\frac{1-\\alpha}{16(3-2\\alpha)^2(5-4\\alpha)(7-4\\alpha)} \n\t\\begin{bmatrix}\n\t-12\\gp_1 & -12\\gp_1+4\\gp_2\\\\\n\t-12\\gp_1+4\\gp_2 & -12\\gp_1+8\\gp_2-4\\gp_3\n\t\\end{bmatrix} \\\\[1ex]\n\t\t&=\\frac{1-\\alpha}{(3- 2\\alpha)^2 (5-4\\alpha) } \\begin{bmatrix}\n\t\\frac{-24 \\alpha^3 +96\\alpha^2 -135\\alpha +69 }{4(7-4\\alpha)} & \\frac{-(2-\\alpha) (1-2\\alpha)}{2} \\\\[2ex]\n\t\\frac{-(2-\\alpha) (1-2\\alpha)}{2} & 2-\\alpha\n\t\\end{bmatrix}.\n\t\\end{align*}\nSince  $(A_n, C_n) = U_n Q$, where $Q$ is as defined in \\eqref{Def-T}, we get\n\t\\[ \\frac{(A_n, C_n) -n (\\nu, \\mu) }{\\sqrt{n}} = \\frac{1}{\\sqrt{n}} \\left(U_n -n {\\mathbf v} \\right)Q \\xrightarrow{~d~}   \\mathcal{N}\\left(\\mathbf{0}, Q^\\top  \\Sigma Q \\right)\n\t=\\mathcal{N}\\left(\\mathbf{0}, S \\right)\n\t.\\]\n This completes the proof.\n\\end{proof}\n\n\nWe end this subsection with the following results on the behaviour of the first and second moments of the limiting joint distribution of cherries and pitchforks in the parameter region, as indicated by their plots in Figure~\\ref{fig:limit:cov}.\n\n\\begin{corollary}\n\t\\begin{enumerate}[\\normalfont(i)]\n\t\t\\item For $0< \\alpha <1$,  $A_n\/C_n  \\ \\stackrel{\\mbox{a.s.}}{\\longrightarrow} \\  1\/2$ as $n\\to \\infty$. That is, the number of pitchforks is asymptotically equal to the number of essential cherries.\n\t\t\n\t\t\\item $ A_n\/n \\ \\stackrel{\\mbox{a.s.}}{\\longrightarrow} \\   \\frac{1-\\alpha}{2(3-2\\alpha)}, $ which decreases strictly from $1\/6$ to $0$, as $\\alpha$ increases from $0$ to $1$.\n\t\t\n\t\t\n\t\t\\item  The limiting variance of $A_n\/\\sqrt{n}$,  $\\tau^2$, decreases strictly from $23\/420$ to $0$, as $\\alpha$ increases from $0$ to $1$.\n\t\t\\item The limiting variance of $C_n\/\\sqrt{n} $, $\\si^2$, increases strictly from $2\/45$\n\t\tto $0.0695$  over $(0, a_0)$ and decreases from $0.0695$ to $0$ over $(a_0, 1)$, where $a_0 =0.7339$,\n\t\t the unique root of $19-48\\alpha+36\\alpha^2-8\\alpha^3 =0$ in $(0,1)$.\n\t\t\n\t\t\\item  The limiting covariance of $A_n\/\\sqrt{n}$ and $C_n\/\\sqrt{n}$ changes sign from negative to positive at $\\alpha =1\/2$. Specifically, it increases from $-1\/45$ \n\t\tto $0.0225$ \n\t\tover $(0, a_1)$ and decreases from $0.0225$\n\t\tover $(a_1, 1)$, where $a_1=0.8688$,\n\t\tthe unique root of $-24\\alpha^4+160\\alpha^3 -370\\alpha^2 +358\\alpha -123=0$ in $(0,1)$.\n\t\\end{enumerate}\n\\end{corollary}\n\n\\begin{figure}[H] \n\t\\label{fig:limit:cov}\n\t\\centering\n\t\\includegraphics[width=0.8\\linewidth,height=0.38\\textheight]{Rplot_New.png}\n\t\\caption{Plot of the limiting  covariances of the joint distribution of cherries and pitchforks with respect to the parameter $\\alpha$ under the Ford model.}\n\\end{figure}\n\n\\subsection{A uniform urn model derived from $U_n$}\n\nFor $\\alpha\\in (0,1)$, consider the diagonal $6\\times 6$ matrix\n$\nT_\\alpha=\\text{diag}(1-\\alpha,1-\\alpha,1-\\alpha,1-\\alpha,\\alpha,\\alpha)\n$ \nand \\[ \\widetilde{U}_n := U_nT_\\alpha = \\left((1-\\alpha) U_{n,1}, \\dots, (1-\\alpha) U_{n,4}, \\alpha U_{n,5}, \\alpha U_{n,6}\\right). \\]\nClearly, there is a one to one correspondence between $U_n$ and $\\widetilde{U}_n= U_nT_\\alpha$ for $\\alpha\\in (0,1)$ and therefore it is sufficient to obtain the limiting results for the urn process $\\widetilde{U}_n$. Note that the off-diagonal elements of the replacement matrix $R_\\alpha$ are not all  non-negative, therefore we will use the limit results from \\cite{Paper1} to  obtain the convergence results for  the urn process $\\widetilde{U}_n$.\n\n\n\\begin{theorem}\\label{Thm:Urn}\nSuppose $\\alpha\\in(0,1)$. Then \t$(\\widetilde{U}_n)_{n\\geq 0}$ is an uniform urn process with replacement matrix $ R_\\alpha = RT_\\alpha$ and\n\\begin{equation}\n\\label{eq:cas:nonuniform}\n\\frac{\\widetilde{U}_n }{n} \\ \\stackrel{\\mbox{a.s.}}{\\longrightarrow} \\  \\widetilde{{\\mathbf v}}_1,\n\\end{equation}\nwhere \\begin{equation}\\label{Leftev1}\n\\widetilde{{\\mathbf v}}_1= \\frac{1}{2(3-2\\alpha)} \\big(2(1-\\alpha)^2, 2(1-\\alpha)^2, (1-\\alpha)^2, 1-\\alpha^2, \\alpha(1-\\alpha), \\alpha(5-3\\alpha) \\big)\n\\end{equation}\nis the normalized left eigenvector of $R_\\alpha$ corresponding to the largest eigenvalue $\\lambda_1=1$. Furthermore,\n\\begin{equation}\n\\label{eq:cweak:nonuniform}\n\\frac{\\widetilde{U}_n -n \\widetilde{{\\mathbf v}}_1}{\\sqrt{n}} \\xrightarrow{~d~}   \\mathcal{N}(\\mathbf{0}, \\widetilde{\\Sigma}),\n\\end{equation}\nwith the polynomials $\\gp_1,\\dots,\\gp_6$ defined in~\\eqref{eq:main:cov:poly} and $\\beta=1-\\alpha$,\n\\begin{equation}\n\\label{eq:covariance:nonuniform}\n\\widetilde{\\Sigma}=\\frac{\\beta}{4(3-2\\alpha)^2(5-4\\alpha)(7-4\\alpha)} \n\\begin{bmatrix*}[r]\n-12\\beta^2\\gp_1 & 4\\beta^2\\gp_2 & -6\\beta^2\\gp_1  & -2\\beta^2\\gp_4 & 2\\alpha\\beta \\gp_2 & -2\\alpha\\beta \\gp_2 \\\\\n4\\beta^2\\gp_2 & -4\\beta^2\\gp_3 & 2\\beta^2\\gp_2  & -2\\beta^2\\gp_6 & -2\\alpha\\beta \\gp_3 & 2\\alpha\\beta \\gp_3   \\\\\n-6\\beta^2\\gp_1 & 2\\beta^2\\gp_2  & -3\\beta^2\\gp_1  & -\\beta^2\\gp_4 & \\alpha\\beta \\gp_2 & -\\alpha\\beta \\gp_2  \\\\\n-2\\beta^2\\gp_4  &  -2\\beta^2\\gp_6  &  -\\beta^2\\gp_4 & \\beta^2\\gp_5 & -\\alpha\\beta \\gp_6 & \\alpha\\beta \\gp_6  \\\\\n2\\alpha\\beta \\gp_2   &  -2\\alpha\\beta \\gp_3  &  \\alpha\\beta \\gp_2& -\\alpha\\beta \\gp_6 & -\\alpha^2 \\gp_3 & \\alpha^2 \\gp_3  \\\\\n-2\\alpha\\beta \\gp_2    &  2\\alpha\\beta \\gp_3   &  -\\alpha\\beta \\gp_2 & \\alpha\\beta \\gp_6  &  \\alpha^2 \\gp_3 & -\\alpha^2 \\gp_3  \n\\end{bmatrix*}.\n\\end{equation}\n\n\\end{theorem}\n\\begin{proof}[Proof of Theorem \\ref{Thm:Urn}]\n\tFirst, observe that at any time $n$, there are $n+2$ pendant edges and $n+1$ internal edges in a rooted tree. That is, \n\\[ U_{n,1}+ U_{n,2}+ U_{n,3}+ U_{n,4}= n+2 \\quad \\text{and } \\quad U_{n,5}+ U_{n,6} = n+1.\\]\nThis gives\n\\begin{align*}\n\\|\\widetilde{U}_n\\|_1\n= (1-\\alpha) \\sum_{j=1}^4 U_{n,j}+ \\alpha  \\sum_{j=5}^6 U_{n,j}  \n= (1-\\alpha) (n+2) + \\alpha (n+1) = n+2 -\\alpha.\n\\end{align*}\nTherefore, from \\eqref{SelecProb} we get,\n\\[ \\mbox{${\\mathbb E}$}[\\chi_{n}| \\mbox{${\\mathcal F}$}_{n-1}] = \\dfrac{U_{n-1} T_{\\alpha}}{\\|U_{n-1}T_{\\alpha}\\|_1}= \\dfrac{U_{n-1} T_{\\alpha}}{ n+1 -\\alpha},\n\\]\nand\n\\begin{align*}\n\\mbox{${\\mathbb E}$}[U_{n}|\\mbox{${\\mathcal F}$}_{n-1}]\n= U_{n-1} + \\mbox{${\\mathbb E}$}[\\chi_{n}|\\mbox{${\\mathcal F}$}_{n-1}] R\n= U_{n-1} + \\dfrac{1}{n+1 -\\alpha} U_{n-1} T_\\alpha R.\n\\end{align*}\nMultiplying both sides by $T_\\alpha$,  we get\n\n\\[\\mbox{${\\mathbb E}$}[\\widetilde{U}_{n}|\\mbox{${\\mathcal F}$}_{n-1}] = \\widetilde{U}_{n-1} + \\left(\\dfrac{1}{\\|\\widetilde{U}_{n-1} \\|_1} \\widetilde{U}_{n-1} \\right) R T_\\alpha. \\]\nHence, $(\\widetilde{U}_n)_{n\\geq 0}$ is a classical uniform urn model with replacement matrix $ R_\\alpha = RT_\\alpha$.\n\n\nNote that  {\\bf (A1)} holds because the general Ford's dynamics on a rooted tree is well defined at every time $n$, thus the corresponding urn model satisfies the assumption of tenability. That is, it is always possible to draw balls  without getting stuck  with the replacement rule.\nNote that $R_\\alpha$ is diagonalisable as \n\\[V R_\\alpha V^{-1}=\\Lambda \\]\nholds with  $ \\Lambda = \\text{diag} \\big(1,0,0,0, -2(1-\\alpha),-(3-2\\alpha)\\big)$,\n\\begin{equation} \\label{Reigen:AalphaR}\nV^{-1}= \\begin{bmatrix}\n1& \\frac{1}{\\beta}&0&0&1& 1-\\alpha\\\\[1ex]\n1& 0&\\frac{1}{\\beta}&0&1& 3-\\alpha \\\\[1ex]\n1& \\frac{-2}{\\beta}&0& \\frac{3}{\\beta}&\\frac{-(2-\\alpha)}{\\beta}&-5+\\alpha \\\\[1ex]\n1 &0&0&\\frac{1}{\\beta}&\\frac{-(2-\\alpha)}{\\beta}&-3+\\alpha \\\\[1ex]\n1 & 0&\\frac{-2}{\\alpha}&\\frac{1}{\\alpha}&1&3-\\alpha \\\\[1ex]\n1 & 0&0&\\frac{-1}{\\alpha} & 1& 1-\\alpha\n\\end{bmatrix}\n\\end{equation}\n\nand \n\t\\setlength{\\arraycolsep}{2.5pt}\n\\medmuskip = 1mu\n\\begin{align}\nV =\\frac{1}{2(3-2\\alpha) }\n\\begin{bmatrix}\\label{Leigen:AalphaR:1}\n2\\beta^2 & 2\\beta^2 & \\beta^2 & (1+\\alpha)\\beta & \\alpha\\beta& \\alpha(5-3\\alpha)\\\\[1ex]\n2\\beta(1+\\alpha-\\alpha^2) & 2\\beta^3 &-(2-\\alpha)\\beta^2 & (2-\\alpha)\\beta^2 & -\\alpha\\beta^2& -\\alpha\\beta(5-3\\alpha) \\\\[1ex]\n2\\alpha\\beta^2 & 2\\alpha(2-\\alpha)\\beta & \\alpha\\beta^2 &-\\alpha\\beta^2 & -\\alpha(3- \\alpha)\\beta & -3\\alpha\\beta^2 \\\\[1ex]\n2\\alpha(2-\\alpha)\\beta & 2\\alpha\\beta^2 & \\alpha(2-\\alpha)\\beta &-\\alpha(2-\\alpha)\\beta & \\alpha^2 \\beta & -3\\alpha(2-\\alpha)\\beta \\\\[1ex]\n2(2-\\alpha)\\beta & -2\\beta^2 & (2-\\alpha)\\beta & -(4-\\alpha)\\beta & -\\alpha\\beta &\\alpha\\beta\\\\[1ex]\n-2\\beta & 2\\beta & -\\beta & \\beta & \\alpha & -\\alpha\n\\end{bmatrix}.