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+{"text":"\\section{Introduction}\n\nA combined measurement of the hyperfine structure (HFS) splittings in hydrogenlike and lithiumlike ions of $^{209}$Bi has been suggested as early as 2001 \\cite{Shabaev:01a} to be a sensitive probe for bound-state strong-field QED in the strongest static magnetic fields available in the laboratory. Such fields exist in the surrounding of heavy nuclei with nuclear spin and a large nuclear magnetic moment. The electron in H-like $^{209}$Bi$^{82+}$, for example, experiences on average a magnetic field of about 30\\,000\\,T, more than 1000 times stronger than available with the strongest superconducting magnet.    \nAccording to \\cite{Shabaev:01a}, a special combination of the ground-state HFS splittings in H-like and Li-like ions ($\\Delta E^{(1s)}$ and $\\Delta E^{(2s)}$, respectively) of the same nuclear species, called the specific difference\n\\begin{eqnarray}\n\\Delta 'E = \\Delta E^{(2s)} - \\xi \\Delta E^{(1s)},\n\\end{eqnarray}\nprovides the best means to test bound-state strong-field QED in the magnetic regime.\nHere, the parameter $\\xi = 0.16886$ \\cite{Shabaev:01a,Volotka:12} is chosen to cancel the contributions of the nuclear-magnetization distribution  (Bohr-Weisskopf effect) to $\\Delta E^{(1s)}$ and $\\Delta E^{(2s)}$.\nThis is required since the uncertainties of these contributions to the HFS splittings are commonly larger than the complete QED contribution and have failed all previous attempts to perform a QED test solely based on the HFS splitting in H-like heavy ions. However, at the time of the proposal \\cite{Shabaev:01a} the experimental uncertainty of the HFS splitting in the Li-like $^{209}$Bi$^{80+}$ extracted \nfrom x-ray emission spectra \\cite{Beiersdorfer:98} was far too high to verify the predictions\nfor $\\Delta 'E$.\nThe first laser spectroscopic observation of the splitting reported in 2014 was orders of magnitude more precise but still limited by systematical uncertainties \\cite{Lochmann:14}. Finally, further improvement in accuracy by more than an order of magnitude was recently reported \\cite{Ullmann:15,Ullmann:17} but the result was surprisingly more than $7\\sigma$ off from the latest theoretical prediction \\cite{Volotka:12}.\nSince the experimental nuclear magnetic moment $\\mu_I$ of $^{209}$Bi enters the calculation of the specific difference, an incorrect value will lead to a proportional change in $\\Delta^\\prime E$, which could be responsible for the discrepancy \\cite{Karr:17}.\nWe also note that in Ref. \\cite{Urrutia:96} the discrepancy between theory and experiment on the HFS splitting in H-like Ho was ascribed to an inaccurate value of the nuclear magnetic moment of ${^{165}}$Ho.\n\nWe have reexamined the literature value $\\mu_I$($^{209}$Bi)\nobtained from nuclear magnetic resonance (NMR) experiments from a theoretical point of view. This has motivated new NMR measurements of bismuth ions in different chemical environments. Results of these experiments  are reported and analyzed applying high-level\nfour-component relativistic coupled cluster theory for advanced chemical shift calculations. We show that our result can completely resolve the hyperfine puzzle established in \\cite{Ullmann:17}. \nThe specific difference $\\Delta ' E$ has, so far, always been calculated using \nthe magnetic moment $\\mu_I(^{209}{\\rm Bi})=4.1106(2)\\mu_N$ tabulated in\n\\cite{Raghavan:89}. \nThis value was obtained using the uncorrected (for shielding effects) experimental value of the magnetic moment $\\mu_I(^{209}{\\rm Bi})=4.03910(19)\\mu_N$ reported in an NMR study \\cite{Ting:53} of bismuth nitrate, Bi(NO$_3$)$_3$, which was then  combined with the shielding constant for the Bi$^{3+}$ cation calculated in \\cite{Johnson:68}.\nIn \\cite{Bastug:96} the self-consistent relativistic molecular Dirac-Fock-Slater calculation of the shielding constant of the Bi(NO$_3$)$_3$ molecule using the Lamb formula \\cite{Lamb:41} was performed.\nThe final value, $\\sigma=17290(60)$\\,ppm, with very small uncertainty was obtained by combining relativistic random phase approximation calculation of the Bi$^{3+}$ cation (17270 ppm) with the molecular correction. The authors concluded that the molecular correction is very small and thus supported the value from \\cite{Raghavan:89}. \n\nHowever, \nthe authors of \\cite{Bastug:96} have not taken into account chemical processes that occur in an aqueous solution of bismuth nitrate molecule Bi(NO$_3$)$_3 \\cdot$5H$_2$O: the compound\ndissociates and the Bi$^{3+}$ cation is surrounded by water molecules (hydration). Neither the completeness nor the exact form of hydration as a function of concentration, \\textit{p}H or temperature is well understood.  \nWhile it was suggested in \\cite{Fedorov:98} that in strongly acidic solutions Bi$^{3+}$ exists as hexaaquabismuth(III)-cation [Bi(H$_2$O)$_6$]$^{3+}$, more recent studies \\cite{Naslund:00} expect that the hydrated form is rather [Bi(H$_2$O)$_8$]$^{3+}$. We found that in both cases the electronic structure of the $n$-coordinated complex significantly differs from the Bi(NO$_3$)$_3$ molecule considered in \\cite{Bastug:96}, which is expected. The molecular environment in \nBi(III\/V)-containing\ncomplexes strongly contributes to the shielding constant and a considerable chemical shift is introduced. \nConsequently, the value of the shielding constant obtained in Ref.\\,\\cite{Bastug:96} cannot be used for the precise extraction of the $^{209}$Bi magnetic moment from the experimental NMR data. \n\nThere is, however, additional NMR data for another Bi containing system: the\nhexafluoridobismuthate(V) anion ($^{209}$BiF$_6^-$) \\cite{Morgan:83}. It has seven atoms and high spatial symmetry.\nAccording to Morgan \\textit{et} al.\\ \\cite{Morgan:83}, a measurement of BiF$_6^-$ with reference to a saturated solution of bismuth nitrate in concentrated nitric acid gave a chemical shift of $-24$\\,ppm. Unfortunately, there is an inconsistency in the reported experimental data of \\cite{Morgan:83}, since the measured frequency ratio is given as $\\nu(^{209} {\\rm BiF}_6^-) \/ \\nu(^1{\\rm H})$ =0.16017649(10). The comparison of this ratio with the one reported in \\cite{Ting:53} indicates a massive chemical shift of about $\\delta \\approx +3200$\\,ppm instead of $-24$\\,ppm. We have performed NMR measurements of both samples to clarify these discrepancies.  \n\n\\section{Experiment}\nSince a dependence of the chemical state of the Bi$^{3+}$ ions in an aqueous solution is expected but details on the sample preparation are missing in the original NMR measurements \\cite{Ting:53}, we performed a systematic study using various bismuth nitrate solutions.\nSamples of ``Bi(NO$_3$)$_3$'' solutions were prepared with concentrations of 2.5\\%, 5\\% and 10\\% Bi$^{3+}$ (wt \\%) in concentrated (65 wt \\%) and diluted aqueous solutions (50, 30, 20, 10 wt \\%) of nitric acid (HNO$_3$). \n\n\\begin{figure}[t]\n\\includegraphics[width=0.98\\linewidth]{Fig1New.pdf}\n\\caption{\\label{fig:Spectra} NMR spectra of Bi(NO$_3$)$_3$ solution (10\\% Bi (wt \\%)) in concentrated nitric acid (gray) and  NMe$_4$BiF$_6$ diluted in acetonitrile (blue).}\n\\end{figure}\n\nBiF$_6^-$ anions were obtained by dissolution of $\\mathrm{(CH_3)_4N^+BiF_6^-}$ (NMe$_4$BiF$_6$)\nin acetonitrile to a saturated solution\n\\cite{Note1}. \n\nAll NMR measurements were performed at an 8.4-T magnet using the same double resonance probe for $^{209}$Bi NMR and $^{1}$H NMR calibration with tetramethylsilane. The sample temperature was stabilized with an accuracy of 1\\,K employing a constant gas flow tempered by an electric heater. Spectra were obtained from the free induction decay following a 90$^\\circ$ pulse of 3.5\\,$\\upmu$s length for $^{209}$Bi. \n\nTypical spectra of the $^{209}$Bi atoms in BiF$_6^-$ and in the nitrate solution are shown in Fig.\\,\\ref{fig:Spectra}. The advantage of BiF$_6^-$ is obvious. It exhibits a much narrower linewidth (200 Hz) and the septet arising from indirect spin coupling of $^{19}$F atoms directly bonded to the bismuth atom assures the chemical environment.\nThe observed ratio of the peak intensities is close to the expected ratio 1\\,:\\,6\\,:\\,15\\,:\\,20\\,:\\,15\\,:\\,6\\,:\\,1 and a spin-spin coupling of 3807(14)\\,Hz was determined, in good agreement with \\cite{Morgan:83}.\nNote that a $^{19}$F spectrum of the sample was taken as well and a decet consistent with the coupling of an $I=9\/2$ nucleus to an octahedral environment of six fluorine atoms was observed. \nThe signal from the nitrate solution is much wider. Even at the highest temperature of 360\\,K, the width of the $^{209}$Bi spectra was 4.4\\,kHz due to the short spin-lattice and spin-spin relaxation times of $\\approx 70$\\,$\\upmu$s. This width limits the accuracy of the $^{209}$Bi resonance frequency in the solution of the nitrate to 1\\,ppm. The chemical shift of Bi$^{3+}$ in the solution of the bismuth nitrate with respect to Bi$^{5+}$ in BiF$_6^-$ is $-106$\\,ppm, larger than the $-24$\\,ppm reported in \\cite{Morgan:83}. Contrary to \\cite{Flynn:59} we found that the variation of the bismuth concentration between mass fractions of 2\\% and about 40\\% (saturation) in nitric acid of 30\\% had no appreciable effect on the measured Larmor frequency as long as temperature and nitric acid concentration were kept constant. \n\n\nVariations of the Bi(NO$_3$)$_3$ sample temperature from 250 to 360\\,K were performed with the sample of 10\\% Bi in concentrated nitric acid (65\\%). We observed a strong linear temperature dependence of the frequency ratio in this range (Fig.\\,\\ref{fig:TempDependence}) with a slope of $+4.69(13)\\times 10^{-7}$\\,K$^{-1}$, corresponding to about 3\\,ppm\/K,\nwhich might be caused by the change of density.\nFor standard NMR conditions at 298.15\\,K a frequency ratio of $\\nu_\\mathrm{^{209}Bi^{3+}}\/\\nu_\\mathrm{H}=0.160699(1)$ was determined, where the given uncertainty is purely statistical. This value is in excellent agreement with 0.160696(6) reported in \\cite{Ting:53}. The temperature dependency of BiF$_6^-$ is 2 orders of magnitude smaller ($\\approx 20$\\,ppb\/K) and of opposite sign. At 298.15\\,K the frequency ratio to the proton is 0.1607167(2) far off from the value provided in \\cite{Morgan:83}. However, the latter matches our value if one simply flips two digits [$0.160{\\bf 17}65(1) \\to 0.160{\\bf 71}65(1)$].  \n\n\\begin{figure}[t]\n\\includegraphics[width=0.98\\linewidth]{Fig2Test3.pdf}\n\\caption{\\label{fig:TempDependence} Temperature and HNO$_3$-concentration dependency of the NMR Larmor-frequency ratios of bismuth and hydrogen. A strong temperature effect is observed for Bi(NO$_3$)$_3$ solutions, here exemplified for a \n10\\% Bi$^{3+}$ (wt \\%) solution in concentrated nitric acid (black), whereas only a minor effect was measured for NMe$_4$BiF$_6$ dissolved in acetonitrile (blue).\nInset: Larmor-frequency ratios\nmeasured by NMR in Bi(NO$_3$)$_3$ solutions with 2.5\\% Bi$^{3+}$ (wt \\%) in nitric acid (HNO$_3$) of various concentrations at 300\\,K. \nThe $y$ axis is identical to the main graph and the gray band represents the total variation.}\n\\end{figure}\n\nFinally, we have studied the resonance position of Bi(NO$_3$)$_3$ as a function of the nitric acid concentration\n(inset in Fig.\\,\\ref{fig:TempDependence}).\nA clear dependence on the acidity is observed for all Bi$^{3+}$ concentrations, covering a range of typically $\\approx 60$\\,ppm. \n\nIn summary, the results clearly demonstrate that a large uncertainty is connected with the extraction of the magnetic moment of $^{209}$Bi from NMR measurements in aqueous solutions of Bi(NO$_3$)$_3$. The influence of the chemical environment was strongly underestimated in theory since the calculations performed to extract the chemical shift do neither account for the temperature nor for the concentration or acidity of the sample. In this respect, BiF$_6^-$ is a much better candidate to obtain a reliable value of the magnetic moment which will be substantiated now also from a  theoretical point of view.  \n\n\\section{Theory}\n\nIn the presence of the external uniform magnetic field \\textbf{B} and nuclear magnetic moment $\\mu_j$ of $j-$th atom in a molecule the corresponding Dirac-Coulomb Hamiltonian includes the following terms:\n\\begin{equation}\n \\label{HB}\nH_B={\\rm \\bf{B}}\\cdot \\frac{c}{2}(\\bm{r}_G \\times \\bm{\\alpha}),\n\\end{equation}\n\\begin{equation}\n \\label{HHFS}\nH_{\\rm hyp}=\\frac{1}{c} \\sum_j \\bm{\\mu}_j\\cdot \\frac{(\\bm{r}_j \\times \\bm{\\alpha})}{r_j^3},\n\\end{equation}\nwhere $\\bm{r}_G = \\bm{r} - \\bm{R_G}$, $\\bm{R_G}$ is the gauge origin, \n$\\bm{r}_j=\\bm{r} - \\bm{R_j}$, $\\bm{R_j}$ is the position of nucleus $j$, and $\\bm{\\alpha}$ are the Dirac matrices.\n\nThe chemical shielding tensor of the nucleus $j$ can be defined as a mixed derivative of the energy with respect to the nuclear magnetic moment and the strength of the magnetic field\n\\begin{equation}\n \\label{SHIELDINGDer}\n\\left.\\sigma^j_{a,b}=\\frac{\\partial^2E}{\\partial\\mu_{j,a}\\partial B_b} \\right|_{\\bm{\\mu}_j=0,{\\rm \\bf{B}}=0}.\n\\end{equation}\nWe are interested in its isotropic part.\n\nIn the one-electron case the shielding tensor (\\ref{SHIELDINGDer}) can be calculated by the sum-over-states method within the second-order perturbation theory with perturbations (\\ref{HB}) and (\\ref{HHFS}). In the relativistic four-component approach the summation should include both positive and negative energy spectra \\cite{Aucar:99}.\nThe part associated with positive energy is called the ``paramagnetic'' term while the part associated with negative energy states is called ``diamagnetic term'' though only their sum is gauge invariant \\cite{Aucar:99}.\n\n\nTo avoid an ambiguity in calculations utilizing finite basis sets due to the choice of the gauge origin $\\bm{R_G}$ one can use the so-called London atomic orbitals (LAOs) method (see e.g.\\ \\cite{Olejniczak:12,Ilias:13} for details).\nIn Refs.\\,\\cite{DIRAC15,Olejniczak:12,Ilias:13} the four-component density functional theory (DFT) using response technique and LAOs has been developed to calculate the shielding constant (\\ref{SHIELDINGDer}).\nTo construct the atomic basis sets for the unperturbed Dirac-Coulomb Hamiltonian calculations one often uses the restricted kinetic balance (RKB) method. However, in the presence of the external magnetic fields the usual relation between the large and small component changes. In Ref.\\,\\cite{Olejniczak:12} the scheme of magnetic balance (MB) in conjunction with LAOs was proposed to take into account the modified coupling which is utilised below.\n\nMost of the chemical shift calculations for heavy atom compounds are performed within the (relativistic) DFT. The drawback of the theory is that it is hard to control the uncertainty of the results as there is no systematic way of improving it. Even combinations with high-level nonrelativistic \\textit{ab initio} wave-function-based calculations are also questionable in the case of heavy atom compounds. In Refs.\\,\\cite{Skripnikov:16b,Skripnikov:15a,Petrov:17b,Skripnikov:17c} it was shown that for such properties as the hyperfine structure constant and the molecular $g$ factor, the relativistic coupled cluster method gives the most accurate results if there are no multireference effects. Therefore, this method has been adopted here to control the uncertainty of the DFT results. \n\n\n\n\n\\section{Electronic structure calculation details}\n\n\nIn the present study we have used atomic basis sets of different qualities.\nThe NZ (where N~$=$~Double, Triple, Quadruple) basis set corresponds to the uncontracted core-valence N-zeta \\cite{Dyall:07,Dyall:12} Dyall's basis set for Bi and augmented correlation consistent polarized valence N-zeta, aug-cc-pVNZ \\cite{Dunning:89,Kendall:92} basis set for light atoms.\nIn the DZC basis set the contracted version of the aug-cc-pVDZ \\cite{Dunning:89,Kendall:92} basis sets were used for light atoms. \n\n\nBased on the nonrelativistic estimates, the hybrid density functional PBE0 \\cite{pbe0} has been chosen because it reproduces the nonrelativistic coupled cluster value rather well.\nGeometry parameters of the BiF$_6^-$ anion have been obtained in the scalar-relativistic DFT calculation using the generalized relativistic pseudopotential method \\cite{Mosyagin:16}.\n\nThe contribution of the Gaunt interaction to the shielding constant was estimated as the difference between  the values calculated at the Dirac-Hartree-Fock-Gaunt and Dirac-Hartree-Fock level of theory within the uncoupled scheme.\n\nNonrelativistic and scalar-relativistic calculations were performed within the {\\sc us-gamess} \\cite{USGAMESS1} and {\\sc cfour} \\cite{CFOUR} codes. Relativistic four-component calculations were performed within the {\\sc dirac15} \\cite{DIRAC15} and {\\sc mrcc} \\cite{MRCC2013} codes. For calculation of the hyperfine-interaction matrix elements and $g$ factors the code developed in Refs.\\,\\cite{Skripnikov:16b,Skripnikov:15b,Skripnikov:15a} was used.\n\n\n\\section{Results and discussion}\n\n\nTable \\ref{BiF6} contains results of the calculation of the BiF$_6^-$ anion.\n\\begin{table}[h]\n\\centering\n\\caption{The values of $^{209}$Bi shielding constants in BiF$_6^-$ in ppm.}\n\\label{BiF6}\n\\begin{tabular}{lccc}\n\\hline                    \n\\hline\nBasis set\/method     & Diamagnetic & Paramagnetic & Total \\\\\n\\hline                    \n\\hline\nDZ-MB-LAO\/DHF    & 8\\,618 & 5\\,768 & 14\\,386  \\\\  \n\nDZ-MB-LAO\/DFT    & 8\\,621 & 3\\,726 & 12\\,347  \\\\     \nTZ-MB-LAO\/DFT    & 8\\,639 & 3\\,733 & 12\\,372  \\\\  \n\\hline    \nDZC-RKB\/DFT      &      & 3\\,848    &    \\\\\nDZC-RKB\/CCSD     &      & 4\\,403    &    \\\\\nDZC-RKB\/CCSD(T)  &      & 4\\,286    &    \\\\\n\\hline    \nQZ-MB-LAO\/DFT    & 8\\,628 & 3\\,763 & 12\\,391  \\\\  \nCorrelation correction &      &  437 &        \\\\  \nGaunt correction       &      &  -37 &        \\\\  \n\\hline                    \nFinal              &      &   & 12\\,792  \\\\  \n\\hline                    \n\\hline                    \n\\end{tabular}\n\\end{table}\nComparing Dirac-Hartree-Fock (DHF) and DFT results in Table \\ref{BiF6} it can be seen that the diamagnetic contribution to $\\sigma(^{209}{\\rm Bi})$ depends only weakly on the correlation effects, while the paramagnetic contribution is strongly affected.\nTo check the accuracy of the latter DFT result we have performed a series of relativistic coupled cluster calculations of $\\sigma(^{209}{\\rm Bi})$ taking into account only the positive energy spectrum. \nComparing values obtained within the coupled cluster with single, double and noniterative triple-cluster amplitudes (CCSD(T)) with that of CCSD shows that the triple amplitudes only slightly contribute to $\\sigma(^{209}{\\rm Bi})$ demonstrating good convergence of the results with respect to the electron correlation treatment \n\\cite{Note2}. \n\nIn the final value of $\\sigma(^{209}{\\rm Bi})$ we include the correlation correction calculated as the difference between the CCSD(T) and PBE0 results.\n\nTo investigate the importance of systematic treatment of the  molecular environment \n we have also performed an additional DHF study of one of the possible hydrated forms of Bi$^{3+}$ in an acidic solution of Bi(NO$_3$)$_3$ -- [Bi(H$_2$O)$_8$]$^{3+}$ cation in comparison with the unsolvated Bi$^{3+}$ cation.\nIt was found that the shielding constant of the $^{209}$Bi$^{3+}$ is significantly larger (by about 20\\% at the DHF level) than that in [$^{209}$Bi(H$_2$O)$_8$]$^{3+}$.\nTherefore, the interpretation of the \\textit{molecular} NMR experiment in terms of the nuclear magnetic moment using a shielding constant obtained for the corresponding ion (as was done in earlier studies) is associated with considerable uncertainties.\n\n\nWe now use the value obtained for $\\nu_\\mathrm{^{209}BiF_6^-}\/\\nu_\\mathrm{H}= 0.1607167(2)$ from our NMR measurements and the shielding constant of $\\sigma(^{209}{\\rm BiF}_6^-)=12\\,792$\\,ppm calculated above to obtain $\\mu_I(\\mathrm{^{209}Bi}) = 4.092(2)\\,\\mu_\\mathrm{N}$ with an uncertainty dominated by theory. \n\nTable \\ref{NewOld} compares the experimental values \\cite{Ullmann:17} of the HFS splittings with the theoretical values calculated with the old [$\\mu_I$(old)$=$4.1106(2)$\\mu_N$] and the new [$\\mu_I$(new)$=$4.092(2)$\\mu_N$] values of the nuclear magnetic moment \\cite{Volotka:12}. The theoretical results include the most elaborated calculation of the Bohr-Weisskopf effect \\cite{Senkov:02}.\n\\begin{table}[]\n\\centering\n\\caption{Theoretical values of $\\Delta E^{(1s)}$ and $\\Delta E^{(2s)}$ (in meV) calculated with old and new nuclear magnetic moment of $^{209}$Bi in comparison with the experimental values \\cite{Ullmann:17}.\nFor the Bohr-Weisskopf effect the most elaborated calculation by Sen'kov and Dmitriev \\cite{Senkov:02} was employed.