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arXiv:1508.07543

The zero-temperature limit of the free energy density in many-electron systems at half-filling

Yohei Kashima

Published Aug 30, 2015• 0 citations
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Abstract

We prove by means of a renormalization group method that in weakly interacting many-electron systems at half-filling on a periodic hyper-cubic lattice, the free energy density uniformly converges to an analytic function of the coupling constants in the infinite-volume, zero-temperature limit if the external magnetic field has a chessboard-like flux configuration. The spatial dimension is allowed to be any number larger than 1. The system covers the Hubbard model with a nearest-neighbor hopping term, on-site interactions, exponentially decaying density-density interactions and exponentially decaying spin-spin interactions. The magnetic field must be included in the kinetic term by the Peierls substitution. The flux configuration and the sign of the nearest-neighbor density-density/spin-spin interactions can be adjusted so that the free energy density is minimum among all the flux configurations. Consequently, the minimum free energy density is proved to converge to an analytic function of the coupling constants in the infinite-volume, zero-temperature limit. These are extension of the results on a square lattice in the preceding work ([Kashima, Y., "The special issue for the 20th anniversary", J. Math. Sci. Univ. Tokyo. 23 (2016), 1-288]). We refer to lemmas proved in the reference in order to complete the proof of the main results of this paper. So this work is a continuation of the preceding work.