\n\\end{align}\nTherefore, $R$ satisfies condition {\\bf (A4)}. Next, {\\bf (A2)} holds because $R_\\alpha$ has eigenvalues \n$$1,\\quad 0,\\quad 0, \\quad 0, \\quad -2(1-\\alpha),\\quad -(3-2\\alpha)$$ which are all real. The maximal eigenvalue $\\lambda_1=1$ is positive with $\\lambda_1>2\\lambda$ holds for all other eigenvalues $\\lambda$ of $R_\\alpha$.\nFurthermore, put $\\mathbf{u}_i={V^{-1}}\\mathbf{e}^\\top_i$ and $\\mathbf{v}_i=\\mathbf{e}_iV$ for $1\\le i \\le 4$. Then {\\bf (A3)} follows by noting that $\\mathbf{u}_1=(1,1,1,1,1,1)^\\top$ is the principal right eigenvector, and\n\\[\\widetilde{{\\mathbf v}}_1= \\frac{1}{2(3-2\\alpha)} \\big(2(1-\\alpha)^2, 2(1-\\alpha)^2, (1-\\alpha)^2, 1-\\alpha^2, \\alpha(1-\\alpha), \\alpha(5-3\\alpha) \\big)\\]  is the principal left eigenvector. \n\n\nSince all the assumptions {\\bf (A1)--(A4)} are satisfied  by the replacement matrix $R_\\alpha$, by Theorem~\\ref{thm1&2:paper1},\n\\eqref{eq:cas:nonuniform} holds. Furthermore, since\n\\begin{equation} \n\\widetilde{\\Sigma} = \\sum_{i,j=2}^6 \\frac{ \\lambda_i \\lambda _j {\\mathbf u}_i^\\top \\mbox{diag}(\\mathbf{v}_1) {\\mathbf u}_j }{1-\\lambda_i -\\lambda_j} \\mathbf{v}_i^\\top \\mathbf{v}_j,\n\\end{equation}\nby~\\eqref{eq:cas:nonuniform} it follows that~\\eqref{eq:cweak:nonuniform} holds.\n\\end{proof}\n\n\n\n\\subsection{Proof of Theorem \\ref{thm:urn:edge}}\n\n\n\n\n\\begin{proof}\n\t\\noindent\n\tObserve that  $\\sum_{i=1}^6U_{n,i} = 3+2n$ (since $2$ balls are added into the urn at every time point), thus the vector of color proportions is $U_n \/(3+2n)$.\n\n\tSince $\\alpha \\in (0,1)$, it follows that  $T_\\alpha$ is invertible and its inverse is \n\t$$\n\tT_\\alpha^{-1}=\\frac{1}{\\alpha(1-\\alpha)}\\mbox{diag}(\\alpha,\\alpha,\\alpha,\\alpha,1-\\alpha,1-\\alpha),\n\t$$\n\twhich is also a diagonal matrix, and so $(T_\\alpha^{-1})^\\top=T_\\alpha^{-1}$.\n\tNote that  we have $U_n = \\widetilde{U}_n T_\\alpha^{-1}$ and consider \n\t$$\n\t{{\\mathbf v}} = \\widetilde{{\\mathbf v}}_1 (T_\\alpha)^{-1} = \\frac{1}{2(3-2\\alpha)} \\big(2(1-\\alpha), 2(1-\\alpha), 1-\\alpha, 1+\\alpha, 1-\\alpha, 5-3\\alpha \\big).\n\t$$\n\tSince $ \\dfrac{\\widetilde{U}_n }{n} \\ \\stackrel{\\mbox{a.s.}}{\\longrightarrow} \\  \\widetilde{{\\mathbf v}}_1$ holds in view of~\\eqref{eq:cas:nonuniform} in Theorem \\ref{Thm:Urn}, \n\n\t\\begin{equation}\n\t\\frac{U_n }{n} \\ \\stackrel{\\mbox{a.s.}}{\\longrightarrow} \\  {{\\mathbf v}},\n\t\\end{equation}\n\twhich concludes the proof of the almost sure convergence in~\\eqref{eq:conv:urn}. \n\t\n\tConsider the covariance matrix $\\widetilde{\\Sigma}$ for $ \\widetilde{U}_n$ as stated in~\\eqref{eq:covariance:nonuniform}, then by straightforward calculation we have \n\t$$\n\t\\Sigma=(T_\\alpha^{-1})^\\top \\widetilde{\\Sigma} T_\\alpha^{-1}=T_\\alpha^{-1}  \\widetilde{\\Sigma} T_\\alpha^{-1}. \n\t$$\n\tTherefore, since\n\t\\[\\frac{\\widetilde{U}_n -n \\widetilde{{\\mathbf v}}_1}{\\sqrt{n}} \\xrightarrow{~d~}  \\mathcal{N} (\\mathbf{0}, \\widetilde{\\Sigma} ) \\]\n\tin view of Theorem \\ref{Thm:Urn}, we get\n\t\\[\\frac{U_n -n {\\mathbf v} }{\\sqrt{n}} \\xrightarrow{~d~}  \\mathcal{N} \\big(\\mathbf{0}, (T_\\alpha^{-1})^\\top \\widetilde{\\Sigma}\\, T_\\alpha^{-1}\\big)=\\mathcal{N} (\\mathbf{0},  \\Sigma). \\]\n\tThis completes the proof.\n\\end{proof}\n\n\n\n\\section{Exact Distributions }\n\\label{sec:exact}\n\nIn this section, we present recursion formulas for exact computation of  the joint distributions of cherries and pitchforks, their means, variances and covariance for fixed $n$ under the Ford model.\n\n\n\nWe begin with the following notation. Given a phylogenetic tree $T$, let $E_{1}(T)$ be the set of pendant edges that are contained in a pitchfork but not a cherry; $E_{2}(T)$ the set of edges in $T$\nthat are contained in a cherry but not in a pitchfork (note that in our notation a cherry contains three leaves);\n$E_{3}(T)$ the set of pendant edges that are contained in neither a cherry nor a pitchfork; and $E_{4}(T)=E(T)\\setminus (E_{1}(T)\\cup E_{2}(T) \\cup E_{3}(T))$.\nIn addition, $E(T)$ can be decomposed into the disjoint union of these four sets of edges.\ni.e., $E(T)=E_1(T)\\sqcup E_2(T) \\sqcup E_3(T) \\sqcup E_4(T)$, where $\\sqcup$ denotes disjoint union.\nLet $C(T), A(T)$ be the number of cherries and pitchforks in a tree $T$. The following result presented in \\cite{WuChoi16}  will be useful later. \n\n\\begin{lemma}\t\\label{lem:edge-set}\n\tSuppose that $T$ is a phylogenetic tree with $n$ leaves. Then we have\n\t\\begin{equation}\n\t\\label{eq:edge:dec}\n\tE(T)=E_{1}(T)\\,\\sqcup\\,E_{2}(T)\\,\\sqcup\\,E_{3}(T)\\,\\sqcup\\,E_{4}(T).\n\t\\end{equation}\n\tIn addition, we have\n\n\t$|E_{1}(T)|=A(T)$,\n\t$|E_{2}(T)|=3(C(T)-A(T))$,\n\t$|E_{3}(T)|=n-A(T)-2C(T)$,\n\tand\n\t$|E_{4}(T)|=n-1+3A(T)-C(T)$.\nFurthermore, suppose that $e$ is an edge in $T$ and $T'=T[e]$. Then\n\t\twe have\n\t\t\\small{\n\t\t\t\\begin{equation*}\n\t\t\tA(T') = \\begin{cases}\n\t\t\tA(T) & \\text{if } e\\in E_3(T)\\cup E_4(T), \\\\\n\t\t\tA(T)-1 & \\text{if } e\\in E_1(T),\\\\\n\t\t\tA(T)+1 & \\text{if } e\\in E_2(T);\n\t\t\t\\end{cases}\n\t\t\t\\mbox{and} \\quad\n\t\t\tC(T') = \\begin{cases}\n\t\t\tC(T) & \\text{if } e \\in E_2(T)\\cup E_4(T), \\\\\n\t\t\t{} & \\\\\n\t\t\tC(T)+1 & \\text{if } e \\in E_1(T)\\cup E_3(T).\n\t\t\t\\end{cases}\n\t\t\t\\end{equation*}\n\t\t}\n\n\\end{lemma}\n\nWe start with the following result on the exact computation of the  joint probability mass function (pmf)  of $A_n$ and $C_n$, \nwhich can be regarded as a generalization of the previous results on the Yule model (e.g. when $\\alpha=0$~\\cite[Theorem 1]{WuChoi16}) and  the uniform model (e.g. $\\alpha=1\/2$~\\cite[Theorem 4]{WuChoi16}).\nA related result for unrooted  trees is presented in~\\cite{CTW19}.\n\n\\begin{theorem} \\label{jointpmf}\nFor $n \\ge 3$, $0 \\le  a\\le n\/3$ and $1\\le b\\le n\/2$, under the Ford model with parameter $\\alpha\\in [0,1]$  we have\n\\begin{eqnarray*}\n&& \\mathbb{P}(A_{n+1}=a, C_{n+1}=b) \\\\&=&  \\frac{2a+ \\alpha(n-a-b-1)}{n-\\alpha} \\mathbb{P}(A_n=a, C_n=b)\n+ \\frac{(1-\\alpha)(a+1)}{n-\\alpha} \\mathbb{P}(A_n=a+1, C_n=b-1) \\\\\n&& \n\\quad\n+ \\frac{(2-\\alpha)(b-a+1)}{n-\\alpha} \\mathbb{P}(A_n=a-1, C_n=b) + \\frac{(1-\\alpha)(n-a-2b+2)}{n-\\alpha} \\mathbb{P}(A_n=a, C_n=b-1).\n\\end{eqnarray*}\n\\end{theorem}\n\\begin{proof}[Proof of Theorem \\ref {jointpmf}]\nFix $n> 3$, and let $T_2,\\dots,T_n,T_{n+1}$ be a sequence of random trees generated by the Ford process, that is, $T_2$ contains two leaves and $T_{i+1}=T_i[e_i]$ for a random edge $e_i$ in $T_i$  chosen according to the Ford model for $2\\leq i \\leq n$. \n\tThen we have\n\t\\begin{align}\n\t\\label{eq:total:yule}\n\t\\mathbb{P}(A_{n+1}=a, C_{n+1}=b) &=\\mathbb{P}(A(T_{n+1})=a, C(T_{n+1})=b) \\notag \\\\\t\n\t&\\hspace{-2cm}\n\t=\\sum_{p,q} \\mathbb{P}(A(T_{n+1})=a, C(T_{n+1})=b\\,|\\,A(T_n)=p,C(T_n)=q) \\mathbb{P}(A(T_n)=p,C(T_n)=q) \\notag \\\\\n\t\\hspace{-1cm}\n\t&\\hspace{-2cm}\n\t=\\sum_{p,q} \\mathbb{P} (A(T_{n+1})=a, C(T_{n+1})=b\\,|\\,A(T_n)=p,C(T_n)=q) \\mathbb{P}(A_{n}=p, C_{n}=q),\n\t\\end{align}\n\twhere the first and second equalities follow from the law of total probability, and the definition of random variables $A_n$ and $C_n$.