\n}\n\\label{NewOld}\n\\begin{tabular}{llll}\n\\hline\n\\hline\n   & \\multicolumn{2}{l}{Theory}                             & Experiment         \\\\\n      & $\\mu_I$(old)  & $\\mu_I$(new)      &    \\\\\n\\hline   \n$\\Delta E^{(1s)}$  & 5112(-5\/+20)                & 5089(-5\/+20)(2)  & 5085.03(2)(9)  \\\\\n$\\Delta E^{(2s)}$  & 801.9(-9\/+34) & 798.3(-9\/+34)(4) & 797.645(4)(14) \\\\          \n\\hline   \n\\hline\n\\end{tabular}\n\\end{table}\n\\begin{figure}[!h]\n\\includegraphics[width=0.98\\linewidth]{deltaPrimeE_NMR_2.pdf}\n\\caption{\\label{fig:CompExpTheory} Specific difference obtained in theory  \\cite{Shabaev:01a,Volotka:12} (red) and experiment \\cite{Lochmann:14,Ullmann:17} (blue). The new nuclear magnetic moment established in this work yields a new value for the specific difference which matches the most recent experimental value within uncertainty. \n}\n\\end{figure}\nThe new magnetic moment has been used to recalculate the specific difference and we obtain \n$\\Delta 'E_{\\mathrm{theo}} = -61.043(5)(30)$\\,meV, where the first uncertainty is due to uncalculated terms and remaining nuclear effects, while the second one is due to the uncertainty of the nuclear magnetic moment obtained in the present work. Revised value of $\\Delta 'E_{\\mathrm{theo}}$ \nis plotted in Fig.\\,\\ref{fig:CompExpTheory} combined with the previous theoretical and experimental data. Theory and experiment are now in excellent agreement and the $7\\sigma$ discrepancy reported in \\cite{Ullmann:17} disappears. \nUnfortunately, the uncertainty of $\\Delta 'E_{\\mathrm{theo}}$ is now 14\\% of the total QED contribution and about 1.5 times larger than the experimental uncertainty.\nHence, an improved value for the nuclear magnetic moment of $^{209}$Bi is urgently required, either from an atomic beam magnetic resonance experiment or from a measurement on trapped H-like ions. The latter will have the advantage that no shielding corrections have to be applied. Such an experiment is planned at the ARTEMIS trap \\cite{Quint:2008} at the GSI Helmholtz Centre in Darmstadt. Only such a measurement combined with an improved determination of the HFS splitting in $^{209}$Bi$^{80+,82+}$ as it is foreseen at SPECTRAP \\cite{Andelkovic:2013} can provide a QED test in the magnetic regime of strong-field QED. Our result also proves that a measurement of the specific difference can also be used to extract the nuclear magnetic moment. \nDoing so results in  $\\mu_I(^{209}\\mathrm{Bi})=4.0900(15)\\,\\mu_{\\mathrm{N}}$ in excellent agreement with the NMR value obtained here. \n\n\\begin{acknowledgments}\n\\section*{Acknowledgments}\nWe thank Petra Th\\\"orle from the Institute of Nuclear Chemistry at the University of Mainz for the preparation of the NMR samples and Dmitry Korolev from Saint-Peterburg State University for valuable discussions.\nThe development of the code for the computation of the matrix elements of the considered operators as well as the performance of all-electron coupled cluster calculations were funded by RFBR, according to Research Project No.~16-32-60013 mol\\_a\\_dk; performance of DFT calculations was supported by the President of Russian Federation Grant No. MK-2230.2018.2. This work was also supported by SPSU (Grants No. 11.38.237.2015 and No. 11.40.538.2017) and by SPSU-DFG (Grants No. 11.65.41.2017 and No. STO 346\/5-1). The experimental part was supported by the Federal Ministry of Education and Research of Germany under Contract No 05P15RDFAA and the Helmholtz\nInternational Center for FAIR (HIC for FAIR).   \n\\end{acknowledgments}\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\noindent\nA BV algebra\nand a QP-structure \nhas been motivated by the structure of \nthe Batalin-Vilkovisky formalism of a gauge theory\\cite{Bat}\nand is its mathematical formulation\n\\cite{Schwarz:1992nx}.\nIn case of a topological field theory of Schwarz type, \na BV formalism has been reformulated to the AKSZ formulation, \nwhich is \na clear construction using geometry of a graded manifold \n\\cite{Alexandrov:1995kv}\\cite{Cattaneo:2001ys}.\nApplication to higher $n+1$ dimensions has been formulated and \nnew topological field theories \nin higher dimensions have been founded\nby applying this construction\n\\cite{Park:2000au}\\cite{Severa:2001}\\cite{Ikeda:2001fq}.\n\n\nIn $n=1$, \na classical QP-structure is \na Poisson structure on a manifold $M$ and is also \na Lie algebroid on $T^*M$ from the explicit construction.\nThis is equivalent to the construction of a Poisson structure\nby the Schouten-Nijenhuis bracket in a classical limit.\nThe topological field theory in two dimensions constructed \nby the AKSZ formulation \\cite{Cattaneo:2001ys}\nis the Poisson sigma model\n\\cite{Ikeda:1993aj}\\cite{Schaller:1994es}\nand the quantization of this model \non disc derives the Kontsevich formula of the \ndeformation quantization on a Poisson manifold\n\\cite{Kontsevich:1997vb}\\cite{Cattaneo:1999fm}.\n\nIn $n=2$, a classical QP-structure is a Courant algebroid\n\\cite{Courant}\\cite{Roy01}.\nThe topological field theory derived \nin three dimensions is\nthe Courant sigma model\n\\cite{Ikeda:2002wh}\\cite{Hofman:2002rv}\\cite{Roytenberg:2006qz}.\n\nHowever structures for higher $n$, more than $2$, \nhave not been understood \nenough apart from BF theories.\n\nIn this paper, we analyze $n=3$ case.\nA QP-structure of degree $3$ leads us to\na new type of algebroid,\nwhich is called a \n\\textbf{Lie algebroid up to homotopy}.\nThe notion of this algebroid\nis defined as\na homotopy deformation of a \nLie algebroid \nsatisfying some integrability conditions.\nWe will prove that a QP-structure of degree $3$\non a N-manifold (nonnegatively graded manifold)\nis equivalent to \na Lie algebroid up to homotopy.\nThis QP-structure defines\na new natural $4$-dimensional topological field \ntheory via the AKSZ construction.\n\nThe paper is organized as follows. \nIn section 2, a BV algebra and a QP-structure of\ndegree $3$ are formulated.\nIn section 3, a QP-structure of degree $3$ is constructed \nand analyzed.\nIn section 4, examples of QP-structures of degree $3$\nare listed.\nIn section 5, the AKSZ construction of \na topological field theory in four dimensions\nis formulated and examples are listed.\n\\footnote{Very recently, Gr\\\"utzmann's paper appears which has overlaps\nwith our paper\n\\cite{Grutzmann}.}\n\n\n\n\\section{QP-manifolds and BV Algebras}\n\\subsection{Classical QP-manifold} \n\\begin{definition}\nA graded manifold\n$\\calM$ is by definition a sheaf\nof a graded commutative algebra over\nan ordinary smooth manifold $M$.\n\\end{definition}\nIn the following, we assume the degrees are nonnegative.\\\\\n\\indent\nThe structure sheaf of $\\calM$ is locally isomorphic to\na graded commutative algebra $C^{\\infty}(U)\\otimes S(V)$,\nwhere $U$ is an ordinary local chart of $M$,\n$S(V)$ is the polynomial algebra over $V$\nand where\n$V:=\\sum_{i\\ge 1}V_{i}$ is a graded vector space\nsuch that the dimension of $V_{i}$ is finite for each $i$.\nFor example, when $V=V_{1}$, $\\calM$ is a vector bundle\nwhose fiber is $V^{*}_{1}$: the dual space of $V_{1}$.\n\\begin{definition}\nA graded manifold $(\\calM,\\omega)$ equipped with\na graded symplectic structure $\\omega$ of degree $n$\nis called a \\textbf{P-manifold} of degree $n$.\n\\end{definition}\nIn the next section, we will study a concrete \nP-manifold of degree 3.\\\\\n\\indent\nThe structure sheaf $C^{\\infty}(\\calM)$ of a P-manifold\nbecomes a graded Poisson algebra.\nThe Poisson bracket is defined in the usual manner,\n\\begin{equation}\\label{gradedpoisson}\n\\sbv{F}{G}=(-1)^{|F|+1}\\iota_{X_F}\\iota_{X_G}\\omega,\n\\end{equation}\nwhere $F,G\\in C^{\\infty}(\\calM)$,\n$|F|$ is the degree of $F$\nand $X_{F}:=\\{F,-\\}$ is the Hamiltonian vector field of $F$.\nWe recall the basic properties of the Poisson bracket,\n\\begin{eqnarray*}\n\\sbv{F}{G}&=&-(-1)^{(|F| - n)(|G| - n)} \\sbv{G}{F},\\\\\n\\sbv{F}{G  H}&=&\\sbv{F}{G} H\n+ (-1)^{(|F| - n)|G|} G \\sbv{F}{H},\\\\\n\\{F,\\{G,H\\}\\}&=&\\{\\{F,G\\},H\\}+(-1)^{(|F|-n)(|G|-n)}\\{G,\\{F,H\\}\\},\n\\end{eqnarray*}\nwhere $n$ is the degree of the symplectic structure and $F,G,H\\in C^{\\infty}(\\calM)$.\nWe remark that the degree of the Poisson bracket is $-n$.\n\\begin{definition}\nLet $(\\calM,\\omega)$ be a P-manifold of degree $n$.\nA function $\\Theta\\in C^{\\infty}(\\calM)$ of degree $n+1$\nis called a \\textbf{Q-structure}, if it is a solution of\nthe \\textbf{classical master equation},\n\\begin{eqnarray}\n\\sbv{\\Theta}{\\Theta}=0.\n\\label{bvaction}\n\\end{eqnarray}\nThe triple $(\\calM,\\omega,\\Theta)$ is called a \\textbf{QP-manifold}.\n\\end{definition}\nWe define an operator $Q:=\\sbv{\\Theta}{-}$,\nwhich is called a homological vector field.\nFrom (\\ref{bvaction}) we have the cocycle condition,\n$$\nQ^2=0,\n$$\nwhich says that the homological vector field\nis a coboundary operator on $C^{\\infty}(\\calM)$ and \ndefines a cohomology called the classical BRST cohomology.\n\n\\subsection{Quantum QP-manifold}\n\\begin{definition}\nA graded manifold is called a quantum BV-algebra\nif it has an odd Laplace operator $\\Delta$, \nwhich is a linear operator on $C^{\\infty}(\\calM)$\nsatisfying $\\Delta^{2}=0$,\nand the graded Poisson bracket is given by\n\\begin{eqnarray}\n\\sbv{F}{G}= \n(-1)^{|F|} \\Delta(FG) - (-1)^{|F|} \\Delta(F){G}-{F}\\Delta(G),\n\\label{Poisson2}\n\\end{eqnarray}\nwhere $F, G \\in C^{\\infty}(\\calM)$.\n\\end{definition}\nIf $n$ is odd, a P-manifold $(\\calM,\\omega)$ has the odd Poisson bracket.\nIf an odd P-manifold $(\\calM,\\omega)$ has a volume form $\\rho$,\none can define an odd Laplace operator $\\Delta$ \n(See \\cite{Khudaverdian:2000zt}):\n\\begin{eqnarray*}\n\\Delta F := \\frac{1}{2}\n(-1)^{|F|} {\\rm div}_{\\rho} X_F.\n\\end{eqnarray*}\nHere a divergence ${\\rm div}_{\\rho}$ \nis a map from a space of vector fields on $\\calM$\nto $C^{\\infty}(\\calM)$ and\nis defined by\n\\begin{eqnarray*}\n\\int_{\\calM} {\\rm div}_{\\rho} X \\ F dv = - \\int_{\\calM} X(F) dv,\n\\label{divergence}\n\\end{eqnarray*}\nfor a vector field $X$ on $\\calM$.\nThe pair $(\\calM, \\Delta)$ is called a \\textbf{quantum P-structure}.\nAn odd Laplace operator has degree $-n$.\n\\begin{definition}\nA function $\\Theta \\in C^{\\infty}(\\calM)$ with \nthe degree $n+1$\nis called a \\textbf{quantum Q-structure},\nif it satisfies a \\textbf{quantum master equation}\n\\begin{eqnarray}\n\\Delta (e^{\\frac{i}{\\hbar}\\Theta}) =0,\n\\label{bvaction2}\n\\end{eqnarray}\nwhere $\\hbar$ is a formal parameter.\nThe triple $(\\calM,\\Delta,\\Theta)$ is called a \\textbf{quantum QP-manifold}.\n\\end{definition}\nFrom the definition of an odd Laplace operator, \nthe equation (\\ref{bvaction2}) is equivalent to \n\\begin{eqnarray}\n\\sbv{\\Theta}{\\Theta} - 2 i \\hbar \\Delta \\Theta =0.\n\\label{qme}\n\\end{eqnarray}\nIf we take the limit of $\\hbar\\to 0$ in (\\ref{qme}), \nwhich is called a classical limit,\nthe classical master equation \n$\\sbv{\\Theta}{\\Theta} =0$\nis derived.\nSince $\\Delta^2=0$,\n$\\Delta$ is also a coboundary operator.\nThis defines a quantum BRST cohomology.\nLet $\\calO^{\\prime} = \\calO e^{\\frac{i}{\\hbar}\\Theta}\n\\in C^{\\infty}(\\calM)$ be a cocycle with respect to $\\Delta$.\nThe cocycle condition \n$\n\\Delta (\\calO^{\\prime}) \n= \\Delta (\\calO e^{\\frac{i}{\\hbar}\\Theta}) = 0\n$\nis equivalent to\n\\begin{eqnarray}\\label{defob}\n\\sbv{\\Theta}{\\calO} - i \\hbar \\Delta \\calO = 0.\n\\end{eqnarray}\nThe solutions of (\\ref{defob})\nare called {\\it observables} in physics.\nIn the classical limit, (\\ref{defob}) is \n$\\sbv{\\Theta}{\\calO} = Q\\calO=0$.\n${\\calO}$ reduces to an element of a classical BRST cohomology.\n\n\\section{Structures and homotopy algebroids}\n\nIn this section, we construct and analyze a classical QP-structure of\n degree $3$ explicitly.\n\n\\subsection{P-structures}\n\nLet $E\\to M$ be a vector bundle over an ordinary smooth manifold $M$.\nThe shifted bundle $E[1]\\to M$ is a graded manifold\nwhose fiber space has the degree $+1$.\nWe consider the shifted cotangent bundle $\\calM:=T^{*}[3]E[1]$.\nIt is a P-manifold of the degree $3$ over $M$,\n$$\nT^{*}[3]E[1]\\to\\calM_{2}\\to E[1]\\to M,\n$$\nwhere $\\calM_{2}$ is a certain graded manifold\n\\footnote{In fact, $\\calM_{2}$ is $E[1]\\oplus E^*[2]$,\nwhich is derived from the result\nin the previous sentence of Remark 3.2.}.\nThe structure sheaf $C^{\\infty}(\\calM)$ of $\\calM$\nis decomposed into the homogeneous subspaces,\n$$\nC^{\\infty}(\\calM)=\\sum_{i\\ge 0}C^{i}(\\calM),\n$$\nwhere $C^{i}(\\calM)$ is the space of functions of degree $i$.\nIn particular, $C^{0}(\\calM)=C^{\\infty}(M)$: the algebra of\nsmooth functions on the base manifold\nand $C^{1}(\\calM)=\\Gamma E^{*}$: the space\nof sections of the dual bundle of $E$.\\\\\n\\indent\nLet us denote by $(x,q,p,\\xi)$\na canonical (Darboux) coordinate on $\\calM$, where\n$x$ is a smooth coordinate on $M$,\n$q$ is a fiber coordinate on $E[1]\\to M$,\n$(\\xi,p)$ is the momentum coordinate\non $T^{*}[3]E[1]$ for $(x,q)$.\nThe degrees of the variables $(x,q,p,\\xi)$ are respectively $(0,1,2,3)$.\\\\\n\\indent\nTwo directions of counting the degree of functions\non $T^{*}[3]E[1]$ are introduced. \nRoughly speaking, these are\nthe fiber direction and the base direction.\n\\begin{definition}(Bidegree, see also Remark 3.3.3 in \\cite{Roy01})\nConsider a monomial $\\xi^{i}p^{j}q^{k}$ on a local chart\n$(U;x,q,p,\\xi)$ of $\\calM$, of which the total degree is $3i+2j+k$.\nThe \\textbf{bidegree} of the monomial is, by definition,\n$(2(i+j),i+k)$.\n\\end{definition}\nThis definition is invariant under the natural coordinate transformation,\n\\begin{eqnarray*}\nx^{\\prime}_{i}&=&x^{\\prime}_{i}(x_{1},x_{2},...,x_{dim(M)}),\\\\\nq^{\\prime}_{i}&=&\\sum_{j}t_{ij}q_{j},\\\\\np^{\\prime}_{i}&=&\\sum_{j}t^{-1}_{ij}p_{j},\\\\\n\\xi^{\\prime}_{i}&=&\\sum_{j}\\frac{\\partial x_{j}}{\\partial x^{\\prime}_{i}}\\xi_{j}\n+\\sum_{jkl}\n(\\frac{\\partial t^{-1}_{jl}}{\\partial x^{\\prime}_{i}}t_{lk}\n+\\frac{\\partial t_{kl}}{\\partial x^{\\prime}_{i}}t^{-1}_{lj})\np_{j}q_{k},\n\\end{eqnarray*}\nwhere $t$ is a transition function. Since $T^{*}[3]E[1]$ is covered\nby the natural coordinates,\nthe bidegree is globally well-defined\n(See also Remark \\ref{shiftremark} below.)\\\\\n\\indent\nThe space $C^{n}(\\calM)$ is uniquely decomposed into\nthe homogeneous subspaces with respect to the bidegree,\n$$\nC^{n}(\\calM)=\\sum_{2i+j=n}C^{2i,j}(\\calM).\n$$\nSince $C^{2,0}(\\calM)=\\Gamma E$ and $C^{0,2}(\\calM)=\\Gamma\\wedge^{2}E^{*}$,\nwe have\n$$\nC^{2}(\\calM)=\\Gamma E\\oplus\\Gamma\\wedge^{2}E^{*}.\n$$\n\\begin{remark}\\label{shiftremark}\nThe P-manifold $T^{*}[3]E[1]$ is regarded as\na shifted manifold of $T^{*}[2]E[1]$.\nThe structure sheaf is also a shifted \nsheaf of the one on $T^{*}[2]E[1]$.\nIn particular, the space $C^{2i,j}$ is \nthe shifted space of $C^{i,j}$ on $T^{*}[2]E[1]$.\n\\end{remark}\nFor the canonical coordinate on $\\calM$,\nthe symplectic structure has the following form:\n$$\n\\omega = \\delta \\bx^i \\delta \\bxi_i + \\delta \\bq^a \\delta \\bp_a,\n$$\nand the associated Poisson bracket\nhas the following expression:\n\\begin{eqnarray*}\n\\sbv{F}{G} &=& \nF \\frac{\\rd}{\\partial \\bx^i} \n\\frac{\\ld }{\\partial \\bxi_{i}} G\n- \nF \\frac{\\rd }{\\partial \\bxi_{i}} \n\\frac{\\ld }{\\partial \\bx^i} G\n+ \nF  \\frac{\\rd}{\\partial \\bq^{a}} \n\\frac{\\ld }{\\partial \\bp_{a}} G\n- \nF \\frac{\\rd }{\\partial \\bp_{a}} \n\\frac{\\ld }{\\partial \\bq^{a}} G,\n\\end{eqnarray*}\nwhere $F,G\\in C^{\\infty}(\\calM)$\nand $\\frac{\\overrightarrow{\\scriptstyle{\\partial}}}{\\partial \\phi}$ and \n$\\frac{\\overleftarrow{\\scriptstyle{\\partial}}}{\\partial \\phi}$\nare the right and left differentiations, respectively.\nNote that the degree of the symplectic structure is $+3$\nand the one of the Poisson bracket is $-3$.\nThe bidegree of the Poisson bracket is $(-2,-1)$, that is,\n$$\n\\{(2i,j),(2k,l)\\}=(2(i+k)-2,j+l-1),\n$$\nwhere $(2i,j)$... are functions with the bidgree $(2i,j)$.\n\n\\subsection{Q-structures}\nWe consider a (classical) Q-structure, $\\Theta$, on the P-manifold.\nIt is required that $\\Theta$ has degree $4$.\nThat is, $\\Theta\\in C^{4}(\\calM)$.\nBecause $C^{4}(\\calM)=C^{4,0}(\\calM)\\oplus C^{2,2}(\\calM)\\oplus C^{0,4}(\\calM)$,\nthe Q-structure is uniquely decomposed into\n$$\n\\Theta=\\theta_{2}+\\theta_{13}+\\theta_{4},\n$$\nwhere the bidegrees of the substructures are\n$(4,0)$, $(2,2)$ and $(0,4)$, respectively.\nIn the canonical coordinate, $\\Theta$ is the following polynomial:\n\\begin{eqnarray}\\label{deftheta}\n\\Theta = f{}_1{}^i{}_{a} (\\bx) \\bxi_i \\bq^a \n+\\frac{1}{2} f_2{}^{ab}(\\bx) \\bp_a \\bp_b\n+\\frac{1}{2} f_3{}^a{}_{bc}(\\bx) \\bp_a \\bq^b \\bq^c\n+\\frac{1}{4!} f_4{}_{abcd}(\\bx) \\bq^a \\bq^b \\bq^c \\bq^d,\n\\end{eqnarray}\nand the substructures are \n\\begin{eqnarray*}\n\\theta_{2}&=&\\frac{1}{2} f_2{}^{ab}(\\bx) \\bp_a \\bp_b,\\\\\n\\theta_{13}&=&\nf{}_1{}^i{}_{a} (\\bx) \\bxi_i \\bq^a+\\frac{1}{2} f_3{}^a{}_{bc}(\\bx) \\bp_a \\bq^b \\bq^c,\\\\\n\\theta_{4}&=&\\frac{1}{4!} f_4{}_{abcd}(\\bx) \\bq^a \\bq^b \\bq^c \\bq^d,\n\\end{eqnarray*}\nwhere $f_{1}$-$f_{4}$ are structure functions on $M$.\nBy  counting the bidegree, one can easily prove that\nthe classical master equation $\\sbv{\\Theta}{\\Theta} = 0$\nis equivalent to the following three identities:\n\\begin{eqnarray}\n\\label{tc1}\n\\{\\theta_{13},\\theta_{2}\\}&=&0,\\\\\n\\label{tc2}\n\\frac{1}{2}\\{\\theta_{13},\\theta_{13}\\}\n+\\{\\theta_{2},\\theta_{4}\\}&=&0,\\\\\n\\label{tc3}\n\\{\\theta_{13},\\theta_{4}\\}&=&0.\n\\end{eqnarray}\nThe conditions (\\ref{tc1}), (\\ref{tc2}) and (\\ref{tc3})\nare equivalent to\n\\begin{eqnarray}\n\\label{fc1}\n&&f{}_1{}^i{}_{b} f_2{}^{ba} = 0,\\\\\n\\label{fc2}\n&&\nf{}_1{}^k{}_{c} \\frac{\\partial f_2{}^{ab}}{\\partial x^k} \n+ f_2{}^{da} f_3{}^b{}_{cd} + f_2{}^{db} f_3{}^a{}_{cd} = 0,\\\\\n\\label{fc3}\n&&\nf{}_1{}^k{}_{b} \\frac{\\partial f{}_1{}^i{}_{a}}{\\partial x^k} \n- f{}_1{}^k{}_{a} \\frac{\\partial f{}_1{}^i{}_{b}}{\\partial x^k} \n+ f{}_1{}^i{}_{c} f_3{}^c{}_{ab} = 0,\\\\\n\\label{fc4}\n&& \nf{}_1{}^k{}_{[d} \\frac{\\partial f_3{}^a{}_{bc]}}{\\partial x^k} \n+ f_2{}^{ae} f_4{}_{bcde}\n- f_3{}^a{}_{e[b} f_3{}^e{}_{cd]} = 0,\\\\\n\\label{fc5}\n&& \nf{}_1{}^k{}_{[a} \\frac{\\partial f_4{}_{bcde]}}{\\partial x^k} \n+ f_3{}^f{}_{[ab} f_4{}_{cde]f} =0,\n\\end{eqnarray}\nwhere $[b \\ c \\ d \\ \\cdots]$ is a skewsymmetrization\nwith respect to indices $b, c, d, \\cdots$, etc.\n\\subsection{Lie algebroid up to homotopy}\nIn this section we study an algebraic structure\nassociated with the QP-structure in 3.1 and 3.2.\n\\begin{definition}\nLet $Q=\\theta_{2}+\\theta_{13}+\\theta_{4}$ be a $Q$-structure\non $T^{*}[3]E[1]$, where $(\\theta_{2},\\theta_{13},\\theta_{4})$\nis the unique decomposition of $\\Theta$.\nWe call the quadruple $(E; \\theta_{2},\\theta_{13},\\theta_{4})$\na \\textbf{Lie algebroid up to homotopy},\nin shorthand, Lie algebroid u.t.h.\n\\end{definition}\nWe should study the algebraic properties of the Lie algebroid up to homotopy.\nLet us define a bracket product by\n\\begin{equation}\\label{braee}\n[e_{1},e_{2}]:=\\{\\{\\theta_{13},e_{1}\\},e_{2}\\},\n\\end{equation}\nwhere $e_{1},e_{2}\\in\\Gamma E$.\nBy the bidegree counting, \n$\\Gamma E$ is closed under this bracket.\nThe bracket is not necessarily a Lie bracket,\nbut it is still skewsymmetric:\n\\begin{eqnarray*}\n[e_{1},e_{2}]&=&\\{\\{\\theta_{13},e_{1}\\},e_{2}\\},\\\\\n&=&\\{\\theta_{13},\\{e_{1},e_{2}\\}\\}+\\{e_{1},\\{\\theta_{13},e_{2}\\}\\},\\\\\n&=&-\\{\\{\\theta_{13},e_{2}\\},e_{1}\\}=-[e_{2},e_{1}],\n\\end{eqnarray*}\nwhere $\\{e_{1},e_{2}\\}=0$ is used.\nA bundle map $\\rho:E\\to TM$ which is called an anchor map\nis defined by the following identity:\n$$\n\\rho(e)(f):=\\{\\{\\theta_{13},e\\},f\\},\n$$\nwhere $f\\in C^{\\infty}(M)$.\nThe bracket and the anchor map \nsatisfy the algebroid conditions (A0) and (A1) below:\n\\begin{description}\n\\item[(A0)]\n$\\rho[e_{1},e_{2}]=[\\rho(e_{1}),\\rho(e_{2})]$,\n\\item[(A1)]\n$[e_{1},fe_{2}]=f[e_{1},e_{2}]+\\rho(e_{1})(f)e_{2}$,\n\\end{description}\nwhere the bracket $[\\rho(e_{1}),\\rho(e_{2})]$\nis the usual Lie bracket on $\\Gamma TM$.\nThe bracket (\\ref{braee}) does not satisfy the Jacobi identity in general.\nSo we should study its Jacobi anomaly, which characterizes\nthe algebraic structure of the Lie algebroid u.t.h.\nThe structures $\\theta_{13}$, $\\theta_{2}$ and $\\theta_{4}$ define\nthe three operations:\n\\begin{itemize}\n\\item $\\delta(-):=\\{\\theta_{13},-\\}$;\na de Rham type derivation on $\\Gamma\\wedge^{\\cdot}E^{*}$,\n\\item $(\\alpha_{1},\\alpha_{2}):=\\{\\{\\theta_{2},\\alpha_{1}\\},\\alpha_{2}\\}$;\na symmetric pairing on $E^{*}$, where $\\alpha_{1},\\alpha_{2}\\in\\Gamma E^{*}$,\n\\item $\\Omega(e_{1},e_{2},e_{3},e_{4}):=\n\\{\\{\\{\\{\\{\\theta_{4},e_{1}\\},e_{2}\\},e_{3}\\},e_{4}\\}$;\na 4-form on $E$.\n\\end{itemize}\nRemark that $\\delta\\delta\\neq 0$ in general.\nBecause the degree of the pairing is $-2$,\nit is $C^{\\infty}(M)$-valued.