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Physical review. B, Condensed matter

Two-dimensional spin-polarized fermion lattice gases

James E. Gubernatis, Douglas J. Scalapino, Robert L. Sugar, W. D. Toussaint

Published Jul 1, 1985• 73 citations
Publisher

Abstract

A model of spin-polarized fermions hopping on a two-dimensional lattice with a nearest-neighbor interaction V is studied. Random-phase-approximation calculations predict that the half-filled system undergoes a density-wave transition for positive values of V, an odd-angular-momentum pairing transition for small negative V, and a condensation-phase-separation transition for more negative values of V. The classical lattice-gas Ising limit matches onto the density-wave transition for V>0 and the condensation transition for V<0. A strong-coupling expansion in powers of the ratio of single-fermion transfer-matrix element t to the two-body interaction V provides the leading corrections to the Ising limit. In order to explore the intermediate-coupling regime, fermion Monte Carlo calculations were carried out and various Green's functions characterizing the quantum correlations evaluated. With use of finite-size scaling techniques, the density-wave and condensation phase boundaries were followed into the intermediate-coupling regime. At these transitions, measured quantities scaled well with the usual Ising indices. Unfortunately, the weak-coupling regime lies beyond the reach of these simulations, and we conclude that a method suitable for the weak-coupling regime is needed to complete the phase diagram.

DOI
10.1103/physrevb.32.103

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A model of spin-polarized fermions hopping on a two-dimensional lattice with a nearest-neighbor interaction V is studied. Random-phase-approximation calculations predict that the half-filled system undergoes a density-wave transition for positive values of V, an odd-angular-momentum pairing transition for small negative V, and a condensation-phase-separation transition for more negative values of V. The classical lattice-gas Ising limit matches onto the density-wave transition for V>0 and the condensation transition for V<0. A strong-coupling expansion in powers of the ratio of single-fermion transfer-matrix element t to the two-body interaction V provides the leading corrections to the Ising limit. In order to explore the intermediate-coupling regime, fermion Monte Carlo calculations were carried out and various Green's functions characterizing the quantum correlations evaluated. With use of finite-size scaling techniques, the density-wave and condensation phase boundaries were followed into the intermediate-coupling regime. At these transitions, measured quantities scaled well with the usual Ising indices. Unfortunately, the weak-coupling regime lies beyond the reach of these simulations, and we conclude that a method suitable for the weak-coupling regime is needed to complete the phase diagram.
A model of spin-polarized fermions hopping on a two-dimensional lattice with a near-neighbor repulsive interaction is simulated with the use of Monte Carlo methods. This system exhibits an Ising-like order-disorder phase transition as the temperature is lowered. Results for the structure factor, staggered susceptibility, and order parameter are obtained, and finite-size scaling on lattices of 4\ifmmode\times\else\texttimes\fi{}4, 6\ifmmode\times\else\texttimes\fi{}6, and 8\ifmmode\times\else\texttimes\fi{}8 is used to estimate Tc{T}_{c}Tc​ and test the nature of the phase transition.
The Holstein model of spinless fermions interacting with dispersionless phonons in one dimension is studied by a Green's function Monte Carlo technique. The ground-state energy, first fermionic excited state, density wave correlations, and mean lattice displacements are calculated for lattices of up to 16 sites, for one fermion per two sites, i.e., a half-filled band. Results are obtained for values of the fermion hopping parameter of t=0.1\ensuremath{\omega}, \ensuremath{\omega}, and 10\ensuremath{\omega}, where \ensuremath{\omega} is the phonon frequency. At a finite fermion-phonon coupling g there is a transition from a metallic phase to an insulating phase in which there is charge-density-wave order. Finite-size scaling is found to hold in the metallic phase and is used to extract the coupling dependence of the Luttinger liquid parameters, ${\mathit{u}}_{\mathrm{\ensuremath{\rho}}}$ and ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$, the velocity of charge excitations, and the correlation exponent, respectively. For free fermions (g=0) and for strong coupling (g2{\mathit{g}}^{2}g2\ensuremath{\gg}t\ensuremath{\omega}) our results agree well with known analytic results. \textcopyright{} 1996 The American Physical Society.
We present a numerically exact study of the Hubbard model with spin-dependent anisotropic hopping on the square lattice using auxiliary-field quantum Monte Carlo method. At half filling, the system undergoes Ising phase transitions on cooling, leading to the formation of Ising-type antiferromagnetic order for repulsive interactions and charge-density wave order for attractive interactions at finite temperatures. By elegantly implementing the sign-problem-free condition and Hubbard-Stratonovich transformations, we achieve significant improvements in precision control of the numerical calculations and obtain highly accurate results of the transition temperatures from weak to strong interactions across representative anisotropies. We further characterize the system by examining the temperature dependence of various thermodynamic properties, including the energy, double occupancy, specific heat, and charge susceptibility. Specifically, we provide unbiased numerical results of the entropy map on temperature-interaction plane, the critical entropy, and the spin, singlon and doublon correlations, all of which are directly measurable in optical lattice experiments. Away from half filling, we explore the behavior of the sign problem and investigate the possible emergence of stripe spin-density wave order in the system with repulsive interaction.
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.
We study the magnetic properties of the two-dimensional Hubbard model with nearest-neighbor (t) and next-nearest-neighbor (t2{t}_{2}t2​) hopping and on-site repulsion U. We first calculate the mean-field phase diagram as a function of band filling and U. Because of the Van Hove singularity in the density of states, we find a ferromagnetic phase extending to zero U for certain band filling. For the half-filled band case, antiferromagnetism sets in at a finite value of U if t2{t}_{2}t2​\ensuremath{\ne}0. We study the behavior of spin-spin correlation functions for small lattices of up to N=64 atoms using Monte Carlo simulations, as well as exact diagonalization for N=4. Our results show enhanced ferromagnetic correlations in some regions, but apparently no ferromagnetic long-range order. In the half-filled case, our numerical results are consistent with a nonzero critical U. For a non-half-filled band our results suggest that there is no long-range order.
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