\n\t\nLet $e_n$ be the edge in $T_n$ chosen in the above Ford process for generating $T_{n+1}$, that is, $T_{n+1}=T_n[e_n]$. Since Lemma~\\ref{lem:edge-set} implies that\n\t\\begin{equation}\n\t\\label{eq:yule:5}\n\t\\mathbb{P}(A(T_{n+1})=a, C(T_{n+1})=b~|~A(T_n)=p,C(T_n)=q)=0 \n\t\\end{equation}\n\tfor $(p,q) \\not \\in\\{(a,b),(a+1,b-1),(a-1,b),(a,b-1)\\}$,\n\tit suffices to consider the following four cases in the summation in (\\ref{eq:total:yule}): case (i): $p=a, q=b$; case (ii): $p=a+1, q=b-1$; case (iii): $p=a-1, q=b$; and case (iv): $p=a, q=b-1$.\n\t\n\t\n\tFirstly, Lemma~\\ref{lem:edge-set} implies that case (i) occurs if and only if $e_n\\in E_4(T_n)$. Using Lemma~\\ref{lem:edge-set} again, it follows that $E_4(T_n)$ contains precisely $2A(T_n)$ pendent edges and $(n-1)+A(T_n)-C(T_n)$ interior edges. Therefore we have\n\t\\begin{align}\n\t&\\mathbb{P}(A(T_{n+1})=a, C(T_{n+1})=b~|~A(T_n)=a,C(T_n)=b)  \\nonumber \\\\\n\t&\\quad \\quad =\\frac{2A(T_n)(1-\\alpha)+\\alpha(n-1+A(T_n)-C(T_n))}{n-\\alpha}=\n\t\\frac{2a+ \\alpha(n-a-b-1)}{n-\\alpha}.\\label{eq:yule:1}\n\t\\end{align}\n\n\n\t\n\tSimilarly, Lemma~\\ref{lem:edge-set} implies that case (ii)\n\toccurs if and only if $e_n\\in E_1(T_n)$.\n\t Using Lemma~\\ref{lem:edge-set} again, it follows that $E_1(T_n)$ contains precisely $A(T_n)$ pendent edges and no interior edges. Therefore we have\n\t\\begin{align}\n\t\\label{eq:yule:2}\n\t\\mathbb{P}(A(T_{n+1})=a, C(T_{n+1})=b~|~A(T_n)=a+1,C(T_n)=b-1)\n\t=\\frac{(a+1)(1-\\alpha)}{n-\\alpha}.\n\t\\end{align}\n\t\n\tNext, Lemma~\\ref{lem:edge-set} implies case (iii) occurs if and only if $e_n\\in E_2(T_n)$. \t Using Lemma~\\ref{lem:edge-set} again, it follows that $E_2(T_n)$ contains precisely $2(A(T_n)-C(T_n))$ pendent edges and $A(T_n-C(T_n)$ interior edges. Thus we have\n\t\\begin{align}\n\t&\\mathbb{P}(A(T_{n+1})=a, C(T_{n+1})=b~|~A(T_n)=a-1,C(T_n)=b) \\nonumber \\\\\n\t&\t\\quad \\quad\n\t=\\frac{2(a-1-b)(1-\\alpha)+\\alpha(a-1-b)}{n-\\alpha}=\\frac{(2-\\alpha)(b-a+1)}{n-\\alpha}.\\label{eq:yule:3}\n\t\\end{align}\n\n\t\n\tFinally, Lemma~\\ref{lem:edge-set} implies case (iv) occurs\n\tif and only if $e_n$ is contained in $E_3(T_n)$. Using Lemma~\\ref{lem:edge-set} again, it follows that $E_3(T_n)$ contains precisely $n-A(T_n)-2C(T_n)$ pendent edges and no interior edges. Hence, it follows that\n\t\\begin{equation}\n\t\\label{eq:yule:4}\n\t\\mathbb{P}(A(T_{n+1})=a, C(T_{n+1}=b)~|~A(T_n)=a,C(T_n)=b-1)\n=\\frac{(1-\\alpha)(n-a-2b+2)}{n-\\alpha}. \n\t\\end{equation}\n\t\n\tSubstituting Eq.~\\eqref{eq:yule:1}--\\eqref{eq:yule:4} into Eq.~\\eqref{eq:total:yule} completes the proof of the theorem.\n\t\\end{proof}\n\n\nTo study the moments of $A_n$ and $C_n$, we present below a functional recursion form of Theorem~\\ref{jointpmf}, whose proof is straightforward and hence omitted here. \n\n\n\\begin{theorem}\n\\label{jointrr}\nLet $\\varphi: \\mathbb{N}\\times \\mathbb{N} \\to \\mathbb{R}$ be an arbitrary function. For $n \\ge 3$,\nunder the Ford model with parameter $\\alpha\\in [0,1]$ we have\n\\begin{eqnarray*}\n(n-\\alpha) \\mbox{${\\mathbb E}$} \\varphi(A_{n+1}, C_{n+1}) &=& \\mbox{${\\mathbb E}$}\\bigg[ \\big\\{\\alpha(n-A_n-C_n-1) +2A_n \\big\\} \\varphi(A_n, C_n) \\\\\n&& +(1-\\alpha)A_n \\varphi(A_n-1, C_n+1) +(2-\\alpha) (C_n-A_n) \\varphi(A_n+1, C_n) \\\\\n&& + (1-\\alpha)(n-A_n-2C_n) \\varphi(A_n, C_n+1) \\bigg ].\n\\end{eqnarray*}\n\\end{theorem}\n\nFor a fix integer $k$, consider the indicating function $I_k(x,y)$ that equals to 1 if $y=k$, and $0$ otherwise. Then by Theorem~\\ref{jointrr} the following result on the distribution of cherries follows.\n\n\\begin{corollary}\n\\label{cherrypmf}\nFor integers $n \\ge 3$ and $0\\le k \\le n\/2$, under the Ford model with parameter $\\alpha\\in [0,1]$ we have\n$$\n(n-\\alpha) \\mathbb{P}(C_{n+1}=k) = [(n-1) \\alpha +2(1-\\alpha) k ] \\mathbb{P}(C_{n}=k)\n+(1-\\alpha)(n-2k +2) \\mathbb{P}(C_{n+1}=k-1).\n$$\n\\end{corollary}\n\nFor the purpose of next section, we end this section by writing the recurrence relation in the following form in the next Corollary.\n\\begin{corollary}\n\\label{recurrence}\nFor $n \\ge 3$, under the Ford model with parameter $\\alpha\\in [0,1]$ we have\n\\begin{eqnarray}\n(n-\\alpha) \\mbox{${\\mathbb E}$} [C_{n+1}] - (n -2+\\alpha) \\mbox{${\\mathbb E}$} [C_n] &=& n(1-\\alpha), \\label{cherrymean}\\\\\n(n-\\alpha) \\mbox{${\\mathbb E}$}[A_{n+1}] - (n-3+\\alpha) \\mbox{${\\mathbb E}$}[A_n] &=& (2-\\alpha) \\mbox{${\\mathbb E}$} [C_n], \\label{forkmean} \\\\\n(n-\\alpha) \\mbox{${\\mathbb E}$} [C_{n+1}^2] - (n-4+3\\alpha) \\mbox{${\\mathbb E}$} [C_n^2] &=& 2(n-1)(1-\\alpha) \\mbox{${\\mathbb E}$} [C_n] + n(1-\\alpha), \\label{cherry2nd}\\\\\n(n-\\alpha)\\mbox{${\\mathbb E}$}[A_{n+1}C_{n+1}] - (n-5+3\\alpha) \\mbox{${\\mathbb E}$} [A_nC_n] &=& (n-1)(1-\\alpha)\\mbox{${\\mathbb E}$}[A_n] + (2-\\alpha) \\mbox{${\\mathbb E}$}[ C_n^2], \\label{cov}\\\\\n(n-\\alpha) \\mbox{${\\mathbb E}$}[A_{n+1}^2]- (n-6+3\\alpha)\\mbox{${\\mathbb E}$}[A_n^2] &=& 2(2-\\alpha)\\mbox{${\\mathbb E}$} [A_nC_n] + (2-\\alpha)\\mbox{${\\mathbb E}$} [C_n] - \\mbox{${\\mathbb E}$} [A_n] \\label{fork2nd}\n\\end{eqnarray}\nwith initial conditions $\\mbox{${\\mathbb E}$}[A_3]= \\mbox{${\\mathbb E}$}[C_3]=\\mbox{${\\mathbb E}$}[A_3^2]=\\mbox{${\\mathbb E}$}[C_3^2]=\\mbox{${\\mathbb E}$}[A_3C_3]=1. $\n\\end{corollary}\n\n\\begin{remark}\nLet $\\mu_n = \\mbox{${\\mathbb E}$} [C_n]$ and \t$\\si_{n}^2=\\mathrm{var}(C_n)$. \nSubstituting $\\mbox{${\\mathbb E}$}[C_{n}^2]=\\si_{n}^2 + \\mu_{n}^2$ into (\\ref{cherry2nd}) and applying (\\ref{cherrymean}), we obtain below a recurrence relation of the $\\si_n^2$, which was also obtained in Ford's thesis (Theorem 60, \\cite{Ford2006}):\n\\begin{eqnarray*}\n(n-\\alpha) \\si_{n+1}^2 - (n-4+3\\alpha) \\si_n^2 \n&=& -\\frac{4(1-\\alpha)^2}{n-\\alpha} \\mu_n^2 + \\frac{2(1-\\alpha)[(1-2\\alpha)n +\\alpha]}{n-\\alpha} \\mu_n\n+ \\frac{\\alpha (1-\\alpha)n(n-1)}{n-\\alpha}. \\label{cherryvar}\n\\end{eqnarray*}\n\\end{remark}\n\n\\section{Higher Order Asymptotic Expansion of the Joint Moments}\n\\label{sec:expansion}\n\nAlthough the leading terms of the first and second moments of the distributions of cherries and pitchforks, $\\mbox{${\\mathbb E}$} [A_n], \\mbox{${\\mathbb E}$} [C_n], \\mathrm{var}(A_n),\\mathrm{var}(C_n)$ and $cov(A_n, C_n)$, can be identified from  Theorem \\ref{Thm:Convg-ChPh}, for better understanding of their asymptotic behaviour \n we derive their  higher order expansions in this Section. \n\n We start with the following result on the first moments. Note that Proposition~\\ref{Prop:firstm} (i) has been obtained in  \\cite{Ford2006}.\n\\begin{proposition\n\\label{Prop:firstm} Under the Ford model with parameter $\\alpha\\in [0,1]$, the following exact expansions hold for $\\mbox{${\\mathbb E}$} [C_n] $ and $\\mbox{${\\mathbb E}$} [A_n]$.\n\\begin{enumerate}[\\normalfont(i)]\n\\item $ \\mbox{${\\mathbb E}$} [C_n]  = \\dfrac{1-\\alpha}{3-2\\alpha} \\ n + \\dfrac{\\alpha}{2(3-2\\alpha)} +x_n, $\nwhere\n\\[x_2= \\frac{(2- \\alpha)}{2(3-2\\alpha)}, \\quad x_3 = \\frac{\\alpha}{2(3-2\\alpha)}, \\quad x_n = \\frac{\\alpha }{2(3-2\\alpha)} \\prod_{i=3}^{n-1} \\frac{i-2+\\alpha}{i-\\alpha},  \\quad n \\ge 4.  \\]\nFurther, as $n \\to \\infty$,\n\\begin{equation}\\label{Order:xn}\nx_n = \\frac{\\alpha\\Ga(3-\\alpha)}{2(3-2\\alpha)\\Ga(1+\\alpha)}n^{-2(1-\\alpha)} \\left(1+o(1) \\right).\n\\end{equation}\n\n\\item $\\mbox{${\\mathbb E}$} [A_n]  = \\dfrac{1-\\alpha}{2(3-2\\alpha)} \\ n + \\dfrac{\\alpha}{2(3-2\\alpha)} + y_n,$\nwhere\n\\[y_2 = \\frac{\\alpha-2}{2(3-2\\alpha)}, ~ y_3=\\frac{1}{2}, ~  y_n =\\frac{1}{2} \\prod_{i=3}^{n-1}\\frac{i-3+\\alpha}{i-\\alpha } + \\frac{(2- \\alpha) \\alpha }{2(3-2\\alpha)} \\frac{n-3}{n-3+\\alpha} \\prod_{i=3}^{n-1}\\frac{i-2+\\alpha}{i-\\alpha }, ~~ n \\geq 4.\\]\nFurther, as $n \\to \\infty$,\n\\begin{equation}\\label{Order:yn}\ny_n  = \\frac{(2- \\alpha) \\Gamma(3-\\alpha) }{2(3-2\\alpha)\\Gamma(\\alpha)} n^{-2(1-\\alpha)} \\left(1+o(1) \\right).