\nThe pairing induces a symmetric\nbundle map $\\partial:E^{*}\\to E$\nwhich is defined by the equation,\n$(\\alpha_{1},\\alpha_{2})=\\bracket{\\partial \\alpha_{1}}{\\alpha_{2}}$,\nwhere $\\bracket{-}{-}$ is the canonical pairing\nof the duality of $E$ and $E^{*}$.\nSince $\\bracket{\\alpha}{e}=\\{\\alpha,e\\}$, we have\n$$\n\\partial\\alpha=-\\{\\theta_{2},\\alpha\\}.\n$$\nBy direct computation, we obtain\n$$\n\\frac{1}{2}\n\\{\\{\\{\\{\\theta_{13},\\theta_{13}\\},e_{1}\\},e_{2}\\},e_{3}\\}\n=[[e_{1},e_{2}],e_{3}]+({\\rm cyclic \\ permutations}),\n$$\nand\n$$\n\\{\\{\\{\\{\\theta_{2},\\theta_{4}\\},e_{1}\\},e_{2}\\},e_{3}\\}\n=-\\partial\\Omega(e_{1},e_{2},e_{3}).\n$$\nFrom Eq.~(\\ref{tc2}), we get an explicit formula of the Jacobi anomaly,\n\\begin{description}\n\\item[(A2)]\n$[[e_{1},e_{2}],e_{3}]+({\\rm cyclic \\ permutations})\n= \\partial\\Omega(e_{1},e_{2},e_{3})$.\n\\end{description}\nIn a similar way, we obtain the following identities:\n\\begin{description}\n\\item[(A3)] $\\rho\\partial=0$,\n\\item[(A4)] $\\rho(e)(\\alpha_{1},\\alpha_{2})=(\\mathcal{L}_{e}\\alpha_{1},\\alpha_{2})\n+(\\alpha_{1},\\mathcal{L}_{e}\\alpha_{2})$,\n\\item[(A5)] $\\delta\\Omega=0$,\n\\end{description}\nwhere $\\mathcal{L}_{e}(-):=\\{\\{\\theta_{13},e\\},-\\}$\nis the Lie type derivation which acts on $E^{*}$.\nAxioms (A3) and (A4) are induced from Eq.~(\\ref{tc1})\nand (A5) is from Eq.~(\\ref{tc3}).\\\\\n\\indent\nThe fundamental relations (\\ref{fc1})--(\\ref{fc5})\ncorrespond to Axioms (A1)--(A5)\\footnote{\nActually, the axiom (A0) depends on (A1) and (A2).\n}.\nThus, the notion of the Lie algebroid up to homotopy\nis characterized by the algebraic properties (A1)--(A5).\nOne concludes that\n\\medskip\\\\\n\\noindent\n{\\em\nThe classical algebra associated with the QP-manifold $(T^{*}[3]E[1],\\Theta)$\nis the space of sections of the vector bundle $E$ with the operations\n$([\\cdot,\\cdot],\\rho,\\partial,\\Omega)$\nsatisfying (A1)--(A5).\n}\n\\medskip\\\\\n\\indent\nIn the next section, we will study some special examples\nof Lie algebroid u.t.h.s.\n\\begin{remark}\n\\normalfont\nIf the pairing is nondegenerate,\nthen the bundle map $\\partial$ is bijective\nand then from (A3) we have $\\rho=0$.\n\\end{remark}\n\\begin{remark}\\label{CDB}\n\\normalfont\n(Higher Courant-Dorfman brackets)\nWe define a bracket on $C^{\\infty}(\\calM)$ by\n$$\n[-,-]_{CD}:=\\{\\{\\Theta,-\\},-\\},\n$$\nwhich is called a Courant-Dorfman (CD) bracket.\nIt is well-known that $[,]_{CD}$ is a Loday bracket (\\cite{Kos}).\nSince the degree of the CD-bracket is $-2$,\nthe total space of degree $i\\le 2$,\n$$\nC^{2}(\\calM)\\oplus C^{1}(\\calM)\\oplus C^{0}(M)\n$$\nis closed under the CD-bracket, in particular,\nthe top space $C^{2}(\\calM)=\\Gamma(E\\oplus\\wedge^{2}E^{*})$\nis a subalgebra.\nIf $\\theta_{2}=0$, the CD-bracket on $E\\oplus\\wedge^{2}E^{*}$\nhas the following form,\n$$\n[e_{1}+\\beta_{1},e_{2}+\\beta_{2}]_{CD}=\n[e_{1},e_{2}]+\\mathcal{L}_{e_{1}}\\beta_{2}-i_{e_{2}}\\delta\\beta_{1}+\\Omega(e_{1},e_{2}),\n$$\nwhere $\\beta_{1},\\beta_{2}\\in\\Gamma\\wedge^{2}E^{*}$.\nThis CD-bracket is regarded as a higher analogue of\nCourant-Dofman's original bracket (cf. \\cite{Courant}).\nWe refer the reader to \nHagiwara \\cite{Hagiwara} and Sheng \\cite{YS}\nfor the detailed study of the higher CD-brackets.\n\\end{remark}\n\n\\section{Examples and twisting transformations}\n\n\\subsection{The cases of $\\theta_{2}=\\theta_{4}=0$}\n\nIn this case, the bracket (\\ref{braee}) satisfies\n(A0), (A1) and the Jacobi identity.\nTherefore, the bundle $E\\to M$ becomes a Lie algebroid:\n\\begin{definition}\n(\\cite{Mackenzie})\nA Lie algebroid over a manifold $M$ is a vector bundle\n$E \\rightarrow M$ with a Lie algebra structure on the \nspace of the sections $\\Gamma(E)$ defined by the \nbracket $[e_1, e_2]$ for $e_1, e_2 \\in \\Gamma(E)$\nand an anchor map\n$\\rho: E \\rightarrow TM$ satisfying (A0) and (A1) above.\n\\end{definition}\n\\indent\nWe take $\\{ e_a \\}$ as a local basis of $\\Gamma E$ and\nlet a local expression of an anchor map\nbe $\\rho(e_a) = f^i{}_{1a}(x) \\frac{\\partial}{\\partial x^i}$\nand a Lie bracket be\n$[e_b, e_c] = f_3{}^a{}_{bc}(x) e_a$.\nThe Q-structure $\\Theta$ associated with\nthe Lie algebroid $E$\nis defined as a function on $T^{*}[3]E[1]$,\n$$\n\\Theta:=\\theta_{13}:=f{}_1{}^i{}_{a} (\\bx) \\bxi_i \\bq^a\n+\\frac{1}{2} f_3{}^a{}_{bc}(\\bx) \\bp_a \\bq^b \\bq^c,\n$$\nwhich is globally well-defined.\nConversely, if we consider $\\Theta:=\\theta_{13}$,\nthe classical master equation\ninduces the Lie algebroid structure on $E$.\n\\medskip\\\\\n\\indent\nLet us consider the case that the bundle is \na vector space on a point.\nA Lie algebroid over\na point $\\mathfrak{g}\\to\\{pt\\}$ is\na Lie algebra $\\mathfrak{g}$.\nThe P-manifold over $\\mathfrak{g}\\to\\{pt\\}$\nis isomorphic to $\\mathfrak{g}^*[2]\\oplus\\mathfrak{g}[1]$\nand the structure sheaf is\nthe polynomial algebra over $\\mathfrak{g}[2]\\oplus\\mathfrak{g}^{*}[1]$,\n$$\nC^{\\infty}(\\calM)=S(\\mathfrak{g})\\otimes \\bigwedge^{\\cdot}\\mathfrak{g}^{*}.\n$$\nThe bidegree is defined by the natural manner,\n$$\nC^{2i,j}(\\calM)=S^{i}(\\mathfrak{g})\\otimes \\bigwedge^{j}\\mathfrak{g}^{*}.\n$$\nThe Q-structure associated with the Lie bracket on $\\mathfrak{g}$ is\n\\begin{eqnarray}\n\\theta_{13}=\\frac{1}{2} f^a{}_{bc} p_a q^b q^c\n\\cong\\frac{1}{2} f^a{}_{bc} p_a \\otimes(q^b \\wedge q^c),\n\\label{liealgebra}\n\\end{eqnarray}\nwhere $p_{\\cdot}\\in\\mathfrak{g}$, $q_{\\cdot}\\in\\mathfrak{g}^{*}$\nand $f^{a}{}_{bc}$ is the structure constant of the Lie algebra.\n\n\\subsection{The cases of $\\theta_{2}\\neq 0$ and $\\theta_{4}=0$}\n\nIn this case, the bracket induced by $\\theta_{13}$\nstill satisfies the Jacobi identity.\n\\medskip\\\\\n\\indent\nWe assume that $\\frak{g}$ is semi-simple.\nThen the dual space $\\frak{g}^{*}$ has a metric,\n$(\\cdot,\\cdot)_{K^{-1}}$, which is\nthe inverse of the Killing form on $\\frak{g}$.\nThe metric inherits the following invariant condition from\nthe Killing form:\n\\begin{equation}\\label{la3}\n(\\mathcal{L}_{p}q_{1},q_{2})_{K^{-1}}\n+(q_{1},\\mathcal{L}_{p}q_{2})_{K^{-1}}=0,\n\\end{equation}\nwhere $\\mathcal{L}_{p}(-)$ is the canonical\ncoadjoint action of $\\mathfrak{g}$ to $\\mathfrak{g}^{*}$.\nEq.~(\\ref{la3}) is a linear version of (A4).\nThus, we obtain a Q-structure,\n\\begin{eqnarray}\n\\Theta:=\nk^{ab}\\bp_a \\bp_b\n+\\frac{1}{2} f{}^a{}_{bc} \\bp_a \\bq^b \\bq^c,\n\\label{killingQ}\n\\end{eqnarray}\nwhere $k^{ab}\\bp_a \\bp_b:=(\\cdot,\\cdot)_{K^{-1}}$.\n\n\\subsection{Non Lie algebra example}\nWe consider the cases that the Jacobi identity is broken.\nLet $(\\mathfrak{g},[\\cdot,\\cdot],(\\cdot,\\cdot)_{K})$\nbe a vector space (not necessarily Lie algebra)\nequipped with a skewsymmetric bracket $[\\cdot,\\cdot]$\nand an invariant metric $(\\cdot,\\cdot)_{K}$.\nThe metric induces a bijection\n$K:\\mathfrak{g}\\to\\mathfrak{g}^{*}$\nwhich is defined by the identity,\n$$\n(p_{1},p_{2})_{K}=\\bracket{Kp_{1}}{p_{2}}.\n$$\nWe define a map from\n$\\mathfrak{g}^{*}$ to $\\mathfrak{g}$ by $\\partial:=K^{-1}$\nand define a 4-form by,\n$$\n\\Omega(p_{1},p_{2},p_{3},p_{4}):=\\Big([[p_{1},p_{2}],p_{3}]+\n{\\rm cyclic \\ permutations},p_{4}\\Big)_{K}.\n$$\n\\begin{remark}\nThe 4-form above is considered to be a higher analogue\nof the Cartan 3-form $([p_{1},p_{2}],p_{3})_{K}$.\n\\end{remark}\nAxioms (A0)--(A4) obviously hold on $\\frak{g}$.\nWe check (A5). It suffices to show (\\ref{tc3}).\nLet us denote by $\\{-,p_{1},p_{2},...,p_{n}\\}$\nthe n-fold bracket $\\{...\\{\\{-,p_{1}\\},p_{2}\\},...,p_{n}\\}$.\nWe already have (\\ref{tc1}) and (\\ref{tc2}).\nFrom $\\{\\theta_{13},\\{\\theta_{13},\\theta_{13}\\}\\}=0$\nand (\\ref{tc2}), we have $\\{\\theta_{13},\\{\\theta_{2},\\theta_{4}\\}\\}=0$.\nSince $\\{\\theta_{13},\\theta_{2}\\}=0$,\nthis is equal to\n$\\{\\theta_{2},\\{\\theta_{3},\\theta_{4}\\}\\}=0$\nup to sign.\nThis gives\n$\\{\\{\\theta_{2},\\{\\theta_{3},\\theta_{4}\\}\\},p_{1},...,p_{5}\\}=0$\nfor any $p_{1},...,p_{5}$.\nFrom $\\{\\theta_{2},p\\}=0$, we have\n$$\n\\{\\theta_{2},\\{\\{\\theta_{3},\\theta_{4}\\},p_{1},...,p_{5}\\}\\}=0.\n$$\nSince $K^{-1}=-\\{\\theta_{2},-\\}$ is bijective, we get\n$$\n\\{\\{\\theta_{3},\\theta_{4}\\},p_{1},...,p_{5}\\}=0,\n$$\nwhich yields the desired relation $\\{\\theta_{3},\\theta_{4}\\}=0$.\n\\begin{proposition}\nThe triple $\\big(\\mathfrak{g},\\partial,\\Omega\\big)$\nis a Lie algebra(oid) up to homotopy.\n\\end{proposition}\n\\subsection{Twisting by $3$-form and the cases of $\\theta_{2}=0$ and $\\theta_{4}\\neq 0$}\nWe introduce the notion of twisting transformation by $3$-form\nbefore studying the cases of $\\theta_{2}=0$.\nGiven a Q-structure $\\Theta$ and a 3-form $\\phi\\in C^{0,3}(\\calM)$,\nthere exists the second Q-structure which is defined by\nthe canonical transformation,\n\\begin{equation}\\label{defgauge}\n\\Theta^{\\phi}:=\\exp(X_{\\phi})(\\Theta),\n\\end{equation}\nwhere $X_{\\phi}:=\\{\\phi,-\\}$ is the Hamiltonian vector field of $\\phi$.\nThe transformation (\\ref{defgauge}) is called a \\textbf{twisting by 3-form},\nor simply twisting.\nBy a direct computation, we obtain\n\\begin{eqnarray*}\n\\label{g1}\\theta^{\\phi}_{2}&=&\\theta_{2},\\\\\n\\label{g2}\\theta^{\\phi}_{13}&=&\\theta_{13}-\\{\\theta_{2},\\phi\\},\\\\\n\\label{g3}\\theta^{\\phi}_{4}&=&\\theta_{4}-\\{\\theta_{13},\\phi\\}+\\frac{1}{2}\\{\\{\\theta_{2},\\phi\\},\\phi\\},\n\\end{eqnarray*}\nwhere $\\Theta^{\\phi}=\\theta^{\\phi}_{2}+\\theta^{\\phi}_{13}+\\theta^{\\phi}_{4}$\nand $X^{i\\ge 3}_{\\phi}(\\Theta)=0$.\nThe twisting by 3-form defines an equivalence relation on the Q-structures.\n\\medskip\\\\\n\\indent\nWe notice that $\\theta_{2}$ is an invariant for the twisting.\nIf $\\theta_{2}=0$, then\n$\\theta_{13}$ is an invariant and\n$$\n\\theta^{\\phi}_{4}=\\theta_{4}-\\delta\\phi,\n$$\nwhere $\\delta\\phi=\\{\\theta_{13},\\phi\\}$.\nThis leads us\n\\begin{proposition}\nThe class of Q-structures which have no $\\theta_{2}$\nis classified into $H^{4}_{dR}(\\bigwedge^{\\cdot}E^{*},\\delta)$\nby the twisting by 3-form.\n\\end{proposition}\n\n\n\\section{AKSZ Construction of Topological Field Theory \nin $4$ Dimensions\n}\n\\subsection{General Theory}\n\n\\noindent\nIn this section, we consider the AKSZ construction of \na topological field theory in $4$ dimensions.\n\nFor a graded manifold $\\calN$,  \nlet $\\calN|_0$ be the degree zero part.\n\nLet $X$ be a manifold in $4$ dimensions\nand $M$ be a manifold in $d$ dimensions.\nLet $(\\calX, D)$ be a differential graded (dg) manifold \n$\\calX$\nwith a $D$-invariant nondegenerate measure $\\mu$, \nsuch that $\\calX|_0 = X$, where\n$D$ is a differential on $\\calX$.\n($\\calM, \\omega, \\Theta$)\nis a QP-manifold of degree $3$ and\n$\\calM|_0 = M$.\nA degree $\\deg (-)$ on $\\calX$ is called the {\\it form degree} and \na degree $\\gh (-)$ on $\\calM$ is called the {\\it ghost number}\n\\footnote{The ghost number $\\gh (-)$ is the degree $|-|$\non $\\calM$ in section 2.}.\nLet \n$\\Map(\\calX, \\calM)$ be\na space of smooth maps from $\\calX$ to $\\calM$.\n$|-| = \\deg (-) + \\gh (-)$ is the degree\non $\\Map(\\calX, \\calM)$ and called the {\\it total degree}.\nA QP-structure on $\\Map(\\calX, \\calM)$\nis constructed from the above data.\n\nSince ${\\rm Diff}(\\calX)\\times {\\rm Diff}(\\calM)$ \nnaturally acts on $\\Map(\\calX, \\calM)$,\n$D$ and $Q$ induce homological vector fields \non $\\Map(\\calX, \\calM)$, \n$\\hat{D}$ and $\\check{Q}$. \n\nTwo maps are introduced.\nAn {\\it evaluation map} \n${\\rm ev}: \\calX \\times \\calM^{\\calX} \\longrightarrow \\calM$ \nis defined as\n\\begin{eqnarray*}\n{\\rm ev}:(z, \\Phi) \\longmapsto \\Phi(z),\n\\end{eqnarray*}\nwhere \n$z \\in \\calX$ and $\\Phi \\in \\calM^{\\calX}$.\n\n\n\nA {\\it chain map} $\\mu_*: \\Omega^{\\bullet}(\\calX \\times \\calM) \n\\longrightarrow \\Omega^{\\bullet}(\\calM)$ is defined as \n$\\mu_* F = \\int_{\\calX} \\mu F$\nwhere $F \\in \\Omega^{\\bullet}(\\calX \\times \\calM)$ \nand \n$\\int_{\\calX} \\mu$  is an integration on $\\calX$\nby the $D$-invariant measure $\\mu$.\nIt is an usual integral for the even degree parts\nand\nthe Berezin integral for the \nodd degree parts. \n\nA (classical) P-structure on $\\Map(\\calX, \\calM)$ is defined as follows:\n\\begin{definition} \nFor a graded symplectic form $\\omega$ on $\\calM$, \na graded symplectic form $\\bomega$ on $\\Map(\\calX, \\calM)$\nis defined as $\\bomega := \\mu_* \\ev^* \\omega$.\n\\end{definition}\nWe can confirm that ${\\bomega}$ satisfies the definition of \na graded symplectic form because $\\mu_* \\ev^*$ preserves\nnondegeneracy and closedness.\nThus $\\bomega$ is a P-structure on $\\Map(\\calX, \\calM)$\nand induces a graded Poisson bracket $\\sbv{-}{-}$ on $\\Map(\\calX, \\calM)$.\nSince $|\\mu_* \\ev^*|=-4$, $|\\bomega| = -1$ and \n$\\sbv{-}{-}$ on $\\Map(\\calX, \\calM)$ has degree $1$ and an odd\nPoisson bracket.\n\n\n\nNext we define a Q-structure $S$ on $\\Map(\\calX,  \\calM)$.\n$S$ is called a {\\it BV action} and\nconsists of two parts $S = S_0 + S_1$.\n$S_0$ is constructed as follows: \nLet $\\omega$ be the odd symplectic form \non $\\calM$.\nWe take a fundamental form $\\vartheta$ such that \n$\\omega= - d \\vartheta$ and\ndefine $S_0 := \\iota_{\\hat{D}} \\mu_* {\\rm ev}^* \\vartheta$.\n$|S_0|=0$ because $\\mu_* {\\rm ev}^*$ has degree $-4$.\n$S_1$ is constructed as follows: \nWe take a Q-structure $\\Theta$ on $\\calM$ and \ndefine $S_1 := \\mu_* \\ev^* \\Theta$.\n$S_1$ also has degree $0$.\n\nWe can prove that\n$S$ is a Q-structure on $\\Map(\\calX,  \\calM)$, \nsince \n\\begin{eqnarray}\n\\sbv{\\Theta}{\\Theta} =0,\n\\Longleftrightarrow \n\\sbv{S}{S} =0\n\\label{classicalmaster}\n\\end{eqnarray}\nfrom the definition of $S_0$ and $S_1$.\n\nA quantum version is \n\\begin{eqnarray}\n\\Delta (e^{\\frac{i}{\\hbar} \\Theta}) =0\n\\Longleftrightarrow \n\\hat{\\Delta} (e^{\\frac{i}{\\hbar} S}) =0,\n\\label{classicalmaster2}\n\\end{eqnarray}\nwhere $\\hat{\\Delta}$ is an odd Laplace operator on $\\Map(\\calX, \\calM)$.\nThe infinitesimal form of the right hand side \nin (\\ref{classicalmaster2}) is \n$\\sbv{S}{S} - 2 i \\hbar \\hat{\\Delta} S =0$,\nwhich is called a {\\it quantum master equation}.\n\\footnote{Discussion for an odd Laplace operator is too naive. \nIn general, the quantum master equation has an obstruction \nexpressed by the modular class \\cite{Lyakhovich:2004kr}.\nWe must regularize an odd Laplace operator\nand a quantum BV action.\n}\n\nThe following theorem has been confirmed\n\\cite{Alexandrov:1995kv}:\n\\begin{thm}\nIf $\\calX$ is a dg manifold and \n$\\calM$ is a QP-manifold,\nthe graded manifold $\\Map(\\calX, \\calM)$\nhas a QP-structure.\n\\end{thm}\n\n\n\\begin{definition}\nA {\\it topological field theory} in $4$ dimensions\nis a triple ($\\calX, \\calM, S$), \nwhere $\\calX$ is a dg manifold with\n$\\dim \\calX|_0 = 4$, \n$\\calM$ is \na QP-manifold\nwith the degree $3$,\nand $S$ is a BV action with the total degree $0$.\n\\end{definition}\n\n\n\nIn order to interpret this theory\nas a `physical' topological field theory,\nwe must take $\\calX= \nT[1]X$.\nThen we can confirm that \na QP-structure on $\\Map(\\calX, \\calM)$ is \nequivalent to the AKSZ formulation of a topological field theory\n\\cite{Cattaneo:2001ys}\\cite{Ikeda:2006wd}.\nWe set $\\calX = T[1]X$ from now.\n\n\n\nIn `physics', a quantum field theory is constructed by quantizing a\nclassical field theory.\nFirst we consider a Q-structure \n$\\sbv{\\cdot}{\\cdot}$ and \na classical P-structure $S$ such that\n\\begin{eqnarray*}\n\\sbv{S}{S} =0.\n\\end{eqnarray*}\nNext we define \na quantum P-structure $\\hat{\\Delta}$ \nand confirm that\n\\begin{eqnarray*}\n\\tilde{\\Delta} (e^{\\frac{i}{\\hbar} S}) =0.\n\\end{eqnarray*}\nFinally we calculate a partition function \n\\begin{eqnarray*}\nZ = \\int_{\\calL} e^{\\frac{i}{\\hbar} S},\n\\end{eqnarray*}\non a Lagrangian submanifold \n$\\calL \\subset \\Map(\\calX, {\\calM})$.\nQuantization is not discussed in this paper.\n\n\n\n\\subsection{Local Coordinate Expression and Examples}\n\\noindent\nA general theory in the previous subsection\nis applied to the local coordinate expression \nin section 3.1 and \na known topological field theory in $4$ dimensions\nis obtained as a special case and \na new nontrivial topological field theory is constructed.\nLet us take a manifold $X$ in $4$ dimensions and\na manifold $M$ in $d$ dimensions.\nLet $E[1]$ is a graded vector bundle on $M$.\nWe take $\\calX = T[1]X$ and $\\calM = T^*[3]E[1]$.\n\nLet $(\\sigma^{\\mu}, \\theta^{\\mu})$ be a local coordinate\non $T[1]X$. $\\sigma^{\\mu}$ is a local coordinate on \nthe base manifold $X$ and \n$\\theta^{\\mu}$ is one on the fiber of $T[1]X$, respectively.\nLet $\\bbx^i$ be a smooth map $\\bbx^i: X \\longrightarrow M$ and \n$\\bbxi_i$ be a section of $T^*[1]X \\otimes \\bbx^*(T^*[3] M)$, \n$\\bbq^a$ be a section of $T^*[1]X \\otimes \\bbx^*(E[1])$ and\n$\\bbp_a$ be a section of $T^*[1]X \\otimes \\bbx^*(T^*[3]E_{\\bbx}[1])$.\nThese are called {\\it superfields}.\nThe exterior derivative $d$ is taken \nas a differential $D$ on $X$.\nFrom $d$, a differential \n$\\bbd = \\theta^{\\mu} \\frac{\\partial}{\\partial \\sigma^{\\mu}}$\non $\\calX$ is induced.\n\nThen a BV action $S$ has the following expression:\n\\begin{eqnarray*}\nS&=&S_0 + S_1,\\\\\nS_0&=&\\int_{\\calX} \\mu \\ (\\bbxi_i \\bbd \\bbx^i\n- \\bbp_a \\bbd \\bbq^a),\\\\\nS_1&=&\\int_{\\calX} \\mu \\ (f{}_1{}^i{}_{a} (\\bbx) \\bbxi_i \\bbq^a \n+ \\frac{1}{2} f_2{}^{ab}(\\bbx) \\bbp_a \\bbp_b\n+ \\frac{1}{2} f_3{}^a{}_{bc}(\\bbx) \\bbp_a \\bbq^b \\bbq^c\n+ \\frac{1}{4!} f_4{}_{abcd}(\\bbx) \\bbq^a \\bbq^b \\bbq^c \\bbq^d).\n\\end{eqnarray*}\n\\medskip\\\\\n\\noindent\n\\textbf{Nonabelian BF theory}.\nLet $\\Theta$ be a Q-structure (\\ref{liealgebra})\nfor a Lie algebra $\\mathfrak{g}$.\n$\\bbxi_i \\bbd \\bbx^i=0$ since $M = \\{pt\\}$.\nIf we define a curvature\n$\\bbF^a = \\bbd \\bbq^a - \\frac{1}{2} f{}^a{}_{bc} \\bbq^b \\bbq^c$,\na Q-structure is\n\\begin{eqnarray*}\n&& S= \\int_{\\calX} \\mu \\ (- \\bbp_a \\bbF^a),\n\\end{eqnarray*}\nwhich is equivalent to a BV formalism for\na nonabelian BF theory in $4$ dimensions.\n\\medskip\\\\\n\\noindent\n\\textbf{Topological Yang-Mills Theory}.\nWe take a nondegenerate Killing form $(\\cdot,\\cdot)_{K}$\nfor a Lie algebra $\\mathfrak{g}$ and consider the Q-structure \n(\\ref{killingQ}).\nA topological field theory constructed from (\\ref{killingQ})\nis \n\\begin{eqnarray*}\n&& S= \\int_{\\calX} \\mu \\ (- \\bbp_a \\bbF^a\n+ k^{ab} \\bbp_a \\bbp_b).\n\\end{eqnarray*}\nThis is equivalent to a topological Yang-Mills theory,\n\\begin{eqnarray*}\n&& S= - \\frac{1}{4} \\int_{\\calX} \\mu \\ k_{ab} \\bbF^a \\bbF^b,\n\\end{eqnarray*}\nif we delete $\\bbp_a$ by the equations of motion.\n\\medskip\\\\\n\\noindent\n\\textbf{Nonassociative  BF Theory}.\nLet us take a non Lie algebra $(\\mathfrak{g},[\\cdot,\\cdot],(\\cdot,\\cdot)_{K})$\nin section 4.3.\nIf we take\n$M = \\{pt\\}$ and $\\calM = \\mathfrak{g^*}[2] \\oplus \\mathfrak{g}[1]$, \n$(\\mathfrak{g},[\\cdot,\\cdot],(\\cdot,\\cdot)_{K})$ leads\na QP-structure with degree $3$.\nIn the canonical basis, it is expressed as\n\\begin{eqnarray*}\n&& f{}_1{}^i{}_{a} (\\bx) = 0, \\qquad \nf_2{}^{ab}(\\bx) = K^{ab}, \\\\\n&& f_3{}^a{}_{bc}(\\bx) = f{}^a{}_{bc}, \\qquad \nf_4{}_{abcd}(\\bx) = K^{-1}_{ae} f{}^e{}_{f[b} f{}^f{}_{cd]},\n\\end{eqnarray*}\nwhere \n$K^{ab} = (p_a, p_b)$ is nondegenerate and \n$[p_a, p_b] = f{}^c{}_{ab} p_c$\nis a nonassociative bracket \nand does not satisfy the Jacobi identity.\nThe AKSZ construction derives a new nontrivial topological field theory\nin $4$ dimensions.\nA BV action $S$ has the following expression:\n\\begin{eqnarray*}\nS&=&\n\\int_{\\calX} \\mu \\ (\n- \\bbp_a \\bbd \\bbq^a\n+ \\frac{1}{2} K{}^{ab} \\bbp_a \\bbp_b\n+ \\frac{1}{2} f{}^a{}_{bc} \\bbp_a \\bbq^b \\bbq^c\n+ \\frac{1}{4!} \nK^{-1}_{ae} f{}^e{}_{f[b} f{}^f{}_{cd]}\n\\bbq^a \\bbq^b \\bbq^c \\bbq^d)\\\\\n&=& - \\frac{1}{4} \\int_{\\calX} \\mu \\ (K_{ab} \\bbF^a \\bbF^b+\n\\frac{1}{3!} K^{-1}_{ae} f{}^e{}_{f[b} f{}^f{}_{cd]}\n\\bbq^a \\bbq^b \\bbq^c \\bbq^d).\n\\end{eqnarray*}\nIt is easily confirmed that $\\sbv{S}{S}=0$. \n\\medskip\\\\\n\\noindent\n\\textbf{Topological $3$-brane on $Spin(7)$-structure}.\nLet $(M, \\Omega)$ be an $8$-dimensional $Spin(7)$-manifold.\nHere $\\Omega$ is a $Spin(7)$ $4$-form, \nwhich satisfies $d \\Omega=0$ and \nthe selfdual condition $\\Omega=*\\Omega$.\nA $Spin(7)$ structure is defined as the subgroup of $GL(8)$ to \npreserve $\\Omega$.\nThe Q-structure on $(TM,\\Omega)$ is given by\n\\begin{eqnarray}\n\\Theta = \\bxi_i \\bq^i \n+ \\frac{1}{4!} \\Omega{}_{ijkl}(\\bx) \\bq^i \\bq^j \\bq^k \\bq^l.\n\\label{multisymQ}\n\\end{eqnarray}\nThe BV action $S$ for (\\ref{multisymQ}) defines the \nsame theory as \nthe topological $3$-brane analyzed in \\cite{Bonelli:2005ti}.\n\n\n\n\n\n\n\\section{Conclusions and Discussion}\n\\noindent\nWe have defined a BV algebra and a QP-structure \nof degree $3$.