DOI
10.48550/arxiv.1508.07543

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We prove by means of a renormalization group method that in weakly interacting many-electron systems at half-filling on a periodic hyper-cubic lattice, the free energy density uniformly converges to an analytic function of the coupling constants in the infinite-volume, zero-temperature limit if the external magnetic field has a chessboard-like flux configuration. The spatial dimension is allowed to be any number larger than 1. The system covers the Hubbard model with a nearest-neighbor hopping term, on-site interactions, exponentially decaying density-density interactions and exponentially decaying spin-spin interactions. The magnetic field must be included in the kinetic term by the Peierls substitution. The flux configuration and the sign of the nearest-neighbor density-density/spin-spin interactions can be adjusted so that the free energy density is minimum among all the flux configurations. Consequently, the minimum free energy density is proved to converge to an analytic function of the coupling constants in the infinite-volume, zero-temperature limit. These are extension of the results on a square lattice in the preceding work ([Kashima, Y., "The special issue for the 20th anniversary", J. Math. Sci. Univ. Tokyo. 23 (2016), 1-288]). We refer to lemmas proved in the reference in order to complete the proof of the main results of this paper. So this work is a continuation of the preceding work.
Renormalization group analysis for multi-band many-electron systems at half-filling at positive temperature is presented. The analysis includes the Matsubara ultra-violet integration and the infrared integration around the zero set of the dispersion relation. The multi-scale integration schemes are implemented in a finite-dimensional Grassmann algebra indexed by discrete position-time variables. In order that the multi-scale integrations are justified inductively, various scale-dependent estimates on Grassmann polynomials are established. We apply these theories in practice to prove that for the half-filled Hubbard model with nearest-neighbor hopping on a square lattice the infinite-volume, zero-temperature limit of the free energy density exists as an analytic function of the coupling constant in a neighborhood of the origin if the system contains the magnetic flux pi (mod 2 pi) per plaquette and 0 (mod 2 pi) through the large circles around the periodic lattice. Combined with Lieb's result on the flux phase problem ([Lieb, E. H., Phys. Rev. Lett. 73 (1994), 2158]), this theorem implies that the minimum free energy density of the flux phase problem converges to an analytic function of the coupling constant in the infinite-volume, zero-temperature limit. The proof of the theorem is based on a four-band formulation of the model Hamiltonian and an extension of Giuliani-Mastropietro's renormalization designed for the half-filled Hubbard model on the honeycomb lattice ([Giuliani, A. and V. Mastropietro, Commun. Math. Phys. 293 (2010), 301-346]).
We consider an extended Hubbard model of interacting fermions on a lattice. The fermion kinetic energy corresponds to a tight-binding Hamiltonian with nearest-neighbor (-t) and next-nearest-neighbor (t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH) hopping matrix elements. In addition to the on-site Hubbard interaction (U) we also consider a nearest-neighbor repulsion (V). We obtain the zero-temperature phase diagram of our model within the Hartree-Fock approximation. We consider ground states having charge- and spin-density-wave ordering as well as states with orbital antiferromagnetism or spin nematic order. The latter two states correspond to particle-hole binding with ${\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}$ symmetry in the charge and spin channels, respectively. For t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH=0, only the charge-density-wave and spin-density-wave states are energetically stable. For nonzero t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH, we find that orbital antiferromagnetism (or spin nematic) order is stable over a finite portion of the phase diagram at weak coupling. This region of stability is seen to grow with increasing values of t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH.
The ground-state properties of the half-filled Hubbard model are investigated in the framework of lattice density functional theory. The single-particle density matrix ${\ensuremath{\gamma}}_{ij\ensuremath{\sigma}}$ is regarded as the central variable of the many-body problem, where iii and jjj refer to the lattice sites and $\ensuremath{\sigma}$ to the spin. The interaction-energy functional $W[{\ensuremath{\gamma}}_{ij\ensuremath{\sigma}}]$ is calculated exactly for representative finite periodic systems by performing exact Lanczos diagonalizations. The relationship between $W[{\ensuremath{\gamma}}_{ij\ensuremath{\sigma}}]$ and the entropy $S[{\ensuremath{\eta}}_{\mathbit{k}\ensuremath{\sigma}}]$ of independent fermions with natural-orbital occupations ${\ensuremath{\eta}}_{\mathbit{k}\ensuremath{\sigma}}$ is analyzed. A simple approximation to the interaction energy of the half-filled Hubbard model is proposed, which takes the form $W=W(S[{\ensuremath{\eta}}_{\mathbit{k}\ensuremath{\sigma}}])$. Using this functional we derive the ground-state energy, kinetic energy, average number of double occupations, charge distribution, magnetic susceptibility, and field-induced spin polarization in one-, two-, and three-dimensional periodic lattices. The limit of infinite dimensions is also explored. The accuracy of the method is assessed by comparison with available exact numerical or analytical results. Goals, limitations, and possible extensions of the domain of applicability of the functional are discussed.
The half-filled extended Hubbard model (including nearest-neighbor interaction) is considered in perturbation theory at weak coupling on a d-dimensional hypercubic lattice (d\ensuremath{\gtrsim}3). It is shown that the exact critical temperature and the exact order parameter in the weak-coupling limit differ from the corresponding Hartree results by a renormalization factor q of order unity. This renormalization factor q is calculated systematically up to O(1/d) in a 1/d expansion. The results depend sensitively on dimension and on the relative strength of the nearest-neighbor interaction. The renormalization factor in d=3, estimated from the 1/d expansion, is q\ensuremath{\simeq}0.282 for the pure Hubbard model, q\ensuremath{\simeq}0.179 near the spin-density-wave to charge-density-wave transition and q\ensuremath{\simeq}0.830 in the limit of pure nearest-neighbor interaction.
Abstract We consider the extended Hubbard model on a two-dimensional square lattice at half-filling. The model is investigated using the strong coupling diagram technique. We sum infinite series of ladder diagrams allowing for full-scale charge and spin fluctuations and the actual short-range antiferromagnetic order for nonzero temperatures. In agreement with earlier results, we find the first-order phase transition in the charge subsystem occurring at v = v c ≳ U/4 with v and U the intersite and on-site Coulomb repulsion constants. The transition reveals itself in an abrupt sign change of a sharp maximum in the zero-frequency charge susceptibility at the corner of the Brillouin. States arising at the transition have alternating deviations of electron occupations from the mean value on neighboring sites. Due to fluctuations, these alternating occupation deviations have short-range order. For the considered parameters, such behavior is found for U ≲ 5t with t the hopping constant. For the insulating case U ≳ 6t, in which the transition is not observed, we find a continuous growth of the Mott gap with v. The evolution of the electron density of states with increasing v is also considered.
A strong-coupling expansion for the antiferromagnetic phase of the Hubbard model is derived in the framework of the slave-boson mean-field approximation. The expansion can be obtained in terms of moments of the density of states of freely hopping electrons on a lattice, which in turn are obtained for hypercubic lattices in arbitrary dimension. The expansion is given for the case of half-filling and for the energy up to fifth order in the ratio of hopping integral t over on-site interaction U, but can straightforwardly be generalized to the non-half-filled case and be extended to higher orders in t/U. For the energy the expansion is found to have an accuracy of better than 1% for U/t\ensuremath{\ge}8. A comparison is given with an earlier perturbation expansion based on the linear-spin-wave approximation and with a similar expansion based on the Hartree-Fock approximation. The case of an infinite number of spatial dimensions is discussed. \textcopyright{} 1996 The American Physical Society.