\n\\end{equation}\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proposition\n\\label{Prop:secondm}\nUnder the Ford model with parameter $\\alpha\\in [0,1]$, the following asymptotic expansions hold for $\\mathrm{var} (C_n)$,  $cov(A_n, C_n)$ and $\\mathrm{var} (A_n)$:\n\\begin{enumerate}[\\normalfont(i)]\n\\item $$\\mathrm{var} (C_n) = \\frac{(1-\\alpha)(2-\\alpha)}{(3-2\\alpha)^2(5-4\\alpha)} \\ n -  \\frac{\\alpha (1-\\alpha)(2-\\alpha)}{(3-2\\alpha)^2(5-4\\alpha)} +\\mbox{${\\mathcal O}$}(n^{-2(1-\\alpha)}).$$\n\\item\n$$cov(A_n, C_n)  = \\frac{-(1-\\alpha)(2-\\alpha)(1-2\\alpha)}{2(3-2\\alpha)^2(5-4\\alpha) } \\ n -\\frac{ \\alpha (1-\\alpha)(2-\\alpha)}{(3-2\\alpha)^2(5-4\\alpha)}  +\\mbox{${\\mathcal O}$}(n^{-2(1-\\alpha)}).$$\n\\item\n$$ \\mathrm{var}(A_n) =  \\frac{(1-\\alpha)(69-135\\alpha+96\\alpha^2-24\\alpha^3)}{4(3-2\\alpha)^2(5-4\\alpha)(7-4\\alpha)} \\ n   + \\frac{3\\alpha(1-\\alpha)(1-2\\alpha)(5-3\\alpha)}{4(3-2\\alpha)^2(5-4\\alpha)(7-4\\alpha)}  +\\mbox{${\\mathcal O}$}(n^{-2(1-\\alpha)}).$$\n\\end{enumerate}\n\\end{proposition}\n\n\n\\begin{remark}\nWhen $n$ is large,  $Cov(A_n, C_n)$ changes sign. Specifically, for $ \\alpha \\in (0, 1\/2)$, $A_n$ and $C_n$ are negatively correlated, which is expected; and  for $ \\alpha \\in (1\/2, 1)$, $A_n$ and $C_n$ are positively correlated, which is unexpected.\n\\end{remark}\n\n\n\n\n\\subsection{Proofs of Propositions~\\ref{Prop:firstm} and~\\ref{Prop:secondm}} \nWe need  the  lemmas below to prove the two propositions.\n\n\\begin{lemma}\\label{Lemma1}\nSuppose a real sequence $\\{X_n, n \\ge n_0\\}$ satisfies the recursion\n\\[ X_{n+1}  = f_n X_n +g_n, \\qquad n \\geq n_0,\\]\nwhere  $\\{f_n, n \\ge n_0\\}$ and $\\{g_n, n \\ge n_0\\}$ are sequences such that for every $ \\ell \\geq n_0$,  $ \\left|\\prod_{i=\\ell}^n f_i \\right| \\leq C (n\/\\ell)^{-a} $ and  $|g_\\ell| \\leq C \\ell^{-b}$,  for some finite $a, b$ and $C>0$. Then, there exists a finite positive constant $C'$ (which depends on $|X_{n_0}|$ and $C$) such that  $|X_n | \\leq C' n^{-q_{a,b}}$ where $q_{a,b} \\coloneqq  \\min\\{a, b-1\\}$.\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma \\ref{Lemma1}]\nIt is easy to verify that the  solution to the given recursion is given by\n\\[X_n =X_{n_0} \\prod_{i= n_0}^{n-1} f_i + \\sum_{i=n_0}^{n-1} g_i  \\prod_{j=i+1}^{n-1}f_j, \\quad n \\ge n_0.\\]\nTherefore,\n\\[ |X_n| \\leq  |X_{n_0}| \\left| \\prod_{i= n_0}^{n-1} f_i \\right| + \\sum_{i=n_0}^{n-1} |g_i| \\left| \\prod_{j=i+1}^{n-1} f_j  \\right|.\\]\nUnder the assumptions of the Lemma,\n\\[ |X_{n_0}| \\left| \\prod_{i= k}^{n-1}f_i \\right|  \\leq  C |X_{n_0}| n^{-a}\n\\leq C' n^{-a}; \\]\nand\n\\begin{align*}\n \\sum_{i=n_0}^{n-1} |g_i| \\left| \\prod_{j=i+1}^{n-1} f_j \\right|\n &\\leq C  \\sum_{i=n_0}^{n-1} |g_i|   (n\/i)^{-a}  \\leq   C^2  \\, n^{-a} \\sum_{i=n_0}^{n-1} i^{-b}  i^{a}  \\leq  C'  \\, n^{-a}   \\, n^{-b+a +1} = C' n^{-b+1}.\n\\end{align*}\nThus\n\\[ |X_n| \\leq  C' \\max( n^{-a} , n^{-b+1})  = C' n^{-q_{a,b}},\\]\nwhere $q_{a,b} = \\min\\{a, b-1\\}$. This completes the proof.\n\\end{proof}\n\n\\begin{lemma} \\label{Lemma2}\nFor finite non-negative integers $l,k$ such that $l\\geq k$, $m\\geq 1$ and  $\\alpha\\in [0,1]$, there exists a positive constant $K=K(\\alpha,l)$  such that\n\\begin{equation} \\label{eq:bound1}\n\t\\left|\\prod_{i=l }^{n-1} \\frac{i-k+m\\alpha}{i-\\alpha}  \\right|   \\leq\n\tK \\left(n\/l\\right)^{-k+(m+1)\\alpha}\n\t~~\\mbox{for all $1\\le l\\le n-1$.}\n\t\\end{equation}\nand as $n\\to \\infty$\n\\begin{equation}\\label{Order:prod}\n\\prod_{i=l }^{n-1} \\frac{i-k+m\\alpha}{i-\\alpha}   =   \\frac{\\Gamma(l-\\alpha) }{\\Gamma(l-k+m\\alpha)} n^{-k+(m+1)\\alpha} \\left(1+o(1) \\right).\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma \\ref{Lemma2}]\nThe bound in \\eqref{eq:bound1} follows from Lemma 2 of \\cite{Paper1}. We now prove \\eqref{Order:prod}. Note that, we can write\n\\[ \\frac{i-k+m\\alpha}{i-\\alpha}  = \\frac{\\Gamma(i+1-k+m\\alpha)\\Gamma(i-\\alpha) }{\\Gamma(i-k+m\\alpha)\\Gamma(i+1-\\alpha)}.\\]\nThus\n\\begin{align}\n\\prod_{i=l }^{n-1} \\frac{i-k+m\\alpha}{i-\\alpha}\n& =  \\prod_{i=l }^{n-1} \\frac{\\Gamma(i+1-k+m\\alpha)\\Gamma(i-\\alpha) }{\\Gamma(i-k+m\\alpha)\\Gamma(i+1-\\alpha) } \\nonumber\\\\\n& =   \\frac{\\Gamma(n-k+m\\alpha) }{\\Gamma(l-k+m\\alpha)}   \\frac{\\Gamma(l-\\alpha) }{\\Gamma(n-\\alpha)} \\label{Prod:Gamma}\\\\\n& =   \\frac{\\Gamma(l-\\alpha)}{\\Gamma(l-k+m\\alpha)}  \\frac{\\Gamma(n+m\\alpha)  }{\\Gamma(n-\\alpha)}   \\prod_{j=1}^k \\frac{1}{n-j+m\\alpha}.\\nonumber\n\\end{align}\n\n\\begin{equation}\\label{Eq:prodk}\n\\prod_{j=1}^k \\frac{1}{n-j+m\\alpha} =n^{-k}\\left(1+o(1) \\right).\n\\end{equation}\nBy  Stirling's approximation formula, $\\Ga(x) = \\sqrt{2 \\pi} \\ x^{x-1\/2} e^{-x} \\left(1+o(1)\\right)$, we have\n\\begin{eqnarray}\n\\frac{\\Ga(n+m\\alpha)}{\\Ga(n-\\alpha)}\n&=& \\frac{\\sqrt{2\\pi} (n+m\\alpha)^{n+m\\alpha-1\/2} \\, e^{-(n+m\\alpha)}}{\\sqrt{2\\pi} (n-\\alpha)^{n-\\alpha-1\/2} \\, e^{-(n-\\alpha)}}\\left(1+o(1) \\right) \\nonumber \\\\\n&=& n^{(m+1)\\alpha} \\frac{(1+ m\\alpha\/n)^{n+m\\alpha-1\/2}}{(1-\\alpha\/n)^{n-\\alpha-1\/2}} e^{-(m+1)\\alpha} \\left(1+o(1) \\right) \\nonumber \\\\\n&=& n^{(m+1)\\alpha} \\left(1+o(1) \\right). \\label{Eq:Str}\n\\end{eqnarray}\nCombining \\eqref{Eq:prodk} and \\eqref{Eq:Str}, we get \\eqref{Order:prod}.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Proposition \\ref{Prop:firstm}]\nRecall  $\\mu_n= \\mbox{${\\mathbb E}$}[C_n]$. By Theorem \\ref{Thm:Convg-ChPh}, $\\mu_n = \\frac{1-\\alpha}{3-2\\alpha} \\ n + \\mbox{${\\mathcal O}$}(1)$. Thus, we write $\\mu_n$  as\n\\begin{equation}\\label{Exp:mu}\n\\mu_n = \\frac{1-\\alpha}{3-2\\alpha} \\ n + \\frac{\\alpha}{2(3-2\\alpha)} +x_n.\n\\end{equation}\nFor simplicity, the dependence of $\\mu_n$ and $x_n$ on $\\alpha$ are  suppressed.\n\nSince $\\mu_2= \\mu_3=1$, we get $x_2 = 1-\\frac{4-3\\alpha}{2(3-2\\alpha)}=\\frac{2-\\alpha}{2(3-2\\alpha)}$\nand\n$x_3 = 1- \\frac{6-5\\alpha}{2(3-2\\alpha)} = \\frac{\\alpha}{2(3-2\\alpha)}.$\nSubstituting (\\ref{Exp:mu}) into (\\ref{cherrymean}) leads to\n$$(n-\\alpha) x_{n+1}- (n-2+\\alpha)x_n=0, \\quad n \\ge 2,  $$\nand hence,\n\\[ x_n=\\begin{cases}\n\\frac{\\alpha }{2(3-2\\alpha)} \\prod_{i=3}^{n-1} \\frac{i-2+\\alpha}{i-\\alpha} & \\quad n \\ge 4, \\\\\n\\frac{\\alpha}{2(3-2\\alpha)} & \\quad n=3, \\\\\n\\frac{(2- \\alpha)}{2(3-2\\alpha)} & \\quad n=2.\n\\end{cases} \\]\nTo prove \\eqref{Order:xn}, we rewrite $x_n$ as follows\n\\[x_n = x_3  \\prod_{i=3}^{n-1} \\frac{i-2+\\alpha}{i-\\alpha} = x_3 \\frac{ \\Gamma(3-\\alpha)}{ \\Ga(1+\\alpha)} \\frac{\\Ga(n-2+\\alpha)}{\\Ga(n-\\alpha)}, \\quad n \\ge 4. \\]\nApply  Lemma \\ref{Lemma2}, \\eqref{Order:xn} holds.\nConsequently,\n\\begin{equation}\\label{Exp:mu_n}\n\\mu_n = \\frac{1-\\alpha}{3-2\\alpha} n + \\frac{\\alpha}{2(3-2\\alpha)} + \\frac{ \\alpha\\Ga(3-\\alpha)}{2(3-2\\alpha) \\Ga(1+\\alpha)} n^{-2(1-\\alpha)} \\left(1+o(1) \\right).\n\\end{equation}\nThis completes the proof of part (i).\n\nThe same method of proof can be used to prove part (ii).\nRecall $\\nu_n =  \\mbox{${\\mathbb E}$}[A_n]$. By Theorem  \\ref{Thm:Convg-ChPh}, $\\nu_n = \\frac{1-\\alpha}{2(3-2\\alpha)} \\ n +\\mbox{${\\mathcal O}$}(1)$, and we write it as\n\n\n\\begin{equation}\\label{Exp:nu}\n\\nu_n = \\frac{1-\\alpha}{2(3-2\\alpha)} \\ n + \\frac{\\alpha}{2(3-2\\alpha)} + y_n,\n\\end{equation}\nwhere, again,  the dependence of $\\nu_n$ and $y_n$ on $\\alpha$ are suppressed.  Substituting (\\ref{Exp:nu}) into (\\ref{forkmean}) leads to\n\\[ y_{n+1}= \\frac{n-3+\\alpha}{n-\\alpha } y_n + \\frac{2- \\alpha}{n-\\alpha } x_n, \\quad n \\ge 4.\\]\nThe solution to this  recurrence relation is given by \n\\[ y_n =y_3 \\prod_{i=3}^{n-1}\\frac{i-3+\\alpha}{i-\\alpha }   + \\sum_{i=3}^{n-1} \\frac{2- \\alpha}{i-\\alpha } x_i \\prod_{j=i+1}^{n-1}\\frac{j-3+\\alpha}{j-\\alpha }.