\nA QP-structure of degree $3$ has been constructed explicitly \nand a Lie algebroid u.t.h.~has been defined as \nits algebraic and geometric structure.\nA general theory of the AKSZ construction of a \ntopological field theory \nhas been expressed and\na new topological field theory in four \ndimensions has been constructed\nfrom a QP-structure.\n\nQuantization of this theory and\nanalysis of a Lie algebroid u.t.h.~will \nshed light on a super Poisson geometry and a quantum field theory.\nThey are future problems. \n\n\n\n\n\n\n\n\n\\section*{Acknowledgements}\n\nThe authors would like to thank Klaus Bering, Maxim Grigoriev, \nCamille Laurent-Gengoux, Yvette Kosmann-Schwarzbach, Kirill Mackenzie, \nDmitry Roytenberg, Alexei Sharapov, Thomas Strobl and Theodore Voronov\nfor their comments and discussion.\nThe author (N.I.)  would like to thank Maskawa Institute for Science and \nCulture, Kyoto Sangyo University for hospitality.\nWe would like to thank to referees for their useful advice.\n\n\\newcommand{\\bibit}{\\sl}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:introduction}\n\nSome million-scale datasets such as movie scripts and social media posts have become available in recent years for building neural dialogue agents~\\cite{lison2016lrec:opensubtitles,henderson2019convai:convdata}. \nSuch large-scale datasets can be expected to improve the performance of dialogue response generation models based on deep neural networks (DNNs) since the combination of DNNs and large-scale training datasets has led to considerable performance improvement in many sentence generation tasks~\\citet{koehn2017nmt:sixchallenges,sennrich2019acl:nmtcasestudy,adiwardana2020arxiv:meena}.\n\n\n\\input{fig-en-preliminary-experiment}\n\n\nIn contrast to the quantity of the data, the quality of the data has often been problematic.\nFor example, OpenSubtitles~\\cite{lison2016lrec:opensubtitles,lison2018lrec:opensubtitles}, the most widely used large-scale English dialogue corpus, was constructed by collecting two consecutive lines of movie subtitles under the simplified assumption that one line of a movie subtitle is one utterance and the next line is the next utterance follow it.\nInevitably, this corpus includes unacceptable utterance pairs from the viewpoint of a conversational sequence, e.g., caused by scene switching or flashback.\nSeveral previous studies have identified such flaws and reported that the corpus is \\emph{noisy}~\\cite{vinyals2015icml:neuralconv,li2016naacl:diversity,baheti2018emnlp:generating-interesting-response}, where \\emph{noisy} refers to unacceptable utterance pairs in this context.\nFigure~\\ref{fig:preliminary experiment} shows the result of our experimental investigation regarding the acceptability rate of the utterance pairs in the OpenSubtitles corpus.%\n    \\footnote{See Appendix~\\ref{a:preliminary experiment} for detailed experimental settings.}\nIt can be noticed from the figure that only half of the utterance pairs can be considered \\textit{acceptable} (i.e., were rated with score $5$: Strongly agree or $4$: Agree), and over 25\\% of utterance pairs are clearly \\textit{unacceptable} (i.e., were rated with score $1$: Strongly disagree or $2$: Disagree) from the human perspective.%\n    \\footnote{See Table~\\ref{tab:preliminary scored pairs} for samples of unacceptable\/acceptable utterance pairs annotated by humans.}\n\nWith this situation, a straightforward research question arises, namely, \\emph{Can we further improve the performance of neural response generation models by ablating unacceptable utterance pairs from training data?}\nTo the best of our knowledge, no previous study has explicitly focused on this question.\nThus, the goal of this paper is to provide an answer to this question.\nFurthermore, it is not clear whether and how one can effectively discover unacceptable utterance pairs within large-scale training datasets.\nThis study explores a way of constructing a scoring method for filtering \\emph{noisy} data filtering to improve the performance of response generation models.\n\n\nTo achieve the set goals, we started with a review of previous arguments about the criteria for identifying appropriate utterances in dialogues and designed our scoring function that is consistent with reflects as much of the community's consensus as possible.\nIn particular, the proposed scoring method estimates the quality of utterance pairs based on the following two aspects: \n(i) the \\textbf{connectivity} between source and target utterances and \n(ii) their \\textbf{content relatedness} (Section~\\ref{sec:idea}).\n\n\nThe contributions of this study are the following:\n\\begin{itemize}\n    \\item We propose a scoring method for estimating the quality of utterance pairs in an unsupervised manner (Section~\\ref{sec:proposed method});\n    %\n    \\item We reveal that our scoring method effectively detects unacceptable utterance pairs, and thus, be appropriate for noisy data filtering (Section~\\ref{sec:experiments});\n    %\n    \\item We empirically prove that our proposed data filtering method improves the performance of neural response generation models (Section~\\ref{sec:case study}); and\n        %\n    \\item We confirm that our noisy data filtering approach is effective across different languages and dataset sizes (Section~\\ref{sec:Japanese}).\n\\end{itemize}\n\n\n\n\n\\section{Task Definition: Noisy Data Filtering}\n\\label{sec:task}\n\nLet $x$ be an \\textbf{utterance} and $y$ be a \\textbf{response} to $x$.\nThen, an \\textbf{utterance pair} can be denoted as we refer to $(x,y)$.\nLet $\\mathcal{D}$ be a dataset that comprising a set of {utterance pairs}, $\\mathcal D = \\{(x,y)\\}$.\nThen, the task can be formulated as ablating unacceptable utterance pairs from $\\mathcal D$ to obtain a less noisy subset $\\mathcal{D}'\\subseteq \\mathcal{D}$, hereinafter referred to as filtered dataset.\n$\\mathcal{D}'$ can then be used to train response generation models. \nThis paper refers to this process as \\textbf{noisy data filtering}, where \\emph{noisy} means unacceptable utterance pairs in this context. \nFurthermore, we establish a function $S\\colon \\mathcal D\\to \\mathbb R$ is used to score the degree of \\textit{acceptability} of each utterance pair $(x,y) \\in \\mathcal D$.\n\n\n\n\n\n\\section{Background}\n\\label{sec:background}\n\n\\paragraph{Response generation using noisy data.}\nThe following two approaches are widely used to address the problem of dialogue response generation noisy dialogue corpora. \nAccording to the \\emph{model approach}, models are trained while handling noise at the same time. \nFor example, \\citet{shang2018ijcai:calibration} proposed a method with a calibration framework and demonstrated its effectiveness on a Chinese corpus.\nAccording to the \\emph{data approach}, training data are pre-processed with the aim of improving their quality before training models.\nIn this study, we take the data approach in light of the success of noisy parallel corpus filtering in machine translation (MT). \nAdditionally, it has become a reasonable strategy to reduce the size of training data since enormous dialogue data has been available.\n\\citet{csaky2019acl:filtering}'s method is most relevant to our study in that it cleanses dialogue corpora.\nHowever, the main goal of their method is to eliminate generic, or boring, responses, whereas the goal of the method proposed here is to eliminate unacceptable utterance pairs. \nThis difference in goals leads to the essential difference in filtering strategies.\n\n\n\n\n\\paragraph{Effectiveness of filtering noisy data in neural machine translation.}\n\nResearchers in the field of neural machine translation (NMT) have recognized that collecting high-quality training data to be equally or even more important than exploring sophisticated model architectures~\\cite{koehn2018wmt:filteringfindings,junczys-dowmunt2018wmt:filtering, morishita2018wmt:ntt}.\nTechniques used in neural response generation and NMT are nearly identical; e.g., sequence-to-sequence models~\\cite{Sutskever2014nips:seq2seq} and Transformers~\\cite{vaswani2017nips:transformer} are often used as base model architectures.\nWe hypothesize that high-quality filtered dialogue data can also improve the performance of dialogue response generators.\nHowever, the straightforward application of methods proposed for filtering noisy data in NMT may not work well due to the different nature of NMT and neural response generation tasks.\nIn particular,  MT data have one-to-one (ignoring paraphrases) correspondence in source and target sentences, whereas dialogues have many-to-many mappings~\\cite{zhao2017acl:onetomany}.\nThe experiments presented in this paper provide an answer to whether NMT filtering methods can perform well in dialogue response generation.\n\n\n\n\n\n\n\\section{Requirements to Utterance Pairs}\n\\label{sec:idea}\n\nIn this section, we investigate the requirements that should be satisfied by an acceptable utterance pair.\n\n\\input{tab-preliminary-scored-pairs.tex}\n\n\\subsection{Criteria for Manual Evaluation}\n\\label{ssec:criteria for manual evaluation}\n\n\nThe instructions for manual evaluation provided by the dialogue community explain the key factors for distinguishing acceptable and unacceptable utterance pairs.\n\nIn many previous studies, human raters were asked to evaluate the \\textbf{connectivity} of utterance pairs.\nFor instance, \\citet{shang2015aclijcnlp:neuralresponding} asked whether a response could be considered as \\textit{an appropriate and natural response to the post}.\n\\citet{xing2017aaai:topicaware} asked whether \\textit{the response can be used as a reply}.\n\\citet{pei2018emnlp:s2spmn} asked whether \\textit{the answer is natural} for the question.\nOther studies have also evaluated the same or similar aspects by using keywords related to the connectivity, such as \\textit{semantically appropriate for}~\\cite{akama2017ijcnlp:generating} or \\textit{coherent with}~\\cite{shen2017acl:conditional-variational}, and \\textit{coherence}~\\cite{lowe2017acl:autoturingtest}.\n\n\n\nAnother frequently used metric is \\textbf{content relatedness}.\nFor instance, \\citet{galley2015aclijcnlp:deltableu} asked human evaluators to evaluate \\textit{each response in terms of their relevance to a given utterance}.\n\\citet{li2016naacl:diversity} asked for the preference of responses \\textit{that were more specific to certain utterances}. \n\\citet{ritter2011:datadriven} suggested that \\textit{an appropriate response should be on the same topic as the utterances}.\nSeveral other studies have also focused on evaluating the \\textit{relevance} between an utterance and its response~\\cite{xu2018naacl:lsdscc,pei2018emnlp:s2spmn,lowe2017acl:autoturingtest}.\n\nIn summary, the most widely used criteria can be categorized into connectivity and content relatedness of utterance pairs. \nIn fact, these two aspects are considered in the field of sociolinguistics as crucial features of conversation~\\cite{sacks1989humanstudies:conversationalrule,sidnell2010book:conversation}.\n \n\n\n\n\n\n\\subsection{Observation}\n\\label{ssec:observation}\n\nFurthermore, we investigated how the two aforementioned aspects can be observed in actual utterance pairs.\nFor this investigation, we use the utterance pairs scored by human raters that were used in our preliminary experiments shown in Figure~\\ref{fig:preliminary experiment}.\nSome examples are shown in Table~\\ref{tab:preliminary scored pairs}.\n\n\nWe observe that typical phrase pair patterns can often be found in utterance pairs with high scores.\nFor example, the pair (\\bgf{\\textit{where is}}, \\bgf{\\textit{at}}) in Table~\\ref{tab:preliminary scored pairs} is one of the typical phrase pair patterns that asks a place and provides an answer to it.\nOther typical examples include (\\textit{why}, \\textit{because}) and (\\textit{what do you want}, \\textit{I want}).\nIn discourse linguistics, such phrase pair patterns are known as the concept of \\textit{cohesive devices}.\nHereafter, we refer to such a typical phrase pair pattern as \\textbf{key phrase pair}.\n\n\nMoreover, in high scored utterance pairs, both an utterance and response are on the \\textbf{same topic}.\nFor example, in the third example listed in Table~\\ref{tab:preliminary scored pairs}, both the utterance and response mention \\texttt{[money]}.\n\n\n\n\n\n\\section{Proposed Method}\n\\label{sec:proposed method}\n\nAs per the discussion in the previous section, each acceptable utterance pair should satisfy the following criteria:\n\\begin{itemize}\n    \\item \\textbf{connectivity} --- existence of key phrase pairs\n    \\item \\textbf{content relatedness} --- topic commonality\n\\end{itemize}\nThis section presents the proposed scoring functions to assess the degree of satisfying the above two criteria in an unsupervised manner.%\n\\footnote{The reason for focusing on an unsupervised approach the lack of data that can provide good supervision for utterance pair evaluation.}\n\n\n\n\\subsection{Connectivity}\n\\label{sec:connectivity}\nLet $f$ and $e$ represent phrases obtained from $x$ and $y$, respectively.\nLet $\\phi(x, y)$ be a function that returns a set of all possible phrase ($n$-gram) pairs obtained from the utterance pair $(x, y)$.\nWe can define a finite set of all possible phrase pairs obtained from the entire dialogue data $\\mathcal{D}$ as $\\overline{\\mathcal{P}}^{}_{\\mathcal{D}} =\\bigcup_{(x,y)\\in\\mathcal{D}} \\phi(x, y)$.\nThen, let $\\mathcal{P}$ represent a set of key phrase pairs (defined in Section \\ref{ssec:observation}).\nWe assume that $\\mathcal{P}$ is a subset of $\\overline{\\mathcal{P}}_{\\mathcal{D}}$, i.e., $\\mathcal{P}\\subseteq \\overline{\\mathcal{P}}_{\\mathcal{D}}$.\n\n\nTo obtain $\\mathcal{P}$, we take advantage of a phrase table extraction technique developed in statistical machine translation, e.g., Moses~\\cite{koehen2017acl:moses}.\nIn this task, we require only some phrase pairs that can contribute to the connectivity of an utterance pair (as mentioned in Section \\ref{ssec:observation}), unlike the translation task where the whole sentence must correspond in mutual.\nAccordingly, in our experiments, we set the null alignment ratio (i.e., probability of no alignment) to $0.5$ and extend the phrase extraction algorithm to include only the explicitly corresponding range as phrases in our table.\n\n\nThen, we define the scoring function $\\Sframe$ to estimate connectivity as: \n\\newcommand{\\LEN}[1]{\\lvert{#1}\\rvert}\n\\begin{align}\n\\label{eq:s_frame}\n    \\Sframe(x,y) := \n    \\sum_{\\mathclap{(f,e) \\in \\phi(x, y)\\cap\\mathcal{P} }} \n    \\hspace{2pt} \n        \\max\\bigl(\\mathrm{nPMI}(f,e),0\\bigr) \\cdot \\frac{\\LEN{f}}{\\LEN{x}} \\cdot \\frac{\\LEN{e}}{\\LEN{y}}\n    \\text{,}\n\\end{align}\nwhere $\\LEN{\\cdot}$ denotes the number of words in the phrase or utterance.\nTo calculate the co-occurrence, we use the normalized pointwise mutual information (nPMI)~\\cite{bouma2009gscl:npmi}, which normalizes the value so that low-frequency phrases do not take an extremely large value.\nNote that we ignore the negative nPMI scores by the $\\max(\\cdot, 0)$ operation because we aim only to consider the positive effect of connectivity. \nThe intuition behind Equation~\\ref{eq:s_frame} is as follows:\n\\begin{itemize}\n    \\setlength{\\itemindent}{0mm}\n\t\\setlength{\\parskip}{0.3mm}\n    %\n    \\item If a phrase pair $(f,e)$ has a high co-occurrence, the association strength of $(x,y)$ including $(f,e)$ might also be high. \n    %\n    \\item If a phrase $f$ or $e$ occupies almost the entire sentence $x$ or $y$, $(f,e)$ is a strong indicator of the association of $(x, y)$.\n    %\n\\end{itemize}\n\n\n\n\\subsection{Content Relatedness}\nLet $\\VEC v(x)$ and $\\VEC v(y)$ be sentence vector of $x$ and $y$, respectively.\nWe compute topic commonality of $x$ and $y$, that is, content relatedness as follows:\n\\begin{align}\n\\label{eq:s_content}\n    \\Scontent(x,y) := \n    \\max\\bigl(\\cos(\\VEC v(x), \\VEC v(y)), 0\\bigr)\n    \\text{.}\n\\end{align}\nCosine similarity between certain kinds of sentence vectors is known to be a good proxy of the topical relatedness of two sentences~\\cite{Conneau2017emnlp:universalrepresentation,subramanian2018iclr:generalpurpose,xu2018emnlp:filtering}.\nFor the same reasons as Equation~\\ref{eq:s_frame}, we ignore the negative $\\cos$ scores by the $\\max(\\cdot, 0)$ operation. \n\n\n\n\\subsection{Summary}\n\nEventually, combining the above two scoring measures, we propose the following function:\n\\begin{align}\n\\label{eq:score}\n    \\Sours(x,y) := \\alpha \\Sframe(x,y) + \\beta \\Scontent(x,y)\n    \\text{,}\n\\end{align}\nwhere $\\alpha,\\,\\beta\\in\\mathbb R_{\\geq 0}$ are hyperparameters that weigh the two viewpoints. \nFor our experiments, we fix $\\alpha$ and $\\beta$ as follows:\n\\begin{align}\n    \\label{eq:score hyp a}\n    & \\alpha \\!=\\! \\frac 1 {\\frac 1 {\\lvert\\mathcal D\\rvert} \\! \\displaystyle \\sum_{\\mathclap{(x,y)\\in\\mathcal D}} \\Sframe(x,y)\\!},\\;\n    \\beta \\!=\\! \\frac 1 {\\frac 1 {\\lvert\\mathcal D\\rvert} \\! \\displaystyle \\sum_{\\mathclap{(x,y)\\in\\mathcal D}} \\Scontent(x,y)}\n    \\text{.}\\!\n\\end{align}\n\n\n\n\n\n\\section{Experiments: Data Scoring}\n\\label{sec:experiments}\n\nIn this section, we describe our experiments that validate the effectiveness of the proposed scoring method.\n\n\\subsection{Experimental Setup}\n\n\\paragraph{Dataset.}\n\\label{ssec:dataset}\n\nWe conducted our experiments on a noisy English dialogue corpus from OpenSubtitles~\\cite{lison2018lrec:opensubtitles} containing roughly $441$M lines.\nAs explained in Section~\\ref{sec:introduction}, this corpus includes many unacceptable utterance pairs (Section~\\ref{sec:introduction}).\nWe first applied several rule-based filtering as rudimentary preprocesses, which are typically used in the related literature.\nThen, we obtained $79,\\!445,\\!453$ utterance pairs as our training data, which excludes our test and validation data\n    \\footnote{See Appendix~\\ref{a:corpus_creation} for details on our data such as the preparation procedure and statistics.}\n\n\n\n\\input{fig-en-correlation-human-3.tex}\n\n\n\\paragraph{Proposed method: detailed setup.}\n\n\nTo compute the connectivity $\\Sframe$, we obtained a phrase table on our training data by using Moses~\\cite{koehen2017acl:moses} with fastAlign~\\cite{dyer2013naacl:fastalign}. \nWe then removed phrase pairs with a low co-occurrence frequency (here, less than 200 times) or composed of the same phrases from the table.\nAs a result, the phrase table included $68,\\!891$ phrase pairs, which were used as the key phrase set $\\mathcal{P}$ as described in Section~\\ref{sec:connectivity}.\n\n\nTo compute the content relatedness $\\Scontent$, we created a sentence vector from pre-trained fastText word embeddings~\\cite{Bojanowski2017tacl:fasttext,mikolov2018lrec:advances-in-wordrep} following \\citet{arora2017iclr:sif}'s method, i.e., using SIF weighting and common component removal.\nTheir method is reported to be useful for computing the relatedness of two given sentences and used in many studies\n~\\cite{marelli2014lrec:sick,marelli2014semeval:compositional-distributional-model,Conneau2017emnlp:universalrepresentation,subramanian2018iclr:generalpurpose, baheti2018emnlp:generating-interesting-response}.\nWe learned common components using $30$K sentences randomly selected from the training costs appropriately. \nWe then removed the first common component for all sentence vectors.\n\n\n\n\\paragraph{Baselines.}\n\nFor comparison, we prepared the following:\n\\begin{itemize}\n    %\n    \\item \\citet{csaky2019acl:filtering}: Entropy-based filtering to remove generic utterances from the training data for promoting less-boring response generation. SRC\/TRG indicates that using the entropy of source\/target utterances.\n    %\n    \\item \\citet{junczys-dowmunt2018wmt:filtering}: Filtering for NMT based on the dual conditional cross-entropy computed by a neural encoder-decoder model. It achieved the best performance on the Parallel Corpus Filtering Task at WMT~2018.%\n        \\footnote{\\url{http:\/\/www.statmt.org\/wmt18\/}}\n    %\n\\end{itemize}\n\n\n\n\n\n\n\\paragraph{Human evaluation.}\n\\label{ssec:scoring eval}\n\nTo validate the ability of the proposed method to estimate the quality of utterance pairs, we measured the correlation between its scores and those assigned by humans through crowdsourcing.\nWe used Amazon Mechanical Turk.%\n    \\footnote{\\url{https:\/\/www.mturk.com\/}}\nWe randomly extracted $200$%\n    \\footnote{Same size as \\citet{sedoc2019naacldemo:chateval,cho2020acl:yesand}.}\nscored utterance pairs and asked native English-speaking crowdworkers to answer the following question for each pair: \\textit{Is the sequence of the two utterances acceptable as a dialogue?} \nWorkers were instructed to provide an answer on a five-point Likert scale (from $5$: Strongly agree to $1$: Strongly disagree)~\\cite{likert1932archivesofpsycho:scale}.