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Physical review. B, Condensed matter

Extended Hubbard model at weak coupling

P. G. J. van Dongen

Published Nov 15, 1994• 63 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.50.14016

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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.
The Hubbard model including nearest neighbor interaction is studied at T=0 on a d-dimensional hypercubic lattice (d\ensuremath{\gg}1) close to half filling. For the model in d=\ensuremath{\infty} we derive the exact result that the ground state at weak coupling is phase separated. Results for lower dimensions are then derived in a 1/d expansion. To obtain these results we first consider possible second order transitions. One then finds that the broken-symmetry phase near half filling is incommensurate. However, the corresponding ground state has negative compressibility and is hence thermodynamically unstable. A Maxwell construction is used to construct the actual phase separated ground state, which consists of homogeneous lower-density and antiferromagnetic or charge density wave higher-density regions. It is shown that both the doping level below which phase separation occurs and the order parameter 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 and turns out to be identical to the renormalization factor previously calculated for the low-temperature thermodynamics at half filling. \textcopyright{} 1996 The American Physical Society.
The Hubbard model including nearest-neighbor interaction is studied at T=0T=0T=0 on a ddd-dimensional hypercubic lattice ($d\ensuremath{\ge}3$) close to half filling. It is shown that the ground state at weak coupling is phase separated, consisting of homogeneous lower-density and antiferromagnetic or charge density wave higher-density regions. Incommensurate phases are unstable. The exact critical doping level and the exact order parameter are renormalized relative to the Hartree results. The renormalization factor at weak coupling turns out to be identical to the result found previously for the same model at half filling.
The extended Hubbard model with on-site interaction U and nearest-neighbor interaction V/d is studied at half filling in high dimensions (d\ensuremath{\gg}1). At small U and V the critical temperature and the order parameter are calculated including the 1/d corrections. The results differ from those obtained in the Hartree approximation by a factor of order unity.$---- At V/U=1/2 a transition is found from an antiferromagnetic to a charge-density-wave phase. A similar transition is found at large U and V, suggesting that the transition is present for all U and V satisfying V/U=1/2.
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 half-filled extended Hubbard model (containing nearest-neighbor interaction) is considered at strong coupling in fourth-order perturbation theory. The ground-state energies in the spin- and charge-density-wave phases and the resulting phase diagram are calculated in various dimensions: here d=1, d=\ensuremath{\infty}, high dimensions (d\ensuremath{\gg}1), and d=2,3 are discussed separately. In d=1 it is shown that, for U\ensuremath{\gtrsim}6, fourth-order perturbation theory leads to excellent agreement with existing Monte Carlo data. Second-order perturbation theory alone is valid only at unrealistically large values of U and V. In d=\ensuremath{\infty}, one needs fourth-order perturbation theory to obtain any nontrivial contribution to the phase diagram at all. As a consequence the fourth-order corrections to the ground-state energy and to the phase diagram are large also in d=2,3. The formalism used to obtain the perturbative expansion is discussed in an appendix.
The Onsager-reaction-field correction to Hartree-Fock theory recently derived by Georges and Yedidia [Phys. Rev. B 43, 3475 (1991)] is evaluated analytically for the half-filled positive-U Hubbard model on a d-dimensional hypercubic lattice. It is shown that the Onsager correction is for all d\ensuremath{\ge}3 irrelevant in the weak-coupling limit, while for d=1 and d=2 it strongly renormalizes the Hartree-Fock result, although it does note completely destabilize the spin-density wave. We examine the perturbation theory for the energy change induced by a small staggered magnetization m to all orders in U, and show that one obtains an infinite series in powers of (U/t)ln[${\mathit{m}}^{\mathrm{\ensuremath{-}}1}$ln(${\mathit{m}}^{\mathrm{\ensuremath{-}}1}$)] for d=1, and in powers of (U/t)ln2{\mathrm{ln}}^{2}ln2[${\mathit{m}}^{\mathrm{\ensuremath{-}}1}$ln2{\mathrm{ln}}^{2}ln2(${\mathit{m}}^{\mathrm{\ensuremath{-}}1}$)] for d=1. The close similarity between d=1 and d=2 is interpreted as evidence that antiferromagnetism in the weak coupling regime is not stable in d=2.