\\]\nSince  $y_3=1\/2$ and the expression for $x_i$ from part (i), we get\n\\begin{align*}\ny_n\n& =y_3 \\prod_{i=3}^{n-1}\\frac{i-3+\\alpha}{i-\\alpha } + \\sum_{i=3}^{n-1} \\frac{2- \\alpha}{i-\\alpha } \\times \\frac{\\alpha }{2(3-2\\alpha)} \\prod_{j=3}^{i-1} \\frac{j-2+\\alpha}{j-\\alpha} \\times \\prod_{j=i+1}^{n-1}\\frac{j-3+\\alpha}{j-\\alpha }  \\\\\n& =\\frac{1}{2}  \\prod_{i=3}^{n-1}\\frac{i-3+\\alpha}{i-\\alpha } +  \\frac{(2- \\alpha) \\alpha }{2(3-2\\alpha)} \\sum_{i=3}^{n-1} \\prod_{j=i+1}^{n-1}\\frac{j-3+\\alpha}{j-\\alpha } \\times \\frac{1}{i-\\alpha}\\times \\prod_{j=3}^{i-1} \\frac{j-2+\\alpha}{j-\\alpha} \\\\\n& =\\frac{1}{2} \\prod_{i=3}^{n-1}\\frac{i-3+\\alpha}{i-\\alpha } + \\frac{(2- \\alpha) \\alpha}{2(3-2\\alpha)} \\sum_{i=3}^{n-1}  \\frac{1}{3-\\alpha}\\prod_{j=4}^{n-1}\\frac{j-3+\\alpha}{j-\\alpha } \\\\\n& =\\frac{1}{2}  \\prod_{i=3}^{n-1}\\frac{i-3+\\alpha}{i-\\alpha } + \\frac{(2- \\alpha) \\alpha}{2(3-2\\alpha)} \\frac{(n-3)}{(3-\\alpha)} \\prod_{j=4}^{n-1}\\frac{j-3+\\alpha}{j-\\alpha }.\n\\end{align*}\nThus, for $n \\geq 5$,\n\\begin{equation}\\label{sol:yn}\ny_n = \\left( \\frac{1}{2}  + \\frac{(2- \\alpha)\\alpha }{2(3-2\\alpha)} \\frac{(n-3)}{(3-\\alpha)}  \\right) \\prod_{j=4}^{n-1}\\frac{j-3+\\alpha}{j-\\alpha }.\n\\end{equation}\nBy  Lemma \\ref{Lemma2},\n\\begin{align*}\ny_n  &=  \\left( \\frac{1}{2}  + \\frac{(2- \\alpha) \\alpha}{2(3-2\\alpha)} \\, \\frac{(n-3)}{(3-\\alpha)} \\right) \\frac{\\Gamma(4-\\alpha) }{\\Gamma(1+\\alpha)} n^{-3+2\\alpha} \\left(1+o(1) \\right)\\\\\n&=  \\frac{(2- \\alpha) \\alpha}{2(3-2\\alpha)} \\frac{\\Gamma(3-\\alpha) }{\\Gamma(1+\\alpha)} n^{-2+2\\alpha} \\left(1+o(1) \\right) \\\\\n&=\\frac{(2- \\alpha)\\Gamma(3-\\alpha) }{2(3-2\\alpha)\\Gamma(\\alpha)}  n^{-2(1-\\alpha)} \\left(1+o(1) \\right)\n\\end{align*}\nas $n \\to \\infty$.\nThis completes the proof of part (ii) and hence the Proposition.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{Prop:secondm}]\n\nThe method of proof is similar to that of Proposition \\ref{Prop:firstm}. \n\nRecall $\\si_n ^2= \\mathrm{var}(C_n)$. From Theorem \\ref{Thm:Convg-ChPh},  we have  $ \\si_n^2 = \\frac{(1-\\alpha)(2-\\alpha)}{(5-4\\alpha)(3-2\\alpha)^2} n+ \\mbox{${\\mathcal O}$}(1)$. \nWe first consider $\\mbox{${\\mathbb E}$}[C_n^2]$. As \n$$\\mbox{${\\mathbb E}$}[C_n^2] = \\mu_n^2 + \\si_n^2 = \\frac{(1-\\alpha)^2}{(3-2\\alpha)^2} n^2 +\\mbox{${\\mathcal O}$}(n),$$ we rewrite it as\n\\begin{equation}\n\\label{CM2}\n\\mbox{${\\mathbb E}$}[C_n^2]  = \\frac{(1-\\alpha)^2}{(3-2\\alpha)^2} \\ n^2 + \\frac{2(1-\\alpha)(1+ 2\\alpha -2\\alpha^2)}{(5-4\\alpha)(3-2\\alpha)^2} \\ n - \\frac{\\alpha (8-17\\alpha+8\\alpha^2)}{4(5-4\\alpha)(3-2\\alpha)^2} +z_n,\n\\end{equation} \nand derive below a recursion on $z_n$. Substituting (\\ref{CM2}) into (\\ref{cherry2nd}) and after   straightforward algebraic simplification, we have  \n$$ (n-\\alpha) z_{n+1} - (n-4+3\\alpha) z_n = 2(1-\\alpha)(n-1) x_n, \\quad n \\ge 2.  $$\nSince $C_2=C_3=1$, we get $ z_2 = \\frac{3(2-\\alpha) (8\\alpha^2 -21\\alpha+14)}{4(3-2\\alpha)^2 (5-4\\alpha)}$  and\n$ z_3 = \\frac{88 \\alpha^3 - 213 \\alpha^2 + 152 \\alpha - 24}{4(3-2\\alpha)^2 (5-4\\alpha)}.$ Consequently, \n\\[ \\si_n^2 = \\frac{(1-\\alpha)(2-\\alpha)}{(5-4\\alpha)(3-2\\alpha)^2} \\ n -  \\frac{\\alpha (1-\\alpha)(2-\\alpha)}{(5-4\\alpha)(3-2\\alpha)^2} + v_n -x_n^2,\\]\nwhere\n \\[v_n=z_n  - \\frac{2(1-\\alpha)}{3-2\\alpha} n x_n - \\frac{\\alpha}{3-2\\alpha} x_n = z_n - \\frac{[2(1-\\alpha) n +\\alpha]}{3-2\\alpha}x_n.\\]\nThen,  for $n\\geq 6$,\n\\begin{align*}\n(n-\\alpha) v_{n+1} \n&= (n-\\alpha)z_{n+1}  - \\frac{[2(1-\\alpha) (n+1) +\\alpha]}{3-2\\alpha} (n-\\alpha)x_{n+1}\\\\\n&=  (n-4+3\\alpha) z_n + 2(1-\\alpha)(n-1) x_n   - \\frac{[2(1-\\alpha) (n+1) +\\alpha]}{3-2\\alpha} (n-2+\\alpha)x_{n}\\\\\n& = (n-4+3\\alpha) v_n  + (n-4+3\\alpha) \\frac{[2(1-\\alpha) n +\\alpha]}{3-2\\alpha}x_n  \\\\\n& \\quad + 2(1-\\alpha)(n-1) x_n   - \\frac{[2(1-\\alpha) (n+1) +\\alpha]}{3-2\\alpha} (n-2+\\alpha)x_{n}\\\\\n&= (n-4+3\\alpha)v_n  -\\frac{2(1-\\alpha)}{3-2\\alpha} x_n.\n\\end{align*}\nEquivalently, \n\\[  v_{n+1} =\\frac{n-4+3\\alpha}{n-\\alpha} v_n  -\\frac{2(1-\\alpha)}{(3-2\\alpha)} \\frac{x_n}{(n-\\alpha)}.\\]\nApplying Lemma \\ref{Lemma1}, with $f_n = \\frac{(n-4+3\\alpha)}{(n-\\alpha)}$,  $g_n  = -\\frac{2(1-\\alpha)x_n}{(3-2\\alpha)(n-\\alpha)}$,  $a = 4-3\\alpha$ and $b= 3-2\\alpha$,  we get  $v_n = \\mbox{${\\mathcal O}$}(n^{-2+2\\alpha})$.\nThis proves part (i) of the proposition.\n\nPart (ii) is proved in a similar fashion.   By Theorem \\ref{Thm:Convg-ChPh}, $ Cov(A_n, C_n) = -\\frac{(1-\\alpha)(2-\\alpha)(1-2\\alpha)}{2(3-2\\alpha)^2(5-4\\alpha)} n+ \\mbox{${\\mathcal O}$}(1).$ \nSince $\\mbox{${\\mathbb E}$}[A_nC_n]= Cov(A_n, C_n)+\\mu_n \\nu_n $, with $\\mu_n$ and $ \\nu_n $ found in Proposition \\ref{Prop:firstm}, we write   \n\\begin{eqnarray}\n\\mbox{${\\mathbb E}$}[A_nC_n] &=& \\frac{(1-\\alpha)^2}{2(3-2\\alpha)^2} \\ n^2 -\\frac{(1-\\alpha)(4-25\\alpha+16\\alpha^2)}{4(5-4\\alpha)(3-2\\alpha)^2} n  -\\frac{\\alpha (8- 17\\alpha +8\\alpha^2 )}{4(5-4\\alpha)(3-2\\alpha)^2}+ t_n. \\label{Eq:E[AnCn]}\n\\end{eqnarray}\nCombining   (\\ref{cov}) and \\eqref{Eq:E[AnCn]}, $t_n$ satisfies the recursion, \n\\begin{equation}\n(n-\\alpha)t_{n+1}-(n-5+3\\alpha)t_n = (2-\\alpha) z_n + (1-\\alpha)(n-1)y_n, \\quad n\\geq 6.\n\\end{equation}\nBy \\eqref{Exp:mu}, \\eqref{Exp:nu} and \\eqref{Eq:E[AnCn]},\n\\begin{eqnarray*}\nCov(A_n, C_n) &=& \\frac{-(1-\\alpha)(2-\\alpha)(1-2\\alpha)}{2(5-4\\alpha)(3-2\\alpha)^2} \\ n -\\frac{ \\alpha (1-\\alpha)(2-\\alpha)}{(5-4\\alpha)(3-2\\alpha)^2}  + w_n -x_n y_n,\n\\end{eqnarray*}\nwhere $w_n = t_n  -\\dfrac{[(1-\\alpha)n +\\alpha]x_n + [2(1-\\alpha)n +\\alpha]y_n}{2(3-2\\alpha)}$. Consider\n\\begin{align*}\n&(n-\\alpha) w_{n+1} \\\\\n&=(n-\\alpha) t_{n+1}  -(n-\\alpha)\\frac{[(1-\\alpha)(n+1) +\\alpha]x_{n+1} + [2(1-\\alpha)(n+1) +\\alpha]y_{n+1}}{2(3-2\\alpha)}\\\\\n&= (n-5+3\\alpha) t_n +(2-\\alpha)z_n+(1-\\alpha)(n-1)y_n  \\\\\n& \\quad - (n-\\alpha) \\frac{[(1-\\alpha)(n+1) +\\alpha]x_{n+1} + [2(1-\\alpha)(n+1) +\\alpha]y_{n+1}}{2(3-2\\alpha)} \\\\\n&= (n-5+3\\alpha) \\left( t_n -\\frac{[(1-\\alpha)n +\\alpha]x_n + [2(1-\\alpha)n +\\alpha]y_n}{2(3-2\\alpha)}\\right) \\\\\n& \\quad +(n-5+3\\alpha) \\frac{[(1-\\alpha)n +\\alpha]x_n + [2(1-\\alpha)n +\\alpha]y_n}{2(3-2\\alpha)}\\\\\n&\\quad +(2-\\alpha)z_n+(1-\\alpha)(n-1)y_n -(n-\\alpha)  \\frac{[(1-\\alpha)(n+1) +\\alpha]x_{n+1} + [2(1-\\alpha)(n+1) +\\alpha]y_{n+1}}{2(3-2\\alpha)}\\\\\n&= (n-5+3\\alpha)w_n +(n-5+3\\alpha) \\frac{[(1-\\alpha)n +\\alpha]x_n + [2(1-\\alpha)n +\\alpha]y_n}{2(3-2\\alpha)}\\\\\n&\\quad +(2-\\alpha)z_n+(1-\\alpha)(n-1)y_n -(n-\\alpha)  \\frac{[(1-\\alpha)(n+1) +\\alpha]x_{n+1} + [2(1-\\alpha)(n+1) +\\alpha]y_{n+1}}{2(3-2\\alpha)}.\n\\end{align*}\nThus $w$ satisfies the following recursion for $n\\geq 6$\n\\begin{align*}\n(n-\\alpha) w_{n+1}  -  (n-5+3\\alpha)w_n &= {\\rm RHS},\n\\end{align*}\nwhere  RHS  is given by \n\\begin{align*}\n& (n-5+3\\alpha) \\frac{[(1-\\alpha)n +\\alpha]x_n + [2(1-\\alpha)n +\\alpha]y_n}{2(3-2\\alpha)}  +(1-\\alpha) (n-1)y_n \\\\\n&\\quad -  (n-\\alpha)\\frac{[(1-\\alpha)(n+1) +\\alpha]x_{n+1} + [2(1-\\alpha)(n+1) +\\alpha]y_{n+1}}{2(3-2\\alpha)} +(2-\\alpha)z_n \\\\\n& = (2-\\alpha)v_n - \\frac{1-\\alpha}{3-2\\alpha} x_n.\n\\end{align*}\nWe omit the straightforward but tedious algebraic simplification steps. Hence,\n\\begin{equation}\\label{Rec:w}\n w_{n+1}  =\\frac{ n-5+3\\alpha}{n-\\alpha} w_n + (2-\\alpha) \\frac{v_n}{n-\\alpha} - \\frac{(1-\\alpha)}{(3-2\\alpha)} \\frac{x_n}{(n-\\alpha)}.\n\\end{equation}\nApplying Lemma \\ref{Lemma1} with $f_n = \\frac{n-5+3\\alpha}{n-\\alpha} $,  $g_n  = \\ (2-\\alpha) \\frac{v_n}{n-\\alpha} - \\frac{(1-\\alpha)}{(3-2\\alpha)} \\frac{x_n}{(n-\\alpha)} $,  $a = 5-4\\alpha$ and $b= 3-2\\alpha$,  we get  $v_n = \\mbox{${\\mathcal O}$}(n^{-2+2\\alpha})$. This proves part (ii).