\nUnqualified workers were filtered out using attention checks.\nEventually, we used the average of the scores provided by five workers as the human score for each pair.\n\n\n\\input{tab-scored-correlation.tex}\n\n\\input{tab-scored-samples.tex}\n\n\n\n\\subsection{Results and Analysis}\n\nTable~\\ref{tab:scored correlation} shows the correlation between human scores and those automatically computed by each method.\nAmong the methods, $\\Sours$ achieved the highest correlation with human scores.\nAdditionally, we also evaluated $\\Sframe$ and $\\Scontent$ as an ablation study of $\\Sours$.\nWe found that both scores were less correlated than $\\Sours$.\nThis result supports the hypothesis that both aspects, namely, connectivity and content relatedness, should be considered when evaluating the quality of utterance pairs.\n\n\n\nFigure~\\ref{fig:en_correlation-human-3} shows the distribution of automatically computed scores corresponding to human scores.%\n    \\footnote{See Appendix~\\ref{a:en-correlation_dist} for the distributions of other methods.}\nAs shown in (c), $\\Sours$ rarely overestimates utterance pairs with low human scores but underestimates those with high human scores.\nThe baseline methods presented in (a) and (b)  do not show such behavior. \nThis behavior unique to $\\Sours$ is safe for the noisy data filtering task since it can successfully detect lower-quality pairs with high precision.\nOn the other hand, improperly underestimating some acceptable pairs (i.e., low recall) is one downside of $\\Sours$, and we discuss its influences in Section~\\ref{ssec:LowRecall}.\nWe emphasize that $\\Sours$ has a desirable property for noisy data filtering in today's situation where a sufficiently large corpus is available; it allows us to obtain a sufficient amount of clean data even if discarding a certain portion of potentially clean data. \nInteresting future work is to investigate how to improve our methods not to underestimate acceptable pairs while maintaining high precision.\nIt is nearly equivalent to develop an unsupervised approach of dialogue evaluation methods, and thus, this direction is a challenging and essential attempt. \n\n\n\nTable~\\ref{tab:scored samples} shows several examples of utterance pairs well-scored by $\\Sframe$, $\\Scontent$, and $\\Sours$.\nNote that the score ranges differ; e.g., human scores are in $[1, 5]$, while $\\Scontent$ is in the range $[0, 1]$.%\n    \\footnote{See Appendix~\\ref{a:score distributions} for score distributions on training data.}\nThus, we discuss relative score values; the comparison of absolute score values across the different methods would be meaningless.\nThese examples demonstrate that the complementary contributions of both $\\Sframe$ and $\\Scontent$ allow $\\Sours$ to provide quality estimations close to human judgments. \n\n\n\n\n\n\\subsection{Discussion on Low Recall Property}\n\\label{ssec:LowRecall}\n\n\\paragraph{What types of pairs cause low recall?}\nSince the proposed method prefers precision over recall, it tends to discard a certain number of acceptable utterance pairs during filtering.\nTo investigate the characteristics of such discarded (yet acceptable) pairs, we analyzed $27$ pairs.%\n    \\footnote{Some examples are listed in Appendix~\\ref{a:post-hoc-analyses}}\nThese pairs were selected from those that obtained a human score of 4.0 or above ($77$ pairs) \\emph{and} were among the worst $50$\\% as scored by $\\Sours$ ($100$ pairs).\nConsequently, we found two potential issues.\nOne is that human annotators may sometimes easily find the connectivity or the content relatedness for the utterance pairs with the low $\\Sours$ scores.\nThis observation indicates that $\\Sframe$ and $\\Scontent$ are still not perfect for scoring functions, and there remains room for improvement.\nThe possible drawbacks we have already noticed in $\\Sframe$ and $\\Scontent$ are that $\\Sframe$ sometimes fails to capture the connectivity because of the limited coverage by a discrete phrase table-based approach, and $\\Scontent$ is not robust for out-of-vocabulary of word vector.\nThe other case is that the human annotators gave high scores, but we found no connectivity and content relatedness in the utterance pairs.\nWe found that some utterance pairs without any connectivity and content relatedness can be judged as acceptable by the human annotators since they can imagine the underlying context and situation of the utterance pairs using human world knowledge, such as commonsense.\nWe think this is a challenging issue that exceeds our focus in this paper, and thus, remains as future work.\n\n\n\n\\input{tab-scored-top-vs-worst-small.tex}\n\n\n\\input{tab-evaluation-results.tex}\n\n\n\n\\paragraph{Does our filtering undermine diversity?}\nOne might think that our method succeeds in filtering by assigning high scores to generic responses such as dull responses.\nThis concern makes sense since it is known that dialogue systems learned from the training data, including many generic utterances, tend to generate bland responses~\\cite{csaky2019acl:filtering}.\nTo answer this interesting question, we confirmed the diversity of utterance pairs with a high score (i.e., remained as training data) and a low score (i.e., removed from training data) in our $\\Sours$ (Table~\\ref{tab:scored-top-vs-worst-small}).%\n    \\footnote{See Appendix~\\ref{a:post-hoc-analyses} for more extensive result.}\nAs a result, there was no significant difference between them.\nTherefore, we conclude that the proposed method does not prefer only generic responses and maintains the diversity of data.\nIt is an essential future attempt to improve the quality of dialogue data further (e.g., more diversity) after using the proposed method to remove unacceptable pairs.\n\n\n\n\n\\input{tab-genresponses.tex}\n\n\n\n\n\\section{Experiments: Response Generation}\n\\label{sec:case study}\n\nThis section reports on the effectiveness of the proposed method for filtering noisy data in neural response generation.\n\n\n\\subsection{Experimental Setup}\n\n\\paragraph{Training.}\n\\label{ssec:TrainingSettings}\n\nWe obtained the filtered training data $\\mathcal{D}'$ by removing utterance pairs with low scores from the original dataset $\\mathcal D$ (approximately $10$\\% or $50$\\% of total utterance pairs were removed).\nAs a response generation model, we used a Transformer~\\cite{vaswani2017nips:transformer} based encoder-decoder model implemented in the \\texttt{fairseq} toolkit~\\cite{ott2019naacldemo:fairseq}.%\n    \\footnote{See Appendix~\\ref{a:training settings} for training details.}\nTransformer has demonstrated high performance in response generation~\\cite{dinan2019iclr:wizard} and other NLP tasks.\n\n\n\\paragraph{Automatic evaluation.}\nHere, we report the following metrics: the average response length in tokens (len), type-token ratio for $\\{1,2\\}$-grams (distinct-$\\{1,2\\}$), and and BLEU-1~\\citep{Papineni2002acl:bleu}.\nThe latter was used as a reference-based metric; while it is widely used in previous studies~\\cite{zhao2017acl:onetomany,baheti2018emnlp:generating-interesting-response,csaky2019acl:filtering}, some studies (e.g., ~\\cite{liu2016emnlp:hownot}) have reported that BLEU-1 may not be highly correlated with the human evaluation of response generation.%\n    \\footnote{See Appendix~\\ref{a:generated response with metrics} for more extensive evaluation results.}\n\n\n\\paragraph{Human evaluation.}\nWe evaluated the quality of the generated responses manually. \nWe asked human evaluators recruited via Amazon Mechanical Turk to evaluate responses that are generated for $100$%\n    \\footnote{Same size as \\citet{shen2017acl:conditional-variational,bao2020acl:plato}.}\ninput utterances randomly sampled from the test data. \nWe used the same task setting and protocol as described in Section~\\ref{ssec:scoring eval} to obtain the human scores for each pair. \nHigher human scores indicate that the better results.\n\n\n\n\\subsection{Results and Analysis}\n\n\nTable~\\ref{tab:evaluation-results-en} shows the results of automatic and human evaluations of the generated responses.\nThe model trained on the data filtered using the proposed method $\\Sours$ produced more than three times as many distinct $\\{1,2\\}$-grams as the model trained on non-filtered data.\nFurthermore, it outperformed the model trained on non-filtered data in the human evaluation, achieving the highest percentage of acceptable responses of 85\\%. \nAdditionally, these results of our $\\Sours$ were better than other baselines.\nTo conclude, these experimental results indicate that the proposed scoring method can help generate diverse responses that are judged as acceptable by humans. \n\nThis experiment provides empirical evidence for supporting our hypothesis that the performance of neural response generation models can be improved by just removing unacceptable utterance pairs from training data, which answers the research question formulated at the start of this paper.\n\n\n\n\n\n\n\n\n\\section{Multilingual Availability}\n\\label{sec:Japanese}\n\n\nWhile the proposed method $\\Sours$ was tested on an English corpus, it can potentially work for other languages as well. \nTo demonstrate this, we selected Japanese dialogue data as another case study.%\n    \\footnote{See Appendix~\\ref{a:japanese} for all experimental results on Japanese.} \nThe linguistic phenomena in Japanese are quite different from those in English, thus making this experiment to be a good test of the applicability of the proposed method to non-English languages.\n\n\n\\paragraph{Japanese dataset.}\nWe prepare the Japanese dialogue data from Japanese OpenSubtitles~\\cite{lison2018lrec:opensubtitles} containing roughly $3$M lines.\nWe obtain $1,\\!893,\\!477$ utterance pairs as our training data, which excludes our test and validation data.%\n    \\footnote{See Appendix~\\ref{a:corpus_creation} for details on our data such as the preparation procedure and statistics.}\n\n\n\\input{tab-scored-correlation-ja.tex}\n\n\\input{fig-ja-correlation-human-ours}\n\n\\input{tab-evaluation-results-ja}\n\n\n\n\n\\subsection{Data Scoring}\n\\label{ssec:JapaneseScoring}\n\n\n\\paragraph{Settings.}\nTo compute $\\Sframe$, we defined a low co-occurrence frequency as less than 20, considering the size of the Japanese corpus, and consequently obtained the key phrase pairs $|\\mathcal{P}|=19,\\!992$.\nTo compute $\\Scontent$, we used pre-trained fastText \\cite{grave2018lrec:wordvec157} and learned common components from all sentences in the training data.\n\n\nFor human evaluation, we used Yahoo!~crowdsourcing%\n    \\footnote{\\url{https:\/\/crowdsourcing.yahoo.co.jp\/}}%\nto hire native Japanese-speaking workers.\nThe task setting and protocol are the same as those for English (Section \\ref{ssec:scoring eval}), regardless of the crowdsourcing platform.\n\n\n\\paragraph{Results and analysis.}\nTable~\\ref{tab:scored correlation_ja} shows the correlation between human scores and those automatically computed by each method.\nOur method $\\Sours$ has the highest correlation with human scores, although the overall result is lower than that obtained for the English dataset.\nFigure~\\ref{fig:ja_correlation-human-onlyours} shows the distribution of our $\\Sours$ corresponding to human scores.\nSimilar to the result obtained for English as presented in Figure~\\ref{fig:en_correlation-human-3} (c), $\\Sours$ rarely overestimates utterance pairs with low human scores but underestimates those with high human scores in Japanese.\n\n\n\n\\subsection{Response Generation}\n\\label{ssec:JapaneseFiltering}\n\n\n\\paragraph{Settings.}\nWe used the same experimental settings described in Section~\\ref{ssec:TrainingSettings} for the preparation of filtered data $\\mathcal{D}'$ and model training.\n\n\n\\paragraph{Results and analysis.}\nTable~\\ref{tab:evaluation-results-ja} shows the results of evaluations of the generated responses.\nThe filtered data generated by $\\Sours$ provided the best results in terms of almost all the metrics, including human evaluation.\nIt supports our hypothesis that the proposed method is also suitable for non-English languages.\n\n\n\n\n\\section{Relationship with Evaluation Metric}\n\\label{sec:relatedWork}\n\nThe proposed method $\\Sours$ maps an utterance pair to a score (scalar value) in terms of the quality of dialogue.\nThat is, formally, our method is similar to the reference-free automatic evaluation metrics for dialogue agents; both of them evaluate the response given an input utterance and also map into a score. \nRecently, the novel reference-free metrics for evaluating generated responses such as USR~\\cite{Mehri2020acl:usr} or \\textsc{MaUde}~\\cite{sinha2020acl:maude} ware developed.\nWhile it is possible to use them as a scoring method for filtering noisy data, in theory, there are some concerns with applying them in practice.\nOne is the difference of the data of interest; since evaluation metrics are intended for responses generated as dialogue, i.e., somewhat valid dialogue data, it is unclear whether they also work for apparently noisy data.\nAnother one is the difference of desired properties; evaluation metrics need to be sensitive to ``how good is it?''\\ while the filtering requires to detect ``is it a dialogue?''\\ with high accuracy.\nIt would be interesting to investigate the effectiveness of reference-free metrics for noisy dialogue data filtering tasks, and vice versa.\nWe leave these investigations for future work.\n\nIn contrast, reference-based metrics require a reference response (i.e., ground truth) when they calculate scores; such metrics include the traditional overlap-based BLEU, ROUGE, METEOR, embedding-based metrics~\\cite{liu2016emnlp:hownot}, and neural network-based RUBER~\\cite{tao2018aaai:ruber} and ADEM~\\cite{lowe2017acl:adem}\nThus, these methods cannot straightforwardly be considered as alternatives to the proposed method, which aims at filtering.\n\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nIn light of the success of noisy corpus filtering in neural machine translation, we attempted to filter out unacceptable utterance pairs from large dialogue corpora in an unsupervised manner.\nThe proposed scoring method estimates the quality of utterance pairs by focusing on the two crucial aspects of dialogue, namely, the \\emph{connectivity} and \\emph{content relatedness} of utterance pairs.\nWe demonstrated that our scoring method has a higher correlation with human judgment than recently proposed methods.\nFurthermore, we provided empirical evidence that our method improves the performance of a response generation model by removing unacceptable utterance pairs from its training data. \nWe hope that this study will facilitate discussions in the dialogue response generation community regarding the issue of filtering noisy corpora.\n\n\n\n\\clearpage\n\\section*{Acknowledgments}\nThis work was supported by JSPS KAKENHI Grant Numbers JP19H04162, JP19J21913.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn the last two decades there has been a growing interest in the study of\nchoreographic solutions (\"choreographies\") of the $n$-body and $n$-vortex \nproblems.\nChoreographies are periodic solutions where the bodies or the vortices \nfollows the same path. \nThe first choreography was discovered numerically for the case of three\nbodies in \\cite{Mo93}, and its existence was rigorously proved in \n\\cite{ChMo00}. \nThe term \"choreography\" was adopted for the $n$-body problem after the work\nof Sim\\'{o} \\cite{Si00}.\nSince then, variational methods \\cite{FeTe04,BT04}, numerical minimization\n\\cite{ChGe02}, numerical continuation \\cite{Ca}, and computer-assisted \nproofs \\cite{KaSi07}, have been used to determine choreographies of the \n$n$-body problem; see also the references in these papers. \nFor vortices in the plane, choreographies have been constructed for $3$ \nand $4$ vortices in \\cite{Bo}. \nFor $n$ vortices in a general bounded domain, choreographic solutions have \nbeen found close to a stagnation point of a vortex \\cite{Ba0}, and close \nto the boundary of the domain \\cite{Ba}.\n\nIn \\cite{Ca}, \\cite{GaIz13}, and \\cite{ChFe08}, chorographies are found in\ndense sets of Lyapunov families that arise from the stationary $n$-polygon \nof bodies in a rotating frame. \nThe existence of these choreographies depends only on the symmetries of \nthe equations in rotating coordinates, {\\it i.e.}, these results can be \nextended to find choreographies of the $n$-vortex problem in the plane, \nin a disk, or in a surface of revolution. \nSuch results can also be extended to the discrete nonlinear Schr\\\"{o}dinger\nequation (DNLSE), which appears in the study of optical waveguide arrays,\nand in Bose-Einstein condensates trapped in optical lattices, among many \nother applications \\cite{Kr}.\n\nWhile much research has been done on choreographies in the $n$-vortex and \n$n$-body problems, we are not aware of its extension to the study of the \nexistence of choreographies in periodic lattices of $n$ sites, as modeled \nby the DNLSE.\nThe interest in $n$-body and $n$-vortex choreographies can perhaps be \nexplained by the fact that variational methods are better suited for \nsingular potentials. \nOn the other hand, the continuation methods used in \\cite{Ca} are very well\nsuited for locating choreographies for other symmetric potentials, such as \nin the DNLSE.\n\nEvidence suggests that linearly stable choreographies in the $n$-body \nproblem only exist for $n=3$, and in the $n$-vortex problem for \n$n=3,\\cdots,7$, as a consequence of the fact that the polygonal relative \nequilibrium of $n$ bodies is unstable for $n\\geq3$ \\cite{Ca}, and for \nvortices for $n>7$ \\cite{GaIz12}. \nOn the other hand, the DNLSE has dense sets of stable choreographies, \nas a consequence of the fact that the DNLSE has stable polygonal relative \nequilibria for all $n$ \\cite{Ga15}.\nWe note that boundary value continuation methods can determine unstable \nchoreographies as easily as stable ones; a property not shared by most \nother techniques.\nThe aim of our paper is to investigate the existence of choreographies \nin the DNLSE using a boundary value continuation method. \nAs illustrative examples we present a selection of choreographies,\nmost of them stable, in a periodic lattice for the case of $n=9$,\n$17$, and $31$ sites.\n\nThe lack of previous work on detecting choreographies in the DNLSE may \nbe related to the difficulties encountered in the measurement of phases \nin physical problems modeled by the DNLSE. \nHowever, such difficulties appear to be surmountable in nonlinear optics.\nWithin the context of nonlinear optics, several predictions of the DNLSE \nhave been found experimentally in the last two decades \\cite{Opt1},\nsuch as the formation of discrete solitons in waveguide arrays. \nThe use of suitable optical techniques, known as laser heterodyne \nmeasurements, to detect the field intensity and the phase \\cite{Opt2,Opt3}, \nmay open the door to experimental observation of stable choreographic \nsolutions. \n\nIn Section~2 we consider Lyapunov families of periodic orbits, and their \nrelation to the existence of choreographies in a periodic lattice of \nSchr\\\"{o}dinger sites. \nIn Section~3 we present methods to continue the Lyapunov families, and we \nexhibit a small selection of the many linearly stable choreographies that\nwe have determined.\nIn Section~4 we discuss our final remarks on choreographies.  \n\\section{Lyapunov families and choreographies}\n\nIn a rotating frame with frequency $\\omega$, $q_{j}(t)=e^{i\\omega t}%\nu_{j}(t)$, the equation that describes the dynamics of a lattice of $n$ \nsites is given by the Hamiltonian system \n\\begin{equation}\n\\dot{u}=\\mathcal{J}\\nabla H_{\\omega}(u), \\quad \\text{where} \\quad\nH_{\\omega}=\\frac{1}{2}\\sum_{j=1}^{n}\\left(  \\frac{1}{2}\\left\\vert\nu_{j}\\right\\vert ^{4}+\\omega\\left\\vert u_{j}\\right\\vert ^{2}-\\left\\vert\nu_{j+1}-u_{j}\\right\\vert ^{2}\\right)  . \\label{Equations}\n\\end{equation}\nThe sites $u_{j}(t)\\in\\mathbb{C}$ satisfy periodic boundary conditions\n$u_{j}(t)=u_{j+n}(t)$.\nThe equation of motion has explicit polygonal equilibrium solutions given \nby%\n\\begin{equation}\na_{j}=ae^{ij(\\alpha\\zeta)},\\qquad\\zeta=\\frac{2\\pi}{n}, \\label{SW1}%\n\\end{equation}\nfor $\\omega(a)=4\\sin^{2}(m\\zeta\/2)-a^{2}$, $\\alpha=1,\\cdots,n$ and $a\\in\n\\mathbb{R}^{+}$. \nThese solutions correspond to relative equilibria in the non-rotating frame given by $q_{j}(t)$for $j=1,\\cdots,n$.\nThe linearized Hamiltonian system at the polygonal equilibrium \n$\\mathbf{a} =(a_{1},...,a_{n})$ is \n$$\\dot{u}=\\mathcal{J}D^{2}H_{\\omega(a)}(\\mathbf{a})u.