\n\nTo prove (iii), we let $\\tau_n^2 =\\mathrm{var}(A_n) = \\frac{(1-\\alpha)(2-\\alpha)}{(3-2\\alpha)^2(5-4\\alpha)} n +\\mbox{${\\mathcal O}$}(1)$ by  Theorem \\ref{Thm:Convg-ChPh}. So, \n$ \\mbox{${\\mathbb E}$}[A_n^2] = \\nu_n^2 + \\tau_n^2= \\frac{(1-\\alpha)^2}{4(3-2\\alpha)^2} n^2 +\\mbox{${\\mathcal O}$}(n)$. We write \n\\[ \\mbox{${\\mathbb E}$}[A_n^2]  = \\frac{(1-\\alpha)^2}{4(3-2\\alpha)^2} n^2 + \\frac{2(1-\\alpha)(1+ 2\\alpha -2\\alpha^2)}{(5-4\\alpha)(3-2\\alpha)^2} n +   \\frac{\\alpha( 5-3\\alpha+\\alpha^2)}{4(3-2\\alpha) (5-4\\alpha)(7-4 \\alpha)} + s_n. \\]\n\nLet $u_n = s_n -  \\dfrac{[(1-\\alpha)n + \\alpha] y_n}{3-2\\alpha}$. Then,\n\\begin{align*}\n(n-\\alpha) u_{n+1} & = (n-\\alpha)  s_{n+1} -  \\frac{[(1-\\alpha)(n+1) + \\alpha] (n-\\alpha)  y_{n+1} }{3-2\\alpha} \\\\\n& = (n-6+3\\alpha)s_n + (2-\\alpha)(2t_n+x_n)-y_n - \\frac{[(1-\\alpha)(n+1) + \\alpha] (n-\\alpha)  y_{n+1} }{3-2\\alpha} \\\\\n& = (n-6+3\\alpha)[ s_n-\\frac{[(1-\\alpha)n + \\alpha] y_n}{3-2\\alpha}]  +  (n-6+3\\alpha) \\frac{[(1-\\alpha)n + \\alpha] y_n}{3-2\\alpha} \\\\\n&\\quad +  (2-\\alpha)(2t_n+x_n)-y_n - \\frac{[(1-\\alpha)(n+1) + \\alpha] (n-\\alpha)  y_{n+1} }{3-2\\alpha} \\\\\n& = (n-6+3\\alpha)u_n +  (n-6+3\\alpha) \\frac{[(1-\\alpha)n + \\alpha] y_n}{3-2\\alpha} \\\\\n&\\quad +  (2-\\alpha)(2t_n+x_n)-y_n - \\frac{[(1-\\alpha)(n+1) + \\alpha] (n-\\alpha)  y_{n+1} }{3-2\\alpha}.\n\\end{align*}\nThus,\n\\begin{align*}\n&(n-\\alpha) u_{n+1} - (n-6+3\\alpha)u_n \\\\\n& =  (n-6+3\\alpha) \\frac{[(1-\\alpha)n + \\alpha] y_n}{3-2\\alpha} -y_n - \\frac{[(1-\\alpha)(n+1) + \\alpha] (n-\\alpha)  y_{n+1} }{3-2\\alpha} +(2-\\alpha)x_n +  2(2-\\alpha)t_n\\\\\n& =  (n-6+3\\alpha) \\frac{[(1-\\alpha)n + \\alpha]y_n }{3-2\\alpha} -y_n - \\frac{[(1-\\alpha)(n+1) + \\alpha] [ (n-3+\\alpha) y_n + (2-\\alpha) x_n] }{3-2\\alpha}\\\\\n& \\quad +  2(2-\\alpha)w_n +(2-\\alpha)x_n + (2-\\alpha) \\frac{[(1-\\alpha)n +\\alpha]x_n + [2(1-\\alpha)n +\\alpha]y_n}{3-2\\alpha}\\\\\n& = y_n \\left\\{  (n-6+3\\alpha) \\frac{[(1-\\alpha)n + \\alpha] }{3-2\\alpha} -1 +(2-\\alpha) \\frac{[2(1-\\alpha)n +\\alpha]}{3-2\\alpha}   - (n-3+\\alpha) \\frac{[(1-\\alpha)(n+1) + \\alpha]   }{3-2\\alpha} \\right\\} \\\\\n&  \\quad +2(2-\\alpha)w_n +(2-\\alpha)  x_n \\left\\{  \\frac{[(1-\\alpha)n +\\alpha] }{3-2\\alpha}  - \\frac{[(1-\\alpha)(n+1) + \\alpha]}{3-2\\alpha}+1\\right\\} \\\\\n& =2(2-\\alpha)w_n - y_n \\frac{\\alpha(2-\\alpha)}{3-2\\alpha} +\\frac{(2-\\alpha)^2}{3-2\\alpha}  x_n.\n\\end{align*}\nEquivalently, \n\\[ u_{n+1} =  \\frac{n-6+3\\alpha}{n-\\alpha} u_n  +  \\frac{\\alpha(2-\\alpha)}{(3-2\\alpha)} \\frac{y_n}{(n-\\alpha)} +\\frac{(2-\\alpha)^2}{(3-2\\alpha)}  \\frac{x_n}{(n-\\alpha)}.\\]\nApply Lemma \\ref{Lemma1} as in the proofs of parts (i) and (ii),  \nwe can conclude that $u_n =\\mbox{${\\mathcal O}$}(n^{-2+2\\alpha})$. This completes the proof of (iii) and hence the Proposition.\n\\end{proof}\n\n\\section*{Acknowledgements} \nThe work of Gursharn Kaur was supported by NUS Research Grant R-155-000-198-114, and that of Kwok Pui Choi by Singapore Ministry of Education Academic Research Fund\nR-155-000-188-114. We thank Chris Greenman and Ariadne Thompson for stimulating discussions on the Ford model. \n\n\n\n\\bibliographystyle{abbrv}     \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe Dirac semi-metals, whose low energy physics can be described by three dimensional (3D) pseudorelativistic Dirac equation with the linear dispersion around the Fermi level \\cite{Burkov2011}, have attracted lots of attention in recent days, owing to their exotic physical properties \\cite{WangZJ-2012-Na3Bi,WangZJ-2013,li2010dynamical,potter2014quantum,Parameswaran2014} and large application potentials in the future \\cite{Abrikosov1998,liang2014,He2014}. Current studies mainly focus on two types of Dirac semi-metals with both inversion symmetry and time-reversal (TR) symmetry. One is achieved at the critical point of a topological phase transition. This type of Dirac semi-metal is not protected by any topology and can be gapped easily via small perturbations \\cite{sato2011-TlBiSSe,wu2013sudden,LiuJP2013}. In contrast, the other type is protected by the uniaxial rotation symmetry \\cite{ChenF2012}, so is quite stable. And according to even or odd parity of the states at the axis of $C_n$ rotation, the symmetry protected Dirac semi-metals can be further classified as two subclasses \\cite{YangBJ2014}. The first subclass has a single Dirac point (DP) at a time-reversal invariant momentum (TRIM) point on the rotation axis protected by the lattice symmetry \\cite{YoungSM2012,Steinberg2014}, while the second one possesses non-trivial band inversion and has a pair of DPs on the rotation axis away from the TRIM points. For the materials of the second subclass (such as Na$_3$Bi \\cite{WangZJ-2012-Na3Bi,liu2014discovery}, Cd$_3$As$_2$ \\cite{WangZJ-2013,liu2014stable,Borisenko2014,yi2014evidence,liang2014,JeonSJ2014,He2014,Narayanan2015}, and some charge balanced compounds \\cite{gibson20143d,du2014dirac}) the non-zero $\\mathbb{Z}_{2}$ number can be well defined at the corresponding two dimensional (2D) plane of the Brillouin zone (BZ) \\cite{Morimoto2014,Gorbar2015}. And due to the non-trivial topology, these stable Dirac semi-metals are regarded as a copy of Weyl semi-metals \\cite{YangBJ2014}. Thus, its Fermi arcs are observed on the specific surfaces \\cite{xu2015}, and a quantum oscillation of the topological property is expected to be achieved in the thin film with the change of thicknesses \\cite{WangZJ-2013}.\n\nIn spite of these successful progresses, the 3D Dirac semi-metal materials either take uncommon lattice structures or contain heavy atoms, which are not compatible with current semiconductor industry. On the other hand, the group \\uppercase\\expandafter{\\romannumeral4} elements, including C, Si, Ge, Sn and Pb, have been widely used in electronics and microelectronics. Generally, for some of the group \\uppercase\\expandafter{\\romannumeral4} elements, the diamond structure is one of the most stable 3D forms at ambient conditions. However, under specific experimental growth conditions, various allotropes with exotic phyiscal and chemical properties are discovered experimentally. For example, the new orthorhombic allotrope of silicon, Si$_{24}$, is found to be a semiconductor with a direct gap of 1.3 eV at the $\\Gamma$ point \\cite{kim2015Si24}; and the 2D forms of silicene \\cite{Seymur2009,Seymur2013-Sil,Seymure-sil-2014}, germanene \\cite{Daviala2014,Chensi-2014} and stanene \\cite{TangPz2014-stanene,Yong2013,zhu2015epitaxial} have been theoretically predicted to exist or experimentally grown on different substrates, which can be 2D topological insulators (TIs) and used as 2D field-effect transistors \\cite{tao2015silicene}.\n\nIn this article, by using \\emph{ab initio} density functional theory (DFT) with hybrid functional \\cite{heyd2003hybrid}, we predict new 3D metastable allotropes for Ge and Sn with staggered layered dumbbell (SLD) structure, named as germancite and stancite; and discover that they are stable Dirac semi-metals with a pair of gapless DPs on the rotation axis of $C_3$ protected by the lattice symmetry. Similar to the conventional Dirac semi-metals, such as Na$_3$Bi and Cd$_3$As$_2$, the topologically non-trivial Fermi arcs can be observed on the surfaces parallel to the rotation axis in the germancite and stancite. And via tuning the Fermi level, we can observe a Lifshitz transition in the momentum space. More importantly for future applications, the thin film of the germancite is found to be an intrinsic 2D TI, and the ultrahigh mobility and giant magnetoresistance can be expected in these compounds due to the 3D linear dispersion.\n\n\\section{Methods}\nThe calculations were carried out by using DFT with the projector augmented wave method \\cite{PhysRevB.50.17953,PhysRevB.59.1758}, as implemented in the Vienna \\textit{ab initio} simulation package \\cite{PhysRevB.54.11169}. Plane wave basis set with a kinetic energy cutoff of $\\mathrm{250~eV}$ and $\\mathrm{150~eV}$ was used for germancite and stancite respectively. The structure is allowed to fully relax until the residual forces are less than $1\\times 10^{-3}~\\mathrm{eV\/\\AA}$. The Monkhorst-Pack $k$ points are $9\\times 9\\times 9$. With the relaxed structure, the electronic calculation of germancite and stancite using hybrid functional HSE06 \\cite{heyd2003hybrid} has been done with and without SOC. The maximally localized Wannier functions \\cite{Mostofi2008685} are constructed to obtain the tight-binding Hamiltonian for the Green's function method \\cite{0305-4608-15-4-009}, which is used to calculate the surface electronic spectrum and surface states.\n\n\\section{Results}\nAs shown in Fig. \\ref{fig:1}, the germancite and stancite share the same rhombohedral crystal structure with the space group of $D_{3d}^6$ ($R\\bar{3}c$) \\cite{PhysRevB.90.085426}, which contains the spacial inversion symmetry and $C_3$ rotation symmetry along the trigonal axis (defined as $z$ axis). In one unit-cell, fourteen atoms bond with each others to form six atomic layers; and in each layer, one dumbbell site can be observed. To clearly visualize the SLD structure in the germancite and stancite, we plot the side view of the hexagonal lattice shown in Fig. \\ref{fig:1}(b) and the top view from (111) direction in Fig. \\ref{fig:1}(c). As the grey shadow shown, the layers containing dumbbell sites stack along (111) direction in the order of $\\cdots B\\bar{A}C\\bar{B}A\\bar{C}\\cdots$. The interlayer interaction is the covalent bonding between adjacent layers, whose bond lengths are almost equal to those of intralayer bonding (the difference is about $0.03$\\AA). Meanwhile, different from the diamond structure, the tetrahedral symmetry is absent in the SLD structure and the coupling here is not typical $sp^3$ hybridization. Furthermore, in order to test the structural stability, we calculate the phonon dispersion for the germancite and stancite shown in Fig. \\ref{fig:1}(e). It can be seen that the frequencies of all modes are positive over the whole Brillouin zone, which indicates that the SLD structures are thermodynamically stable. Furthermore, compared with the other experimentally discovered metastable allotropes of Ge and Sn \\cite{guloy2006guest,kiefer2010synthesis,PhysRevLett.110.085502,ja304380p,Ceylan2015407,PhysRevB.34.362}, the germancite and stancite share the same order of magnetite of the mass density and cohesive energies (see Supplemental Information for details), so we expect the germancite and stancite could be composed in the future experiments.\n\n\n\\begin{figure}\n\\centerline{ \\includegraphics[clip,width=0.8\\linewidth]{Figure1.eps}}\n   \\caption{(Color online) (a) The unit cell of the SLD structure with three private lattice vectors set as \\textbf{a$_{1,2,3}$}. The balls in different colors stand for the same kind of atoms in different layers. (b) The side view and (c) top view of the SLD structure. The layers containing dumbbell (DB) structures are labelled. The letters ($A,B,C$) denote the positions of DB sites and the sign of bar is applied to distinguish between two trigonal lattices transformed to each other by inversion. As an example, the top view of two adjacent layers (marked by dashed blue lines) is shown. The DB structures are labeled by the grey shadow shown in the top view of a single layer, and the atoms in one DB structure are represented by grey balls. (d) The 3D Brillouin zone (BZ) of germancite and stancite.  The four inequivalent TRIM points are $\\Gamma$ (0,0,0), $L$ (0,$\\pi$,0), $F$ ($\\pi$,$\\pi$,0) and T ($\\pi$,$\\pi$,$\\pi$). The hexagon and square, connected to $\\Gamma$ by blue lines, show the 2D BZs projected to (111) and (2$\\bar{1}\\bar{1}$) surfaces respectively, and the high-symmetry $k$ points are labelled. (e) The phonon dispersion of germancite and stancite along high symmetry lines of 3D BZ.}\n\\label{fig:1}\n\\end{figure}\n\nThe calculated electronic structures of the germancite and stancite around the Fermi level are shown in Fig. \\ref{fig:2}(a), in which the solid lines and the yellow shadow stand for the bulk bands with and without spin-orbit coupling (SOC) respectively. It could be observed that: when the SOC effect is not included, the germancite is a conventional semi-metal whose bottom of the conduction bands and top of valence bands touch at the $\\Gamma$ point with the parabolic dispersions; while for stancite, it is a metal whose band touching at the $\\Gamma$ point is higher than the Fermi level. When the SOC effect is fully considered, our calculations indicate both germancite and stancite to be 3D Dirac semi-metals with a pair of DPs in the trigonal rotation axis (DP at (0,0,$\\pm k_{z0}$)). Therefore, the low energy physics of this kind of materials can be described by the 3D Dirac-type Hamiltonian. And the schematic band structure based on the effective $k\\cdot p$ model (see Supplemental Information for details) for germancite and stancite is shown in Fig. \\ref{fig:2}(c), in which the pair of 3D DPs is clear.\n\nTo understand the physical origin of the 3D gapless Dirac Fermions in the SLD structure, we plot the schematic diagram of the band evolution for the germancte and stancite in Fig. \\ref{fig:2}(b). In contrast to isotropic coupling in the diamond structure, the hybridizations in the layered SLD structure are anisotropic, in which the inter-layer couplings are relatively weaker than intra-layer couplings and the $p_z$ and $p_{x\\pm iy}$ states are splited.  Furthermore, based on our calculations, the kind of anisotropic coupling will further shift down the anti-bonding state of $s$ orbital which is even lower than the bonding states of the $p_{x\\pm iy}$ orbitals at the $\\Gamma$ point. So the band inversion occurs at the $\\Gamma$ point even without SOC effect, and the SOC herein just removes the degeneracy of $p_{x\\pm iy}$ orbitals around the Fermi level. In the 2D BZ which contains the $\\Gamma$ point and is perpendicular to the $\\Gamma$-$\\text{T}$ direction, the non zero $\\mathbb{Z}_{2}$ topological number can be well defined. On the other hand, the $C_{3v}$ symmetry along the $\\Gamma$-$\\text{T}$ line contains one 2D ($\\Lambda_{4}$) and two degenerate 1D ($\\Lambda_{5}$, $\\Lambda_{6}$) irreducible representations for its double space group \\cite{koster1963properties}. As shown in the Fig. \\ref{fig:2}(b), the two crossing bands at the Fermi level belong to $\\Lambda_{5}+\\Lambda_{6}$ and $\\Lambda_{4}$ respectively.  So there is no coupling and a TR pair of 3D DPs can be observed at the Fermi level along the $\\Gamma$-$\\text{T}$ direction.\n\n\\begin{figure}\n\\centerline{ \\includegraphics[clip,width=0.8\\linewidth]{Figure2.eps}}\n         \\caption{(Color online) (a) The band structures of germancite (left) and stancite (right) along high symmetry lines with the corresponding DOS around the Fermi level (dashed horizontal line). In the k-path $\\text{T}$-$\\Gamma$, the size of the red dots represents the contribution from the atomic $s$ and $p_z$ orbitals. The cyan dots are the Dirac points at (0,0,$k_{z0}$), where $k_{z0}\\approx 0.08 $ \\AA$^{-1}$ and $\\approx 0.18 $ \\AA$^{-1}$ respectively. Shaded regions denote the calculated energy spectrum without SOC. (b) Schematic diagrams of the lowest conduction bands and highest valence bands from the $\\text{T}$ point to the $\\Gamma$ point for germancite and stancite. The black lines present the SOC effect at the $\\text{T}$ and $\\Gamma$ point. Between them, the red and blue lines denote doubly degenerate bands belonging to different irreducible representations, where the solid\/dashed red line is for germancite\/stancite. And the crossing points (solid cyan dots) correspond to those gapless Dirac points in (a) respectively. (c) Schematic band dispersion based on the effective $k\\cdot p$ model for germancite and stancite. The $k_{\\perp}$ direction refers to any axis perpendicular to the $k_{z}$ direction in the momentum space and the color becomes warmer, as the energy increases.