$$\nIn \\cite{Ga15} it is proved that the matrix \n$\\mathcal{J}D^{2}H_{\\omega (a)}(\\mathbf{a})$ \nhas a pair of imaginary eigenvalues $\\pm i\\nu_{k}$ for each\n$k\\in\\{1,\\cdots,n-1\\}$ such that\n\\begin{equation}\n\\frac{a^{2}}{2\\cos\\alpha\\zeta\\sin^{2}k\\pi\/n}<1. \\label{In}%\n\\end{equation}\nIt is also proved in \\cite{Ga15} that for each $a$ \nand $k\\in\\{1,\\cdots,n-1\\}$ such that (\\ref{In}) holds, \nthe equilibrium $\\mathbf{a}$ has a \\emph{global family} \nof periodic solutions that arises from the normal modes \nof the polygonal equilibrium, and has the form\n\\begin{equation}\nu_{j}(t)=e^{ij\\alpha\\zeta}u_{n}\\left(  \\nu t\\pm jk\\zeta\\right)  \\text{,}\n\\label{TW}%\n\\end{equation}\nwhere $u_{n}(t)=a+\\mathcal{O}(b)$ is a $2\\pi$-periodic function,  \n$\\nu =\\nu_{k}+\\mathcal{O}(b)$ is the frequency and $b$ is a parameterization of the local family. \nThe traveling waves (\\ref{TW}) form two-dimensional families parametrized \nby the amplitude $a$ and the bifurcation parameter $b$. In the non-rotating frame, after rescaling time, these periodic solutions \nare traveling waves of the form%\n\\[\nq_{j}(t)=e^{i\\omega t\/\\nu}e^{ij\\alpha\\zeta}u_{n}\n     \\left(  t\\pm jk\\zeta\\right) ,\\qquad\\omega=\\omega(a)\\text{.}%\n\\]\nWe say that a Lyapunov orbit is $\\ell:m$ resonant if $\\ell$ and $m$ are\nrelatively prime such that%\n\\[\n\\frac{\\omega}{\\nu}=\\frac{\\ell}{m}\\text{,\\qquad}k\\ell-\\alpha \n  m\\in n\\mathbb{Z}\\text{.}%\n\\]\nSuch frequencies $\\nu$ are dense in the set of real numbers.\nFor an $\\ell:m$ resonant orbit, we have%\n\\[\nq_{n}(t)=e^{it\\omega\/\\nu}u_{n}(t)~.\n\\]\nSince $e^{it\\omega\/\\nu}=e^{it\\ell\/m}$ is $2\\pi m$-periodic, the function\n$q_{n}(t)=e^{it\\omega\/\\nu}u_{n}(t)$ is $2\\pi m$-periodic. \nAlso, since%\n\\begin{equation}\n q_{n}(t-2\\pi)=e^{i(t-2\\pi)\\omega\/\\nu}u_{n}(t-2\\pi)=e^{-i2\\pi\\ell\/m}%\n q_{n}(t),\\label{symq}%\n\\end{equation}\nthe orbit of $q_{n}(t)$ is invariant under rotation by $2\\pi\/m$.\nThe solutions satisfy\n\\begin{align*}\nq_{j}(t)  &  = e^{it(\\omega\/\\nu)}u_{j}(t)\n       =e^{it(\\omega\/\\nu)}e^{ij\\alpha\\zeta }u_{n}(t+jk\\zeta)\\\\\n&  = e^{it(\\omega\/\\nu)}e^{i\\alpha j\\zeta}e^{-i(\\omega\/\\nu)\n  \\left( t+jk\\zeta\\right)  }q_{n}(t+jk\\zeta)=e^{-ij\n  \\left(  (\\omega\/\\nu)k-\\alpha\\right) \\zeta}q_{n}(t+jk\\zeta)\\text{.}%\n\\end{align*}\nUsing the facts that $\\omega\/\\nu=\\ell\/m$ and $\\zeta=2\\pi\/n$, we have\n\\[\nj\\left(  k\\frac{\\omega}{\\nu}-\\alpha\\right)  \\zeta=2\\pi j\n   \\left(  \\frac{\\ell k-\\alpha m}{mn}\\right)  =2\\pi j \\left( \\frac{r}{m} \\right) ,\n\\]\nwith $r=(k\\ell-\\alpha m)\/n\\in\\mathbb{Z}$ by assumption. \nSince $\\ell$ and $m$ are relatively prime we can find $\\ell^{\\ast}$, \nthe $m$-modular inverse of $\\ell$. \nSince $\\ell\\ell^{\\ast}=1$ mod $m$, it follows from the symmetry\n(\\ref{symq}) that\n\\[\nq_{n}(t-2\\pi jr\\ell^{\\ast})=e^{-i2\\pi j(r\/m)}q_{n}(t).\n\\]\nThen%\n\\begin{equation}\nq_{j}(t)=e^{-i2\\pi j(r\/m)}q_{n}(t+jk\\zeta)\n   = q_{n}(t+j(k-rn\\ell^{\\ast})\\zeta).\n\\end{equation}\nThus in the non-rotating frame, an $\\ell:m$ resonant Lyapunov orbit \nis a choreography satisfying%\n\\[\nq_{j}(t)=q_{n}(t+j\\tilde{k}\\zeta)\\text{,}%\n\\]\nwhere \n$\\tilde{k}=k-(k\\ell-\\alpha m)\\ell^{\\ast}$ \nwith $\\ell^{\\ast}$ the $m$-modular inverse of $\\ell$. \nThe period of the choreography is $m~T_{\\ell :m}$ with%\n\\[\nT_{\\ell:m}=\\frac{2\\pi}{\\nu}=2\\pi\\omega(a)\\left(  \\frac{\\ell}{m}\\right)  .\n\\]\nThe choreography is symmetric under rotation by $2\\pi\/m$, and it winds\naround a center $\\ell$ times.\n\\section{Computational Results}\n\\label{sec:2}\nWe have computed families of periodic solutions that arise directly or\nindirectly from the circular polygonal relative equilibrium of the DNLSE.\nThese families are computed by numerical continuation using boundary value \nformulations. \nIn this article we present numerical results for several choices of the \nnumber of sites $n$ in the DNLSE, namely for $n=9$, $n=17$, and $n=31$,\nwith various values of the amplitude parameter $a$.\nIn our boundary value setting the DNLS differential equations are \nformulated as\n\\begin{align*}\nu_{k}^{\\prime}(t)  &  =-iT\\left(  u_{k-1}-2u_{k}+u_{k+1}+\\left\\vert\nu_{k}\\right\\vert ^{2}u_{k}+\\omega u_{k}\\right) \\\\\n&  +p_{1}(4u_{k}-4u_{k}^{3}-2\\bar{u}_{k+1})+p_{2}(u_{k+1}-\\bar{u}_{k-1}),\n\\end{align*}\nwhere $u_{k}(t)=x_{k}(t)+iy_{k}(t)$ for $k=1,2,\\cdots,n$ and\n\\[\nu_{0}(t)\\equiv u_{n}(t)\\quad,\\quad u_{n+1}(t)\\equiv u_{1}(t).\n\\]\nHere $T=2\\pi\/\\nu$ is the period of a periodic orbit, so that the scaled time\nvariable $t$ takes values in the fixed time interval $[0,1]$. The parameters\n$p_{1}$ and $p_{2}$ are \\textit{unfolding parameters} that are necessary to\ntake care of invariances related to the presence of two conserved quantities.\nThe parameters $p_{1}$ and $p_{2}$ are always part of the unknowns solved in\nthe Newton iterations. However, upon converge their values are zero up to\nnumerical accuracy.\nThe boundary conditions that we impose always include the periodicity\nconditions\n\\[\nu_{k}(1)-u_{k}(0)=0,\\quad k=1,2,\\cdots,n.\n\\]\nAdditional constraints can be used to fix certain quantities along \nsolution families, provided other appropriate parameters are allowed \nto vary. \nIn particular, we can fix the $y$-coordinate of the $n$th site at time \n$t=0$, {\\it i.e.},%\n\\[\ny_{n}(0)=0.\n\\]\nThis constraint can be viewed as a phase condition that is sometimes more\nconvenient than an integral phase condition of the type mentioned below. \nIt can also be useful to fix the $x$-coordinate of the $n$th site \nat time $t=0$, which is accomplished by adding the boundary condition\n\\[\nx_{n}(0)-x_{n}^{0}=0,\n\\]\nwhere $x_{n}^{0}$ is a parameter that can be kept fixed.\nFor convenience, constraints of this form can also be used to keep track \nof such quantities. \nFor example, the parameter $x_{n}^{0}$, when free to vary, trivially keeps \ntrack of $x_{n}(0)$. \nSuch constraints can also fix (or to keep track of) one of the conserved \nquantities $E$ or $A$, or the resonance ratio $T\/T_{0}$, \nwhere $T_{0}=2\\pi\/\\omega$ is the period of the rotating frame. \nThis is accomplished by adding one of the following constraints:\n\\[\n\\left\\vert u_{k}-u_{k+1}\\right\\vert ^{2}-\\left\\vert u_{k}\\right\\vert\n^{4}-\\omega\\left\\vert u_{k}\\right\\vert ^{2}-E=0~,~~\\mbox{where}\\quad\n\\omega=4\\sin^{2}(\\pi\/n)-a^{2},\n\\]%\n\\[\n\\sum_{k=1}^{n}\\left\\vert u_{k}\\right\\vert ^{2}-A=0,\\quad\\mbox{or}\\quad\nT\/T_{0}-r=0.\n\\]\nThe boundary value formulation can also contain integral constraints, \nsuch as the phase condition\n\\[\n\\int_{0}^{1}x_{n}(t)~\\tilde{x}_{n}^{\\prime}(t)\n          +y_{n}(t)~\\tilde{y}_{n}^{\\prime }(t)~dt=0,\n\\]\nwhich here is applied to the $n$th site only, and where \n$(\\tilde{x}_{n}^{\\prime}(t),\\tilde{y}_{n}^{\\prime}(t))$ represents \nthe time-derivative of a reference solution, which typically is the \npreceding solution in the numerical continuation process. \nAnother integral constraint sets the average $y$-coordinate \nof the $n$th site to zero, namely,\n\\[\n\\int_{0}^{1}y_{n}(t)~dt=0.\n\\]\nThe purpose of this constraint is to remove the rotational invariance of\nperiodic solutions. \nThere are more general integral constraints for fixing the phase and for \nremoving invariances. \nHowever the ones listed above are simple, and appropriate in the current \ncontext.\n\nWe now briefly outline the computational procedure that we have used to \nlocate the periodic orbits shown in Figure~\\ref{fig1} and in\nFigure~\\ref{fig2}, for which corresponding data is given in Table~1.\nAs a starting procedure we follow a family of periodic solutions that \nemanates from the polygonal equilibrium, namely a family that arises \nfrom a conjugate pair of purely imaginary eigenvalues of the equilibrium.\nThis starting procedure is relatively standard, and essentially the same \nas the one used for Hopf bifurcation. \nInitially only a small portion of the periodic solution family is computed;\nin fact only a few continuation steps are taken. \nA minor adjustment of the standard starting procedure ensures that the \nsmall-amplitude starting solution satisfies in particular the conditions \n$y_{n}(0)=0$ and $\\int_{0}^{1} y_{n}(t)~dt = 0$.\nKeeping $y_{n}(0)=0$, the small-amplitude starting solution is followed \nuntil $x_{n}(0)$ reaches a specified target value, for which we have used \nvalues such as $x_{n}^{0}=-0.04$, $x_{n}^{0}=0.005$, and $x_{n}^{0}=0.0001$\n(see Table~1).\nThe free continuation parameters in this step include $T$, $x_{n}^{0}$,\n$p_{1}$, and $p_{2}$. \nThe resulting periodic orbit has the property that all nine solution \ncomponents pass near the origin in the complex plane. \nThe subsequent main computational step then consists of keeping $x_{n}^{0}$ \nfixed at the value $x_{n}^{0}$, while allowing the amplitude parameter \n$a$ to vary. \nSpecifically, the free continuation parameters now include \n$T$, $a$, $p_{1}$, and $p_{2}$. \nIn the quest for locating interesting, stable  periodic solutions of the \nDNLSE, there are multiple variations on the continuation scheme outlined \nabove. \nFor example, one can start from a selected periodic solution from the main\ncomputational step, now keeping the conserved quantity $H$ fixed. \nYet another variation that we use is to follow periodic solutions found \nin the main computational step above, keeping the resonance ratio \n$T\/T_0$ fixed. Here $T$ is the period of the periodic orbit and $T_{0}$ \nis the period of the rotating frame.\nIn fact, along all families of periodic solutions we monitor the value \nof the ratio $T\/T_{0}$.\nSpecifically we are interested in rational values of $T\/T_{0}$, for which \nthe orbits correspond to a choreographies in the non-rotating frame.\nAs discussed in Section~2, there is a countably infinite number of such\nchoreographies along families of periodic orbits, provided that the \nthe orbits possess certain symmetries, and provided the period $T$ is \nnot constant.\nMoreover, as already mentioned above, such choreographies can subsequently\nbe continued with varying amplitude parameter $a$, while keeping the ratio\n$T\/T_{0}$ fixed at a choreographic value.\nThus the number of choreographies then becomes in fact uncountably infinite.\n\nTwo choreographies for $n=9$ are shown in Figure~\\ref{fig1}, namely \nin the top-right panel and center-right panel of that Figure.\nIn addition, all twelve orbits shown in Figure~\\ref{fig2} correspond to\nchoreographies.\nSpecifically, in Figure~\\ref{fig1}, the top-left panel shows a resonant \norbit in the rotating frame.\nThe coloring of this orbit is according to its nine components, \nand evidently it consists of nine separate closed curves.\nThe top-right panel of Figure~\\ref{fig1} shows the same orbit in the\nnon-rotating frame, where all components follow a single curve, {\\it i.e.},\nthe orbit is a choreography.\nThe coloring of the choreography is also according to its nine components.\nHowever, since all components follow the same curve, the color of this\ncurve changes gradually to a uniform final color, as one complete orbit\nis traversed.\n\nSimilarly, the center-left panel of Figure~\\ref{fig1} shows an orbit in\nthe rotating frame, while the center-right panel shows the same orbit in\nthe non-rotating frame, where it evidently corresponds to a choreography.\nThe bottom panels of Figure~\\ref{fig1} show two resonant orbits in the\nnon-rotating frame.\nAs the coloring indicates, neither of these two orbits is a choreography.\nHowever, both can be designated as a {\\it partial choreography}, since\neach consists of three separate closed curves, and each of these three\ncurves is traversed by three components.\nFinally, Figure~\\ref{fig2} shows a selection of twelve orbits in the\nnon-rotating frame, each of which is a choreography.\nThese choreographies are visually appealing, and particularly interesting \nto watch in animations.\n\n\\section{Conclusions and discussion}\n\\label{sec:3}\nThe DNLS equations with $n$ sites, and with periodic boundary conditions, \nhave Lyapunov families of periodic solutions that arise from polygonal \nequilibria in the rotating frame. These Lyapunov families can be \nparameterized by the rotational frequency $\\omega$ and the frequency \n$\\nu$ of the Lyapunov orbit. \nWhen the ratio of the freqencies $\\omega$ and $\\nu$ is $l:m$ resonant, \n{\\it i.e.}, when $\\omega\/\\nu=l\/m$, then the Lyapunov orbit corresponds \nto a choreography that is symmetric with respect rotations by $2\\pi\/m$, \nand that has winding number $l$. \nFor fixed $\\omega$ this resonance condition is satisfied for a dense set \nof rational frequencies $\\nu$.\nThus if the frequency range of $\\nu$ along a Lyapunov family contains \nan interval, then there is an infinite number of Lyapunov orbits that \ncorrespond to choreographies.\n\nIn this paper a robust and highly accurate boundary value technique with \nadaptive meshes has been used to continue the Lyapunov families. \nThe presence of two conserved quantities, namely the amplitude $A$ and the \nenergy $E$, is dealt with by using two unfolding parameters. \nThe formulation also allows continuation of solution families with fixed \n$E$, $A$, or $\\omega\/\\nu$. \nFor example, by fixing $\\omega\/\\nu=l\/m$ one can compute a continuum of\nchoreographies, each of which is symmetric with respect to rotation by\n$2\\pi\/m$, and has winding number $l$.  We have included a small sample \nof the infinitely many stable choreographies that can be computed \nin this manner.\n\nIn principle, physical observation of stable choreographies appears to be\npossible with heterodyne optical techniques.\nSuch techniques have been used experimentally to record both the phase \nand the amplitude of a coherent optical signal source, such as that at \nthe end of a waveguide array, as modeled by the DNLSE. \nThe basic idea is to study the optical field composed of a reference and \na source field.\nThis optical technique has allowed the confirmation of predictions from\nthe Lorenz model, which describes to a good degree the dynamics of the \n$NH_{3}$-laser \\cite{Opt2}. \nSimilarly, heterodyne optical techniques have been used to study optical \nfields in two-dimensional light sources \\cite{Opt3}, which are precisely \nthe geometries that support stable choreographies.\n\\clearpage\n\\begin{figure}[ptb]\n\\begin{center}\n\\resizebox{16.0cm}{!}{  \n\\includegraphics{f1a.png}~\n\\includegraphics{f1b.png} }\n\\end{center}\n\\par\n\\vskip-.90cm\\noindent\n\\par\n\\begin{center}\n\\resizebox{16.0cm}{!}{  \n\\includegraphics{f1c.png}~\n\\includegraphics{f1d.png} }\n\\end{center}\n\\par\n\\vskip-.90cm\\noindent\n\\par\n\\begin{center}\n\\resizebox{16.0cm}{!}{  \n\\includegraphics{f1e.png}~\n\\includegraphics{f1f.png} }\n\\end{center}\n\\par\n\\vskip-.5cm\\noindent\n\\par\n\\caption{ \nAll solutions in this figure are for $n=9$ and, to numerical \naccuracy, linearly stable.\nData are given in Table~1.\nTop-Left:\nA periodic solution in the rotating frame, of resonance 1:10. \nTop-Right:\nThe corresponding periodic solution in the non-rotating frame, \nwhere it is a choreography. \nCenter-Left: \nA periodic solution in the rotating frame, of resonance 23:1. \nCenter-Right:\nThe corresponding choreography.\nBottom-Left:\nA partial choreography in the non-rotating frame, of resonance 2:5.\nBottom-Right: \nA partial choreography in the non-rotating frame of resonance 5:8.\n}\n\\label{fig1}\n\\end{figure}\n\\clearpage\n\\begin{figure}[ptb]\n\\begin{center}\n\\resizebox{17.0cm}{!}{  \n\\includegraphics{f2a.png}~~\n\\includegraphics{f2b.png}~~\n\\includegraphics{f2c.png} }\n\\end{center}\n\\par\n\\vskip-.90cm\\noindent\n\\par\n\\begin{center}\n\\resizebox{17.0cm}{!}{  \n\\includegraphics{f2d.png}~\n\\includegraphics{f2e.png}~\n\\includegraphics{f2f.png} }\n\\end{center}\n\\par\n\\vskip-.90cm\\noindent\n\\par\n\\begin{center}\n\\resizebox{17.0cm}{!}{  \n\\includegraphics{f2g.png}~\n\\includegraphics{f2h.png}~\n\\includegraphics{f2i.png} }\n\\end{center}\n\\par\n\\vskip-.90cm\\noindent\n\\par\n\\begin{center}\n\\resizebox{17.0cm}{!}{  \n\\includegraphics{f2j.png}~\n\\includegraphics{f2k.png}~\n\\includegraphics{f2l.png} }\n\\end{center}\n\\par\n\\vskip-.5cm\\noindent\n\\par\n\\caption{ \nThe choreographies in the first two rows are for $n=9$,\nthe choreographies in the third row are for $n=17$, \nand the choreographies in the last row are for $n=31$.\nAll but two choreographies in this Figure are stable.\nThe two unstable choreographies are on the right in rows 1 and 2.\n}\n\\label{fig2}%\n\\end{figure}\n\\clearpage\n\\vskip.0cm\\noindent\n\\begin{table}[H]\n\\begin{center}\n\\begin{large}\n\\begin{tabular}{|c|c|c||c|c|c|c|c|c|c|}\n\\hline\n      &   &   &   &       &          &         &          &         &   \\cr\nFigure&Row-&Orbit-&$n$&$T\/T_0$& a    & $T$     &   $T_0$  &$x_n(0)$ &S\/U\\cr\n      &Col.&Label&&        &          &         &          &         &   \\cr\n\\hline\n      &1-1& 1 & 9 & 1:10  & 0.651774 & 14.5773 &  145.773 & -0.04   & S \\cr\n      &1-2& 2 & 9 & 1:10  & 0.651774 & 14.5773 &  145.773 & -0.04   & S \\cr\n   1  &2-1& 3 & 9 & 23:1  & 0.657102 & 4000.00 &  173.913 &  0.005  & S \\cr\n      &2-2& 4 & 9 & 23:1  & 0.657102 & 4000.00 &  173.913 &  0.005  & S \\cr\n      &3-1& 5 & 9 & 2:5   & 0.520316 & 12.7459 &  31.8649 & -0.04   & S \\cr\n      &3-2& 6 & 9 & 5:8   & 0.396319 & 12.6334 &  20.2134 & -0.04   & S \\cr\n\\hline\n      &1-1& 7 & 9 & 1:10  & 0.647930 & 13.0635 &  130.635 & -0.04   & S \\cr\n      &1-2& 8 & 9 & 1:10  & 0.646671 & 12.6423 &  126.353 & -0.04   & S \\cr\n      &1-3& 9 & 9 & 1:10  & 0.627791 & 8.51610 &  85.1505 & -0.04   & U \\cr\n      &2-1&10 & 9 & 2:11  & 0.510285 & 5.50498 &  30.2774 & -0.04   & S \\cr\n      &2-2&11 & 9 & 2:11  & 0.531986 & 6.17839 &  33.9811 & -0.04   & S \\cr\n   2  &2-3&12 & 9 & 2:11  & 0.565906 & 7.73660 &  42.5513 & -0.04   & U \\cr\n      &3-1&13 &17 & -8:9  & 0.576588 & 28.2933 & -31.8299 &  0.005  & S \\cr\n      &3-2&14 &17 & -8:9  & 0.578005 & 28.0608 & -31.5684 &  0.005  & S \\cr\n      &3-3&15 &17 & -6:11 & 0.505528 & 28.4406 & -52.1411 &  0.005  & S \\cr\n      &4-1&16 &31 & -15:16& 0.421561 & 43.0673 & -45.9385 &  0.0001 & S \\cr\n      &4-2&17 &31 & -15:16& 0.421549 & 43.0706 & -45.9420 &  0.0001 & S \\cr\n      &4-3&18 &31 & -15:16& 0.420348 & 43.3913 & -46.2840 &  0.0001 & S \\cr\n\\hline\n\\end{tabular}\n\\end{large}\n\\caption{\nData for the orbits in Figures~1 and 2.\nHere $n$ is the number of sites,\n$a$ the amplitude parameter, \n$T$ the period of the orbit,\n$T_0$ the period of the rotating frame,\n$x_n(0)$ the $x$-component of the $n$th site at time zero,\n\"S\" stands for \"Stable or almost stable\", and\n\"U\" stands for \"Unstable\".\n}\n\\end{center}\n\\end{table}\n\n\\section*{Acknowledgments}\nThis work was supported by NSERC (Canada), BUAP and CONACYT (M\\'{e}xico).\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSpin crossover (SCO) is a phenomenon which takes place when the metal ion changes its spin state between low spin (LS) and high spin (HS) configuration under the effect of external perturbation such as pressure, magnetic field, temperature, or light irradiation. The SCO can be observed in transition metal compounds (often in the 3d-metal oxides with $d^4$-$d^7$ electronic configurations) \\cite{Halder, Brooker, LyubutinUFN, Ohkoshi} or in transition metal complexes, like metalorganic molecules or molecular assemblies \\cite{Saha-Dasgupta}. Free inertial molecular switches to store and process information in fast computational devices were the primary interest for SCO. In the nanotechnology certain properties of the SCO are of the interest for quantum transport and a new generation of sensors and displays \\cite{Jureschi}. The SCO in Fe-containing oxides is also important for the understanding the physical properties of the Earth's mantle \\cite{Wentzcovitch, Hsu, Liu, Ovchinnikov12, Sinmyo}.\n\nAt first glance the SCO is a problem of an individual ion and results from the competition of the Hund intra-atomic exchange interaction and the crystal field value determined by surrounding ions. Nevertheless, the effective interaction between magnetic ions due to electron-phonon, exchange, and quadrupole couplings results in cooperative effects, which provide different hysteresis phenomena and play an important role in practical applications and understanding the origin of the SCO. There are many papers where the cooperative effects have been treated within the Ising model \\cite{Jureschi, Wajnflasz, Bari, Nishino03, Timm, Banerjee, Paez-Espejo}. In all these studies the effective exchange interaction is postulated phenomenologically within the Ising or Heisenberg model with empirical exchange parameters. In the last decade the cooperative effects in SCO have been studied by the density functional theory \\cite{Marbeuf}, molecular dynamics \\cite{Nishino07, Boukheddaden}, and Monte Carlo simulations \\cite{Konishi, Miyashita}. The interplay of electron hopping between neighboring ions with the orbital structure of different spin multiplets also results in spin-orbital cooperative effects in strongly correlated transition metal oxides \\cite{Sboychakov}.