}\n\\label{fig:2}\n\\end{figure}\n\nDue to the non-trivial topology of 3D Dirac semi-metals, the projected 2D DPs and Fermi arcs are expected to be observed on some specific surfaces for the germancite and stancite. As shown in the Fig. \\ref{fig:3}, by using the surface Green's function method \\cite{0305-4608-15-4-009}, we study the electronic spectrum on the (111) and (2$\\bar{1}\\bar{1}$) surface whose BZs are perpendicular and parallel to the $\\Gamma$-$\\text{T}$ direction respectively. For the BZ of (111) surface, the pair of 3D DPs project to the $\\widetilde{\\Gamma}$ point as 2D Dirac cones (see Fig. \\ref{fig:3}(a) and (d)); when the coupling between two projected 2D DPs is considered, a finite band gap could be easily obtained. Furthermore, besides the projected Dirac cones, we also observe the trivial surface states in the germancite and stancite ($\\alpha_{1,2}$ states in the Fig. \\ref{fig:3}(a) and (d)) which mainly originate from the dangling bonds on the (111) surface.\n\n\\begin{figure}\n\\centerline{ \\includegraphics[clip,width=0.8\\linewidth]{Figure3.eps}}\n         \\caption{(Color online) The electronic spectrum on the $(111)$ surface and its corresponding Fermi surface for (a) germanctie and (d) stancite respectively. Two bulk DPs are projected to the $\\widetilde{\\Gamma}$ point. The electronic spectrum on the $(2\\bar{1}\\bar{1})$ surface and its corresponding Fermi surface for (b) germanctie and (e) stancite respectively. The cyan dots label the projected DPs and the yellow dot represents the band crossing at the $\\bar{\\Gamma}$ point. On the Fermi surface, the Fermi arcs connect two projected DPs (cyan dots). For stancite $(2\\bar{1}\\bar{1})$ surface, the constant-energy contour is at $\\epsilon_f-5.2$ meV, slightly away from the Fermi level, to distinguish the Fermi arcs. Stacking plots of constant-energy contours at different energies on its $(2\\bar{1}\\bar{1})$ surface of (c) germanctie and (f) stancite respectively. The Fermi level is set to be zero.}\n\\label{fig:3}\n\\end{figure}\n\nFor the (2$\\bar{1}\\bar{1}$) surface of the germancite and stancite, the electronic structures are quite different. Because the BZ of (2$\\bar{1}\\bar{1}$) surface is parallel to the $\\Gamma$-$\\text{T}$ direction, the pair of 3D DPs are projected to different points (0,0,$\\bar{\\pm k_{z0}}$) which are marked by the cyan dots in the Fig. \\ref{fig:3}(b) and (e). Between the projected DPs, a pair of the Fermi arcs could be observed clearly, which share the helical spin-texture and are not continuous at the projected points. This Fermi arcs originate from the non-trivial $\\mathbb{Z}_{2}$ topology in the Dirac semi-metals. On any 2D plane in the bulk whose BZ is perpendicular to the $\\Gamma$-$\\text{T}$ direction with $-k_{z0}<k_z<k_{z0}$, the $\\mathbb{Z}_{2}$ number is +1. Thus, in real space, the corresponding ``edge state\" exist on the boundary. In the moment space, the BZ of the ``edge state'' corresponds to the line parallel to $\\bar{Y}$-$\\bar{\\Gamma}$-$\\bar{Y}$ with $-\\bar{k_{z0}}$$<$$\\bar{k_z}$$<$$\\bar{k_{z0}}$, and its Fermi surface should be two points. After concluding all the contributions of planes with  $\\mathbb{Z}_{2}$=1, the Fermi surface becomes a pair of the Fermi arcs on the BZ of (2$\\bar{1}\\bar{1}$) surface which connect the projected DPs. At the same time, on the (2$\\bar{1}\\bar{1}$) surface, the other surface states contributed by the dangling bond also exist. Via tuning the Fermi level, we could observe the hybridization between the non-trivial surface states and Fermi arcs (see Fig. \\ref{fig:3}(c) and (f)), so a Lifshitz transition is found on the Fermi surface. Additionally, because the Fermi surface contours on the (2$\\bar{1}\\bar{1}$) surface contain roughly the same wave vector (see the yellow arrow in Fig. \\ref{fig:3} (e)), the charge density wave or surface reconstruction is possible to be observed here. However, the surface coupling will not break the TR symmetry or change the bulk topology, the pair of Fermi arcs always exist.\n\n\n\\section{Discussion and Conclusion}\nBecause of the compatibility with the traditional semiconductor devices and dissipationless edge transport, the realization of the quantum spin Hall (QSH) effect in the thin film of Ge attracts lots of attention recently. In a recent proposal \\cite{Zhang2013-Ge-TI}, the non-trivial topology of the 2D thin film is induced by the large build-in electric field in the semiconductor interface, which may be difficult to control in real experiments. However, owing to the non-trivial topology of the Dirac semi-metal, the germancite (111) film may provide an opportunity for obtaining the QSH insulator. As discussed above, the topologically non-trivial band inversion occurs around the $\\Gamma$ point in the germancite. So if we build 2D film with the proper thickness along the (111) direction, the band inversion may be restored at the $\\widetilde{\\Gamma}$ point and this thin film would become a QSH insulator. Figure \\ref{fig:4} shows the electronic structure for germancite (111) film with the thickness of 14.5 nm (i.e., 16 layer). A small band gap (5.6 meV) opens at the $\\widetilde{\\Gamma}$ point due to the quantum confinement. To confirm our prediction, we calculate its $\\mathbb{Z}_{2}$ number from the evolution of the Wannier charge centers (see Supplemental Information for details). It is found that the (111) thin film of the germancite is a 2D TI without applying the external electric field.\n\n\\begin{figure}\n\\centerline{ \\includegraphics[clip,width=0.8\\linewidth]{Figure4.eps}}\n         \\caption{(Color online) (a) Band structure of 16 layer germancite (111) film. The topologically nontrivial gap at the $\\widetilde{\\Gamma}$ can be seen in the inset. (b) Schematic device consisting of three germancite thin films with different thickness. The middle one is a QSH insulator, whereas the other two are topologically trivial. In the lower panel, the purple and green vectors stand for the spin-polarized current at the interfaces.}\n\\label{fig:4}\n\\end{figure}\nIn conclusion, from DFT calculations with the hybrid functional, we predict the germancite and stancite with SLD structure are stable topological Dirac semi-metals protected by the rotation symmetry. And it is found that the Fermi arcs coexist with the trivial surface states on the surface plane parallel to the rotation axis of $C_3$, and a Lifshitz transition is observed when the Fermi level is tuned. Furthermore, we discover the (111) thin film of the germancite is a 2D TI without applying the external electric field which is important for future applications. Experimentally, the metastable allotropes of germanium has been synthesized through the oxidation of Ge$_9^{4-}$ Zintl anions in ionic liquids under ambient conditions \\cite{guloy2006guest}. And owing to similar density and cohesive energy, we expect the germancite and stancite could be synthesized via the similar methods in the future.\n\n\n\\begin{acknowledgments}\nWe would like to thank S. Cahangirov and L. Xian for useful discussions. W.C. and W.D. acknowledge support from the Ministry of Science and Technology of China (Grant Nos. 2011CB606405 and 2011CB921901) and the National Natural Science Foundation of China (Grant No. 11334006). A.R. acknowledges financial support from the European Research Council Grant DYNamo (ERC-2010-AdG No. 267374) Spanish Grants (FIS2010-21282-C02-01), Grupos Consolidados UPV\/EHU del Gobierno Vasco (IT578-13) and EC project CRONOS (280879-2 CRONOS CP-FP7). P.T. and S.-C.Z. acknowledge NSF under grant numbers DMR-1305677 and FAME, one of six centers of STARnet, a Semiconductor Research Corporation program sponsored by MARCO and DARPA. The calculations were done on the ``Explorer 100'' cluster system of Tsinghua University.\n\\end{acknowledgments}\n\nW.C., P.T., W.D., S.-C.Z, and A.R. conceived and designed the project. W.C. and P.T. performed the $ab~initio$ calculations and theoretical analysis. P.T. and W.C. wrote the manuscript with the help from other authors. All authors contributed to the discussions. W.C. and P.T. contribute equally to this work.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}