\n\nIn conventional magnetic insulators only the ground term $E_0$ of magnetic cation in the multielectron configuration $d^n$ with some spin value $S_0$ is involved in the formation of the Heisenberg Hamiltonian as the effective low-energy model. The important difference of the magnetism in SCO systems is that at least two different terms, usually HS and LS, are involved in the formation of the effective low energy model. This is a reason for the non-Heisenberg model effects that will be discussed in this paper. Recently we have developed a general approach to construct the effective exchange interaction model that takes into account the contribution of the excited terms of the magnetic cation \\cite{Gavrichkov17} and found that the interatomic exchange interaction results in the SCO to be the first order phase transition \\cite{Nesterov}. For arbitrary $d^n$ configuration we cannot write down analytically the parameters of the effective Hamiltonian that contains the interatomic exchange as well as the interatomic hopping of excitons, the excitations between HS and LS terms.\n\nIn this paper we study more simple toy model with two electronic orbitals and the Coulomb interaction in the Kanamori approach \\cite{Kanamori}. Within the generalized tight binding (GTB) method \\cite{Ovchinnikov89, Gavrichkov00} to the electronic structure of strongly correlated systems we provide the exact diagonalization of the local intraatomic part of the Hamiltonian, construct the Hubbard operators using a set of the exact local eigenstates, and write down the total Hamiltonian as the multiorbital Hubbard model. This model describes a magnetic insulator with the energy gap $E_g$ between the occupied valence and empty conductivity bands. Two electrons per site form the HS triplet and LS singlets with the SCO at increasing the crystal field splitting between two orbitals (for example, by external pressure). We should mention that similar models under different names (the two-band Hubbard model or the extended Falicov-Kimball model) have been intensively discussed in the literature, see the review paper \\onlinecite{Kunes15}. We write down explicitly the matrix elements of the exchange and exciton hopping contributions, which are beyond the conventional Heisenberg model. The other non-Heisenberg model effect is related to a structure of the local Hilbert space, which contains for our model 3 magnetic eigenstates for HS with $S=1$ and 3 singlets with $S=0$. Within the two-band Hubbard model similar strong coupling approach \\cite{Werner, Suzuki, Nasu} has also revealed two terms with $S=1$ and $S=0$ for electronic concentration $n_e=2$ and the intersite interaction matrix elements (see also Refs.~\\onlinecite{Balents, Kaneko, Kunes14}). The main object of all these papers is the possible excitonic condensation in systems of strongly correlated electrons. In our paper we restrict our interest to the SCO systems and possible non Heisenberg effects. The presence of the additional LS states does not allow introducing the Brillouin function in the mean field (MF) approximation. A small number of electrons in our toy model ($n_e=2$ per site) allows us to study the model's phase diagram applying a cluster mean field (CMF) approach in order to go beyond the standard MF. In this way we can obtain qualitative information about the model's phase diagram and explore the validity of approximation by considering different cluster sizes as well as discuss the short-order effects, which are also different from the conventional Heisenberg model, in the vicinity of the first order transition from the HS antiferromagnetic phase into the LS non magnetic phase due to local nature of SCO.\n\nThe paper is organized as follows. In section \\ref{sec:2} we describe the two-orbital Kanamori model, the effective low energy Hamiltonian containing HS and LS states, and interatomic exchange interaction and exciton hopping. In section \\ref{sec:3} we briefly remind the CMF theory. The non-Heisenberg model and short-order effects in the vicinity of spin crossover are discussed in section \\ref{sec:4}. In section \\ref{sec:5} we discuss the main results. \n\n\\section{\\label{sec:2}Two-band Kanamori model}\n\nThe multielectron states for the $d^n$-configuration in the cubic crystal field can be obtained from the Tanabe-Sugano diagrams \\cite{Tanabe, Sugano}, which demonstrate stability of the HS terms for small value of the crystal field $10Dq$, and that the crossover of the HS and LS terms takes place for $d^4$-$d^7$ electronic configurations with increasing the crystal field value stabilizing the LS state. Beyond the crystal field theory, the SCO may also happen due to increasing the cation-anion $p$-$d$ hybridization \\cite{Ovchinnikov07}. The minimal multielectron model to discuss SCO is the two-orbital tight-binding model that includes two single electron levels $\\varepsilon_1$ and $\\varepsilon_2$ with interatomic hopping $t_{i,j}$ and the local Coulomb interaction for electron concentration $n_e=2$. Its Hamiltonian is given by\n\\begin {equation}\nH = H_t + H_{Coulomb}.\n\\label {eq:1}\n\\end {equation}\nThe interatomic term\n\\begin {eqnarray}\nH_t &=& \\varepsilon_1 \\sum\\limits_{i, \\sigma}{a_{i1\\sigma}^{\\dag}}{a_{i1\\sigma}^{}} + \\varepsilon_2 \\sum\\limits_{i, \\sigma}{a_{i2\\sigma}^{\\dag}}{a_{i2\\sigma}^{}} \\nonumber \\\\\n&+& t_1 \\sum\\limits_{\\left\\langle i, j \\right\\rangle, \\sigma} {a_{i1\\sigma}^{\\dag}}{a_{j1\\sigma}^{}} + t_2 \\sum\\limits_{\\left\\langle i, j \\right\\rangle, \\sigma} {a_{i2\\sigma}^{\\dag}}{a_{j2\\sigma}^{}} \\nonumber \\\\\n&+& t_{12} \\sum\\limits_{\\left\\langle i, j \\right\\rangle, \\sigma}\\left( {a_{i2\\sigma}^{\\dag}}{a_{j1\\sigma}^{}} + {a_{i1\\sigma}^{\\dag}}{a_{j2\\sigma}^{}} \\right)\n\\label {eq:2}\n\\end {eqnarray}\ndescribes the intraband $t_1$ and $t_2$ hoppings and the interband hopping $t_{12}$ of electrons between the nearest neighbor sites with the single electron energies $\\varepsilon_1$ and $\\varepsilon_2 = \\varepsilon_1 + \\Delta$, where $\\Delta$ is the crystal field value. The local Coulomb interaction within the Kanamori approach contains different matrix elements, the intraorbital $U$ and interorbital $V$, as well as the Hund coupling $J$ and the interband coupling $J'$:\n\\begin{widetext}\n\\begin {eqnarray}\nH_{Coulomb} = U \\sum\\limits_{i, \\lambda}{a_{i\\lambda\\uparrow}^{\\dag}}{a_{i\\lambda\\downarrow}^{\\dag}}{a_{i\\lambda\\uparrow}^{}}{a_{i\\lambda\\downarrow}^{}} + V \\sum\\limits_{i, \\lambda\\neq\\lambda'}{a_{i\\lambda\\uparrow}^{\\dag}}{a_{i\\lambda'\\downarrow}^{\\dag}}{a_{i\\lambda\\uparrow}^{}}{a_{i\\lambda'\\downarrow}^{}} + V \\sum\\limits_{i, \\lambda>\\lambda'}{a_{i\\lambda\\sigma}^{\\dag}}{a_{i\\lambda'\\sigma}^{\\dag}}{a_{i\\lambda\\sigma}^{}}{a_{i\\lambda'\\sigma}^{}} \\nonumber \\\\\n+ J \\sum\\limits_{i, \\lambda>\\lambda',\\sigma}{a_{i\\lambda\\sigma}^{\\dag}}{a_{i\\lambda'\\sigma}^{\\dag}}{a_{i\\lambda'\\sigma}^{}}{a_{i\\lambda\\sigma}^{}} + J \\sum\\limits_{i, \\lambda\\neq\\lambda'}{a_{i\\lambda\\uparrow}^{\\dag}}{a_{i\\lambda'\\downarrow}^{\\dag}}{a_{i\\lambda'\\uparrow}^{}}{a_{i\\lambda\\downarrow}^{}} + J' \\sum\\limits_{i, \\lambda\\neq\\lambda'}{a_{i\\lambda\\uparrow}^{\\dag}}{a_{i\\lambda\\downarrow}^{\\dag}}{a_{i\\lambda'\\uparrow}^{}}{a_{i\\lambda'\\downarrow}^{}} \n\\label {eq:3}\n\\end {eqnarray}\n\\end{widetext}\nIn the limit  $\\Delta=0$ and for one electron per site this model transforms in the Kugel-Khomskii model for charge ordering \\cite{Kugel82}. In this paper we will consider this model only for homopolar case $n_e=2$. As we have mentioned in the introduction, similar models have been studied recently to find the excitonic insulator phase.\n\nFor zero interatomic hopping there are 6 exact two-electron states. The triplet ($S=1$)\n\\begin{equation}\n\\left|\\sigma\\right\\rangle=\\begin{cases}\n               a^{\\dag}_{1\\uparrow}a^{\\dag}_{2\\uparrow}\\left|0\\right\\rangle, \\sigma = +1 \\\\\n               \\frac{1}{\\sqrt{2}}\\left(a^{\\dag}_{1\\uparrow}a^{\\dag}_{2\\downarrow}+a^{\\dag}_{1\\downarrow}a^{\\dag}_{2\\uparrow}\\right)\\left|0\\right\\rangle, \\sigma = 0 \\\\\n               a^{\\dag}_{1\\downarrow}a^{\\dag}_{2\\downarrow}\\left|0\\right\\rangle, \\sigma = -1\n            \\end{cases}\n\\label {eq:4}\n\\end{equation}\ntriply degenerate HS-term $\\left|\\sigma\\right\\rangle$ with the energy $E_{HS}=2\\varepsilon_1+\\Delta+V-J$ is the ground state for the crystal field  (Fig.\\ref{fig:1}, red dashed line), for $\\Delta>\\Delta_c$   the singlet ($S=0$) LS state \n\\begin{equation}\n\\left|S\\right\\rangle=C_1\\left(\\Delta\\right)a^{\\dag}_{1\\uparrow}a^{\\dag}_{1\\downarrow}\\left|0\\right\\rangle - \\sqrt{1-C^2_1\\left(\\Delta\\right)}a^{\\dag}_{2\\uparrow}a^{\\dag}_{2\\downarrow}\\left|0\\right\\rangle,\n\\label {eq:5}\n\\end{equation}\nwhere $C_1\\left(\\Delta\\right)=J'\/\\sqrt{{J'}^2 - \\left(2\\varepsilon_1+U-E_{LS}\\right)^2}$, with the energy $E_{LS}=2\\varepsilon_1+\\left(\\Delta+U\\right)-\\sqrt{\\Delta^2-{J'}^2}$ becomes the ground state (Fig.\\ref{fig:1}, green dotted line). The crossover occurs at $\\Delta=\\Delta_c=\\sqrt{\\left(U-V+J\\right)^2-{J'}^2}$. There are two more singlets,\n\\begin{equation}\n\\left|S_1\\right\\rangle=\\frac{1}{\\sqrt{2}}\\left(a^{\\dag}_{1\\uparrow}a^{\\dag}_{2\\downarrow}-a^{\\dag}_{1\\downarrow}a^{\\dag}_{2\\uparrow}\\right)\\left|0\\right\\rangle\n\\label {eq:6}\n\\end{equation}\nwith the energy $E_{S_1}=2\\varepsilon_1+\\Delta+V+J$ and \n\\begin{equation}\n\\left|S_2\\right\\rangle=\\left(\\sqrt{1-C^2_1\\left(\\Delta\\right)}a^{\\dag}_{1\\uparrow}a^{\\dag}_{1\\downarrow}+C_1\\left(\\Delta\\right)a^{\\dag}_{2\\uparrow}a^{\\dag}_{2\\downarrow}\\right)\\left|0\\right\\rangle\n\\label {eq:7}\n\\end{equation}\nwith the energy $E_{S_2}=2\\varepsilon_1+\\left(\\Delta+U\\right)+\\sqrt{\\Delta^2-{J'}^2}$, which are excited for all parameters; they are shown by the solid black lines in Fig.\\ref{fig:1}.\n\\begin{figure}\n\\includegraphics{Fig1}\n\\caption{\\label{fig:1}The crystal field dependence of the two-electron local eigenstates. The red dashed line shows the ground HS term for $\\Delta < \\Delta_c$, the green dotted line indicates the ground LS term for $\\Delta > \\Delta_c$, black solid lines correspond to the high-energy singlets. Calculation has been carried out for the following parameters: $U=3 \\text{eV}$, $V=1 \\text{eV}$, $J=0.7 \\text{eV}$, and $J=0.7 \\text{eV}$.}\n\\end{figure}\n\nTo treat the intersite electron hopping we use the GTB approach \\cite{Ovchinnikov89, Gavrichkov00, Korshunov}, which is a version of cluster perturbation theory. We introduce the Hubbard $X$-operators $X^{pq}=\\left|p\\right\\rangle\\left\\langle q\\right|$, where where $\\left|p\\right\\rangle$ and $\\left|q\\right\\rangle$ are the eigenstates of the Hamiltonian (\\ref{eq:1}) at $t_{\\lambda\\lambda'}=0$ with different numbers of electrons $n_e = 1,2,3$. A single electron creation\/annihilation operator at site $i$ with an orbital index $\\lambda$ as well as any other local operator is given by a linear combination of the Hubbard operators \\cite{Hubbard}:\n\\begin{equation}\na_{i\\lambda\\sigma}=\\sum\\limits_{pq}\\left|p\\right\\rangle\\left\\langle p\\right\\rangle a_{i\\lambda\\sigma} \\left|q\\right\\rangle \\left\\langle q\\right| = \\sum\\limits_{pq}\\gamma_{\\lambda\\sigma}\\left(pq\\right)X^{pq}_i.\n\\label {eq:8}\n\\end{equation}\nThe number of different quasiparticles $\\left(pq\\right)$ is finite, one can numerate them by the number $m$, which is the quasiparticle band index, then $a_{i\\lambda\\sigma}=\\sum\\limits_m \\gamma_{\\lambda\\sigma}\\left(m\\right)X^{m}_i$ $\\left(a^{\\dag}_{i\\lambda\\sigma}=\\sum\\limits_m \\gamma^*_{\\lambda\\sigma}\\left(m\\right){X_i^{m}}^{\\dag}\\right)$.\n\nIn the $X$-operator representation the Hamiltonian (\\ref{eq:1}) can be written exactly as \n\\begin{equation}\nH = \\sum\\limits_{i,p}E_p X^{pp}_i + \\sum\\limits_{\\left\\langle i,j\\right\\rangle}\\sum\\limits_{mn}t^{mn}{X_i^{m}}^{\\dag} X^{n}_j.\n\\label {eq:9}\n\\end{equation}\nHere $E_p$ is the energy of the term $\\left|p\\right\\rangle$, $t^{mn} = \\sum\\limits_{\\sigma,\\lambda,\\lambda'}t_{\\lambda\\lambda'}\\gamma^*_{\\lambda\\sigma}\\left(m\\right)\\gamma_{\\lambda'\\sigma}\\left(n\\right)$ is the intersite hopping matrix element. We would like to emphasize that the Hamiltonian (\\ref{eq:9}) is the general multielectron Hamiltonian that is valid for any complete and orthonormalized set of local eigenstates, all microscopic details are given by the structure of local eigenstates.\n\nFor number of electrons $n_e=2$ the Hamiltonian (\\ref{eq:9}) results in the Mott-Hubbard insulator ground state with the insulator band gap $E_g$. The localized magnetic moment at each site is HS for $\\Delta<\\Delta_c$ and LS for $\\Delta>\\Delta_c$. To obtain the interatomic exchange interaction we apply the method developed for the Hubbard model \\cite{Chao} and generalized for arbitrary set of local eigenstates in \\cite{Gavrichkov17} (see also Refs. \\onlinecite{Kunes15, Nasu}). The idea is to construct the effective Hamiltonian excluding the interband interatomic hopping. Contrary to the general case, in our toy model we can write down the exchange interaction analytically. The effective Hamiltonian is equal to\n\\begin{equation}\nH_{eff} = H_{s} + H_{ex}.\n\\label{eq:10}\n\\end{equation}\nHere the first term is the spin Heisenberg-type Hamiltonian, while the second term describes the non-Heisenberg intersite hopping of the local excitons. This Hamiltonian acts within the Hilbert space that contains four states: three $S=1$ triplet states $\\left|-\\right\\rangle$, $\\left|0\\right\\rangle$, $\\left|+\\right\\rangle$ and the singlet state $\\left|s\\right\\rangle$. The spin part is given by\n\\begin{equation}\nH_{s} = \\frac{J}{2} \\sum\\limits_{\\left\\langle i,j\\right\\rangle}\\left({\\bf S}_i {\\bf S}_j-\\frac{1}{4}n_in_j\\right)-\\varepsilon_s \\sum\\limits_i X^{ss}_i,\n\\label{eq:11}\n\\end{equation}\nwhere the superexchange parameter is \n\\begin{equation}\nJ=4\\left(t^2_{11}+2t^2_{12}+t^2_{22}\\right)\/E_g,\n\\label{eq:12}\n\\end{equation}\n${\\bf S}_i$ is the $S=1$ spin operator, in the Hubbard operators given by $S^+_i=\\sqrt{2}\\left(X^{+0}_i + X^{0-}_i\\right)$, $S^-_i=\\sqrt{2}\\left(X^{0+}_i + X^{-0}_i\\right)$, $S^z_i=\\sqrt{2}\\left(X^{++}_i - X^{--}_i\\right)$, and $n_i=q_e\\left(X^{++}_i + X^{--}_i + X^{00}_i + X^{ss}_i\\right)$ is the number of electrons operator, $q_e=2$ is the number of electrons per site, in our homopolar case the completeness of our two-electron exact set of eigenvectors looks like\n\\begin{equation}\nX^{++}_i + X^{--}_i + X^{00}_i + X^{ss}_i = 1,\n\\label{eq:13}\n\\end{equation}\nso $n_i=2$. The last term in the Hamiltonian $H_s$ (\\ref{eq:11}) is the non-Heisenberg contribution of the nonmagnetic LS state with the spin gap value $\\varepsilon_s=E_{HS}-E_{LS}$. This is the local exciton energy. Below we will assume the linear dependence of the crystal field parameter on the external pressure: $\\Delta = \\Delta(0) + aP$  due to the linear decrease of crystal volume under the pressure.\n\nThe creation\/annihilation of the local excitons is given by the Hubbard operators $X^{\\sigma s}_i$ (from the initial LS state $\\left|s\\right\\rangle$ in the final HS state $\\left|\\sigma\\right\\rangle$, and $X^{s\\sigma}_i$ corresponds to the back excitation. These excitons describe the fluctuations of multiplicity, the term used many years ago in the paper \\cite{Vonsovskii}. We consider this term is the appropriate one in the spin crossover physics, the term spin fluctuations in magnetism usually means the change of a spin projection for the same value of the spin. The second part of the effective Hamiltonian (\\ref{eq:10}) describes the intersite exciton hopping\n\\begin{eqnarray}\nH_{ex} = \\frac{J_{ex}}{2}\\sum\\limits_{\\left\\langle i,j \\right\\rangle, \\sigma} \\left[ X^{\\sigma s}_i X^{s\\sigma}_j + X^{s\\sigma}_i X^{\\sigma s}_j \\right. \\nonumber \\\\\n\\left. - \\left(-1\\right)^{\\left|\\sigma\\right|} \\left(X^{\\sigma s}_i X^{\\bar{\\sigma}s}_j + X^{s \\sigma}_i X^{s \\bar{\\sigma}}_j    \\right) \\right],\n\\label{eq:14}\n\\end{eqnarray}\nwhere the exciton hopping parameter is \n\\begin{equation}\nJ_{ex}=4\\left(t^2_{12}-t_{11}t_{22}\\right)\/E_g.\n\\label{eq:15}\n\\end{equation}\nOne can note that due to the orthogonality of the HS and LS terms they do not mix locally, but the exciton hopping mix them non locally. The first line in Eq.~\\ref{eq:14} describes the intersite single particle exciton hopping, while the second line corresponds to the creation and annihilation of the biexciton pair. We can compare the exciton hopping parameter $J_{ex}$ with similar terms in the effective low-energy models in the literature. In the paper~\\onlinecite{Kunes14} the biexciton excitation is possible only due to the interband cross-hopping matrix element $t_{12}$.In the paper~\\onlinecite{Nasu} the cross-hopping is not considered, nevertheless the biexciton hopping is possible due to the product $t_1t_2$. As we can see from Eq.~\\ref{eq:15}, we have both contributions.\n\nLet us compare two nonlocal parameters of the effective Hamiltonian (\\ref{eq:10}), the values of the exchange $J$ (\\ref{eq:12}) and exciton hopping $J_{ex}$ (\\ref{eq:15}). We consider four different sets of the electron hopping parameters:\\\\*\nA) in the limit $\\Delta = \\infty$, $t_{12}=t_{22}=0$, we get $J=4t^2_{11}\/E_g$ and $J_{ex}=0$ as in the single-band Hubbard model \\cite{Anderson}, \\\\*\nB) symmetrical hopping parameters $t_{11}=t_{22}=t_{12}=t$, then the exchange value $J=16t^2\/E_g$ is proportional to the superexchange parameter from the Hubbard model, while the exciton hopping $J_{ex}=0$,\\\\*\nC) $t_{12}=0$, then $J=4\\left(t_{11}^2+t_{22}^2\\right)\/E_g$ and $J_{ex}=-4t_{11}t_{22}\/E_g$, they have opposite signs,\\\\*\nD) $t^2_{12} \\gg t_{11}t_{22}$, then $J=8t_{12}^2\/E_g$ and $J_{ex}=4t_{12}^2\/E_g$, they are of the same order in magnitude.\\\\*\nThese examples and the general expression for the superexchange parameter $J$ demonstrate that antiferromagnetic type of superexchange takes place in our model for all electron hopping parameters, while the hopping of excitons may be positive, negative, and zero.\n\nIn the rest of the paper the unimportant term $n_i n_j=4$ for our homopolar case will be omitted from the Hamiltonian. Due to qualitative aim of our paper we will study the effects of the non-Heisenberg contributions and short-order fluctuations given by the spin part (\\ref{eq:11}) of the effective Hamiltonian (\\ref{eq:10}) with antiferromagnetic exchange parameter, neglecting the exciton dispersion given by the hopping term (\\ref{eq:14}). We will restrict ourselves by the symmetrical set B of the hopping parameters, so the exciton hopping parameter~\\ref{eq:15} will be zero. Nevertheless, basic exciton processes are still taken into account due to LS term $-\\varepsilon_s \\sum\\limits_{i} X^{ss}_i$ in the Hamiltonian (\\ref{eq:11}), which introduces some new non-Heisenberg model effects. Let us illustrate this statement using a simple example. Within MF approximation the Hamiltonian is given by \n\\begin{equation}\nH_{MF}=\\sum\\limits_{m=-1}^{1}E_m\\sigma X^{mm} - \\varepsilon_s X^{ss},\n\\label{eq:16}\n\\end{equation}\nwhere $m=-1,0,1$ are triplet states, $z$ is the number of nearest neighbors, $E_m=Jz\\sigma m$, so the 3 triplet energy levels $E_m$ are $Jz\\sigma$, 0, $-Jz\\sigma$, $\\sigma$ is the positive sublattice magnetization. Thus, the MF magnetization is\n\\begin{equation}\n\\sigma = \\frac{\\exp(\\beta J z \\sigma) - \\exp(-\\beta J z \\sigma)}{\\sum\\limits_{m=-1}^{1}{\\exp(-\\beta J z \\sigma m)} + \\exp(\\beta \\varepsilon_s)},\n\\label{eq:17}\n\\end{equation}\nwhich deviates from the Brillouin function due to the LS term. From the other hand, let us consider the exciton Green functions\n\\begin{equation}\nG^m_{ij}=\\left\\langle \\left\\langle X_i^{sm}|X_j^{ms}\\right\\rangle\\right\\rangle,\n\\label{eq:18}\n\\end{equation}\nwhich describe three types of excitons. After writing down the equations of motion and decoupling them using Tyablikov approximation, we have obtained\n\\begin{equation}\nG_m\\left(E+i\\delta\\right)=\\left(E-\\varepsilon_s+E_m+i\\delta\\right)^{-1}.\n\\label{eq:19}\n\\end{equation}\nThus, the three excitons with spin projection $m=+1,0,-1$ will have the energies $E_{ex}\\left(m\\right)=E_m-E_s$. This way, at finite temperature the occupation numbers of our HS sublevels can be found from the equation\n\\begin{equation}\nn_m=\\left(n_s - n_m\\right)f_B\\left(E_{ex}\\left(m\\right)\\right),\n\\label{eq:20}\n\\end{equation}\nwhere $f_B\\left(E\\right)$ is the Bose-Einstein distribution function. Together with the completeness condition~\\ref{eq:13} we have the full set of MF equations exactly the same as we obtain from Eq.~\\ref{eq:17}. This way, we see that in the simplest approximation the exciton process are present in the system and give consistent values for the occupation numbers. Below, instead of MF we will use its cluster generalization, in which all possible positions of singlets within the cluster are taken into account.\n\n\\section{\\label{sec:3}Cluster mean field theory}\n\nDue to the LS term, the problem given by the Hamiltonian (\\ref{eq:10}) cannot be straightforwardly treated by the approaches that work well for the Heisenberg model, like Tyablikov approximation \\cite{Bogolyubov, Fu-Cho, Valkov82, Du}, or more sophisticated Green's function approaches \\cite{Kondo, Plakida, Junger04, Junger09}. The simplest approach is to use a MF theory given by the Eq.~\\ref{eq:17}. However, the Heisenberg term contains spin fluctuations, which are neglected within the standard MF consideration. To go beyond MF we use its cluster generalization, the self-consistent CMF, which has been applied to various quantum spin models \\cite{Valkov06, Valkov07, Brzezicki11, Albuquerque, Brzezicki12, Ren, Gotfryd, Ray, Morita, Koga, Singhania}. We believe that CMF method is suitable for a qualitative study of the toy model we consider at a wide range of temperatures and pressure and it is better anyhow than the single site MF. The approach captures short-range effects, which will be discussed in the next section, and allows treating HS and LS terms equally within a cluster. We note that at high temperature close to second-order phase transition the approach can be considered as only qualitative since it does not capture long-range fluctuations. At zero temperature, as will be presented below, CMF provides results which fall into reasonable agreement with more rigorous approaches.\n\nWithin the CMF approach the lattice is covered by translations of a cluster to treat the intracluster interactions by exact diagonalization, whereas the interactions between spins $f$ and $f'$ belonging to different clusters are approximated within MF as ${\\bf S}_f{\\bf S}_f' \\approx S_f^z \\left\\langle S_f'^z\\right\\rangle + \\left\\langle S_f^z\\right\\rangle S_f'^z- \\left\\langle S_f^z\\right\\rangle \\left\\langle S_f'^z\\right\\rangle$. Thus, after applying the translational invariance the problem reduces to a single cluster in a MF determined by parameters $\\left\\langle S_i^z\\right\\rangle$, which are determined self-consistently by iterative diagonalizations ($i$ runs over boundary sites of a cluster). In our calculations we suppose the mean-fields to be in Neel antiferromagnetic ordering, since there are no competing exchange parameters, but there is a competition between the exchange and the spin gap $\\varepsilon_s$, which may be rescaled to pressure. In the main part of the paper we take $J$ as an energy unit and explore the $\\varepsilon_s-T$ phase diagram, where $T$ is temperature. For each value of $\\varepsilon_s$ and $T$ we compare the free energies of the system in magnetic and non-magnetic phases to decide, which of them is realized. A tolerance factor for convergence of $\\left\\langle S_i^z\\right\\rangle$ was set $10^{-5}$. We use full diagonalization at finite temperatures and Lanczos at $T=0$. Since we are dealing with basis consisting of three HS and one LS states, computationaly reasonable sizes of a cluster are $N_c\\lesssim10$, where $N_c$ is number of sites, in the former case and $N_c\\lesssim20$ in the latter. So, we mostly use a $2\\times2$ cluster to illustrate the main physics, but also compare the results using $3\\times2$, $4\\times2$, and $2\\times2\\times2$ clusters to study the finite-size effects of our calculations at finite temperature and clusters $4\\times3$ and $4\\times4$ at zero temperature.\n\n\\section{\\label{sec:4}Non-Heisenberg behavior and short order effects in the vicinity of spin crossover}\n\nIn the main part of this chapter we will discuss the results of our CMF calculations with the spin Hamiltonian (\\ref{eq:11}) in the most interesting regime  $\\varepsilon_s\\sim J$. To compare staggered magnetization obtained with different clusters we will consider the magnetization $m$ on a bulk site, which we define as located as close as possible to the center of a cluster. As known, Fe-based SCO compounds in ambient conditions are 3D magnets. In our cluster calculations it is more numerically practical to consider 2D case, since in 3D only $2\\times2\\times2$ cluster is available. We can use small $2\\times2$ cluster for the main results as well as compare $2\\times2$ CMF with larger clusters. Although in 2D the Mermin-Wagner theorem prohibits an ordered state for the spherically symmetric Hamiltonian (\\ref{eq:11}), in the case of MF-based approach the results for 2D and 3D are qualitatively identical.\n\nAn important quantity characterizing SCO is a HS (LS) concentration. It is accessible in experiments on X-ray emission \\cite{Lin} and $\\text{M}\\ddot{\\text{o}}\\text{ssbauer}$ spectroscopy \\cite{Lyubutin12}. We show in Fig.~\\ref{fig:2} the LS concentration $n_{LS}$ dependence on spin gap and temperature obtained by $2\\times2$ exact diagonalization. It is qualitatively similar to the obtained experimentaly in Ref.~\\cite{Lin} and calculated within MF approaches \\cite{Sturhahn, Lyubutin12} and first-principle studies \\cite{Tsuchiya}. SCO takes place at $\\varepsilon_s = 1.5$ instead of $\\varepsilon = 0$ since intracluster exchange interaction stabilizes the HS state and larger crystal field (pressure) is required to reach SCO. Another effect of correlations is the curvature of the isolines of $n_{LS}$ at low temperatures as shown by colors in Fig.~\\ref{fig:2}. If to neglect the exchange correlations and take the value $J=0$, all lines of the constant value for LS\/HS concentrations will be the straight lines going from the SCO critical point $\\varepsilon_s=0$ \\cite{Sturhahn, Ovchinnikov11}.\n\n\\begin{figure}\n\\includegraphics{Fig2}\n\\caption{\\label{fig:2}The map of the LS occupation number obtained with $2\\times2$ cluster exact diagonalization.}\n\\end{figure}\n\nAs shown in Fig.\\ref{fig:3}(a), at $\\varepsilon_s\\sim-10$ almost Heisenberg behavior of magnetization with temperature is observed, because the system is in the HS state. Thus, a second-order transition from magnetic to nonmagnetic state is realized with heating. From Fig.\\ref{fig:3}(b) one can see that for $\\varepsilon_s\\sim-10$ the population of the LS is zero at low temperature, that provides the conventional Heisenberg model behavior. The nonmagnetic HS phase is the paramagnetic one. With increasing $\\varepsilon_s$ thermal fluctuations enhance LS population, so the second-order transition Neel temperature decreases. At $\\varepsilon_s=0$ the magnetic transition with heating is still the second order, but the paramagnetic moment is reduced by approximately $20\\%$ of the LS states. At $\\varepsilon_s = \\varepsilon_s^*\\approx 1.87$ there is a tricritical point. Increasing $\\varepsilon_s$ further leads to a first-order phase transition to nonmagnetic state caused by the change of the ground state from HS to LS, as seen from Fig.~\\ref{fig:3}(b). The maximal value of magnetization in Fig.~\\ref{fig:3}(a) is $m=0.9528$, instead of $m=1$. This is the manifestation of quantum shortening of spin, which is taken into account partially within CMF by calculating spin-fluctuation terms within a cluster. The non magnetic phase of Fig.\\ref{fig:3}(a) can be qualitatively viewed as HS to the left of the $n_{LS}=0.5$ dashed line, which comes out close to the tricritical point, and LS to the right.\n\nThe distribution of LS density in Fig.\\ref{fig:3}(b) is related to the Curie constant in paramagnetic susceptibility\n\\begin{equation}\nC = \\mu^2\\left(1-n_{LS}\\right)S\\left(S+1\\right),\n\\label{eq:21}\n\\end{equation}\nwhere $\\mu = \\frac{\\mu_B^2}{3k_B}$. The temperature dependence of $C$ is shown in Fig.~\\ref{fig:4} for different values of the spin gap. Equation (\\ref{eq:16}) makes sense for the paramagnetic phase above the Neel temperature indicated in Fig.~\\ref{fig:4}(a) by dashed lines. Using parameters extracted from the anvil-cell experiments on ferropericlase \\cite{Lyubutin12, Lyubutin13} we can estimate the corresponding values of pressure $P$ by assuming that the spin gap defines pressure as $\\varepsilon_s-\\varepsilon_s^c= \\alpha_\\Delta(P-P_c)$, where $\\alpha_\\Delta=7.8\\text{meV}\/\\text{GPa}$, the critical pressure $P_c$ is $55\\text{GPa}$ and taking into account the pressure dependence of the exchange integral is $J\\left(P\\right)=J_0\\left(1+\\frac{2\\alpha_t}{t}P\\right)$, where $J_0$ is taken to be $18\\text{K}$ and $\\frac{2\\alpha_t}{t}=0.0122 1\/\\text{GPa}$. This way, for each value of $\\varepsilon_s$ we show corresponding pressure values $\\frac{\\Delta P}{P_c} = \\frac{\\left(P-P_c\\right)}{P_c}$. Note that within this set of parameters the exchange integral value is chosen to reproduce the real compound's Neel temperature and the critical pressure is aligned with our critical value of the spin gap for a more convenient qualitative discussion of our results in a context of experimental data as discussed below. Few percent below the critical pressure there is simply a drop of an effective magnetic moment with temperature. Around percent below $P_c$ an effective magnetic moment is almost temperature independent. Very close to critical pressure the LS component at the Neel temperature is already significant and thermal fluctuations lead mainly to increase of the HS component. Above the critical pressure, as shown in Fig.~\\ref{fig:4}(b), increasing pressure leads to slowdown in temperature growth of an effective magnetic moment.\n\n\\begin{figure}\n\\includegraphics{Fig3}\n\\caption{\\label{fig:3}(a) Average staggered magnetization $m$ and (b) LS occupation number obtained with $2\\times2$ CMF. The arrow shows the position of a tricritical point. The dashed line is the $n_{LS} = 0.5$ isoline.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics{Fig4}\n\\caption{\\label{fig:4}Temperature dependence of the Curie constant defined by Eq.\\ref{eq:16} for different values of spin gap (pressure) obtained with $2\\times2$ CMF (a) below, (b) above the critical pressure. The dashed lines indicate the values of the Neel temperature for the data of the same color, $\\mu^2$ of Eq.~\\ref{eq:16} is set equal to one.}\n\\end{figure}\n\nTo explore finite-size effects of our CMF calculations we now turn to comparison of magnetization obtained within different clusters and within the Tyablikov approximation (or RPA) in the Heisenberg limit. Within the Heisenberg model RPA is known to provide results in a decent agreement with numerically exact quantum Monte Carlo \\cite{Junger09, Yasuda05}. From Fig.~\\ref{fig:5} it is seen that inclusion of nearest correlations leads to an appearance of zero fluctuations in $m$ and a substantial decrease in Neel temperature when comparing MF with $2\\times2$ CMF. At zero temperature the bulk magnetization seems to gradually approach the RPA value 0.8168, for example for $4 \\times 3 $ (not shown) and $4 \\times 4$ clusters we obtain $m=0.886$ and $m=0.88$.  In 2D the Neel temperature is zero in RPA, since it satisfies to the Mermin-Wagner theorem, unlike (C)MF, where the symmetry of the cluster's (site's) Hamiltonian is lowered artificially. Analogous comparison in 3D is shown in Fig.~\\ref{fig:6}: the Neel temperature is approximately 1.5 times higher within MF that within RPA and 1.33 times higher with $2\\times 2\\times 2$ CMF. This way, in terms of staggered magnetization's and Neel temperature's values we obtain intermediate results between RPA and MF. In our CMF calculations in the 2D case the bulk site magnetization $m(N_c)$ as a function of the number of sites  turned to be proportional to $\\sqrt{N_c}$. Least square extrapolation gave the result $m_{\\infty}\\approx0.81$, which is similar to the RPA value (see Fig.~\\ref{fig:7}).\n\n\\begin{figure}\n\\includegraphics{Fig5}\n\\caption{\\label{fig:5} Bulk site's magnetization calculated in the Heisenberg limit in 2D within MF, CMF with different rectangular clusters, and RPA.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics{Fig6}\n\\caption{\\label{fig:6}The same as in Fig.~\\ref{fig:5} in 3D within MF, $2\\times2\\times2$ CMF, and RPA.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics{Fig7}\n\\caption{\\label{fig:7}Extrapolation to the infinite system limit for the bulk magnetization $m(N_c)$ and the critical value of the spin gap $\\varepsilon^c_s(N_c)$ in the Heisenberg limit.}\n\\end{figure}\n\n\\begin{table}\n\\caption{\\label{tab:1}\nTricritical $\\varepsilon^*_s$ and critical $\\varepsilon^c_s$ values of the spin gap for different clusters within CMF.}\n\\begin{ruledtabular}\n\\begin{tabular}{c|cccccc}\n&\n\\multicolumn{1}{c}{\\textrm{$MF$}}&\n\\multicolumn{1}{c}{\\textrm{$2\\times2$}}&\n\\multicolumn{1}{c}{\\textrm{$3\\times2$}}&\n\\multicolumn{1}{c}{\\textrm{$4\\times2$}}&\n\\multicolumn{1}{c}{\\textrm{$4\\times3$}}&\n\\multicolumn{1}{c}{\\textrm{$4\\times4$}}\\\\\n\\hline\n$\\varepsilon^*_s $ & $\\approx 1.59$ & $\\approx 1.87$ & $\\approx 1.93$ & $\\approx 1.98$ & ---\\footnote{Have not been calculated for this cluster.} & ---\\\\\n$\\varepsilon^c_s $ & 2 & 2.148 & 2.175 & 2.189 & 2.217 & 2.232\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\nNext, we compare average staggered magnetization obtained with different clusters and MF at different values of spin gap in Fig.~\\ref{fig:8}. Phase diagrams obtained within different clusters are very similar. Besides the decrease in Neel temperature there is an increase in tricritical value of a spin gap $\\varepsilon^*_s$ and the critical value $\\varepsilon^c_s$, at which the first-order phase transition occurs, as it is shown in Table~\\ref{tab:1}. The increase in $\\varepsilon^c_s$ with cluster's size is related to the lowering of the cluster's ground state energy in magnetic phase with increasing size, because the main competition is between states with 0 and $N_c$ singlets per cluster. Similarly to the case of magnetization, we observe $1\/\\sqrt{N_c}$ behavior of $\\varepsilon^c_s(N_c)$ or the ground-state energy $E_0$ with opposite sign in the Heisenberg limit (see Fig.~\\ref{fig:7}). By least squares extrapolation for $E_0(N_c)$ we found $E_0(\\infty) \\approx -2.31$, which is similar to the value $E_0(\\infty) \\approx -2.33$ from the quantum Monte Carlo \\cite{Harada} and density matrix renormalization group \\cite{Ramos} studies. The size dependence of $\\varepsilon^*_s$ and $\\varepsilon^c_s$ shows the most crucial change when going from MF to 4-site CMF with predictable behavior when increasing the system size. Thus, the part of the phase diagram obtained at finite temperature close to the first order transition with small clusters from 4 to 8 sites can be considered as semi-quantitative.\n\n\\begin{figure}\n\\includegraphics{Fig8}\n\\caption{\\label{fig:8} Bulk site's magnetization obtained within (a) MF, (b) $2\\times 2$, (c) $3\\times 2$, and (d) $4\\times 2$ CMF. The black line shows MF second-order transition line. Arrows show the position of a tricritical point.}\n\\end{figure}\n\nAlthough within standard MF approach qualitatively correct magnetic phase diagram is obtained, it provides no information about short-range correlations in the system. In Fig.~\\ref{fig:9} we show transverse antiferromagnetic nearest-neighbor spin correlations $C_{\\bot}=-\\left\\langle \\left(S^+_0S^-_1 + S^-_0S^+_1\\right)\\right\\rangle$ and longitudinal ones $C_{\\parallel}=-\\left\\langle S^z_0S^z_1\\right\\rangle$.  At $\\varepsilon_s<\\varepsilon^c_s$ the longitudinal correlations are always decreasing with temperature, but transverse ones are increasing with temperature at low values of spin gap, reaching maximum at Neel points and lowering in a paramagnetic phase. A non-Heisenberg effect is that at $\\varepsilon_s>\\varepsilon^c_s$ the spin correlations show a reentrant behavior. At low temperature they are zero, then increasing with heating due to thermal excitement of triplet states. When temperature is increased further, the correlations lower again.\n\n\\begin{figure}\n\\includegraphics{Fig9}\n\\caption{\\label{fig:9}(a) Transverse $C_{\\bot}$ and (b) longitudinal $C_{\\parallel}$ nearest-neighbor spin correlations, obtained within $2\\times 2$ CMF.}\n\\end{figure}\n\nFinally, we use parameters from the anvil-cell experiments on ferropericlase (Mg,Fe)O ~\\cite{Lyubutin12, Lyubutin13} used above to model its magnetization dependence on pressure and temperature. The exchange parameter value and its linear pressure dependence at low pressure in the HS state were obtained by fitting the experimental data from the paper~\\onlinecite{Lyubutin12}. The magnetization's phase diagram is presented in Fig.~\\ref{fig:10}(a). Heisenberg behavior is realized in a broad range of pressure, where the Neel temperature scales linearly with pressure and reaches its maximum. At $P\\approx P_c$ the Neel temperature drops discontinuously to zero due to a phase transition of the first order. Deviation from Heisenberg behavior is realized at $P \\gtrsim 51 \\text{GPa}$ at $T=0$ and at $P \\gtrsim 45 \\text{GPa}$ at room temperatures, as it is seen from spin correlations in Fir.~\\ref{fig:10}(b). The non magnetic phase can be qualitatively identified as HS to the left of the black line, which denotes $50\\%$ of maximal effective magnetic moment, and LS to the right. Our phase diagram is consistent with experimental data and model calculations of Refs.~\\cite{Lyubutin12, Lyubutin13}. This shows that the microscopic Hamiltonian we have studied is capable of capturing the main physics of spin crossover in ferropericlase.\n\n\\begin{figure}\n\\includegraphics{Fig10}\n\\caption{\\label{fig:10} (a) Average sublattice magnetization $m$ calculated for ferropericlase parameters from Ref.~\\cite{Lyubutin12} by $2\\times 2$ CMF. The black line is the $n_{LS} = 0.5$ isoline. (b) Transverse spin correlations for the same set of parameters.}\n\\end{figure}\n\n\\section{\\label{sec:5}Discussion}\n\nTo sum up, in order to study non-Heisenberg effects due to SCO we have derived an effective Hamiltonian for the two-orbital Kanamori model. The parameters of the effective Hamiltonian have been written down analytically. It contains HS and LS states, and interatomic exchange interaction, as well as the exciton hopping and the biexciton creation and annihilation processes. As it can be seen within simple MF, due to the presence of LS states the MF magnetization within this model is not described by the Brillouin function. The effective Hamiltonian has been studied within CMF approximation. As we have shown by comparing our results between different cluster sizes and to other methods in the special case, our results are of qualitative character at high temperatures, but we expect them to be semi-quantitative within an interesting region close to first-order transition. We have obtained a magnetic $\\varepsilon_s-T$ phase diagram of the model with antiferromagnetic and paramagnetic phases. At very low spin gap  values $\\varepsilon_s$  the magnetization's temperature dependence is almost Heisenberg-like. Increasing $\\varepsilon_s$ leads to reduction of the Neel temperature and paramagnetic moments (or the Curie constant in the paramagnetic susceptibility) due to thermal population of LS states. Up to a tricritical point $\\varepsilon^*_s$ the phase transition line is second-order one and from $\\varepsilon^*_s$ to a critical value of quantum phase transition $\\varepsilon^c_s$ it is first-order. Few percent below $\\varepsilon_s$ there occurs a drastic change in the temperature dependence of the Curie constant in paramagnetic susceptibility. At $\\varepsilon_s>\\varepsilon^c_s$ the magnetic moment and the Curie constant are zero at zero temperature and they increase with heating because of growing population of HS states. From quantitative point of view we expect our results for the magnetic phase diagram to be between simple MF (closer to MF) and RPA, which has not been rigorously developed yet in the case when LS states must be taken into account. However, we have shown that the results of CMF calculations shall approach correct values with further increase in cluster's size, thus showing predictable behavior. Using cluster approach has allowed us to predict another non-Heisenberg effect, which is a reentrant behavior of the temperature dependence of spin correlation functions at $\\varepsilon_s>\\varepsilon^c_s$. For the $P-T$ magnetic phase diagram that we have obtained for ferropericlase the non-Heisenberg behavior is realized at $P \\gtrsim 51 \\text{GPa}$ at $T=0$ and $P \\gtrsim 45 \\text{GPa}$ at $T\\approx300K$, which is a realistic pressure and temperature interval for a more detailed experimental investigation of this compound and for observing the non-Heisenberg effects.\n\n\\begin{acknowledgments}\nThe authors thank the Russian Scientific Foundation for the financial support under the grant 18-12-00022.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}