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Physical review. B, Condensed matter

Monte Carlo study of a two-dimensional spin-polarized fermion lattice gas

Douglas J. Scalapino, Robert L. Sugar, W. D. Toussaint

Published May 1, 1984• 31 citations
Publisher

Abstract

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 ${T}_{c}$ and test the nature of the phase transition.

DOI
10.1103/physrevb.29.5253

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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.
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.
To demonstrate the application of computational physics in modern statistical physics, this paper investigates the critical behavior of a finite-size ferromagnetic system based on the two-dimensional square-lattice Ising model.A nearest-neighbor Ising Hamiltonian is constructed, and periodic boundary conditions are adopted to reduce finite-boundary effects.The Metropolis Monte Carlo algorithm is used to sample spin configurations at different temperatures, while energy, magnetization, specific heat, magnetic susceptibility and Binder cumulant are calculated to describe the transition from an ordered state to a disordered state.The simulation results show that the system exhibits clear long-range order at low temperatures, where the order parameter remains high.As temperature increases, thermal fluctuations become stronger, magnetization decreases rapidly, and the system gradually evolves into a disordered state.The specific heat curve shows a pronounced peak at approximately (T = 2.269), while the susceptibility peak shifts slightly toward the high-temperature side due to finite-size effects and the discrete temperature grid.Overall, the simulated critical region is consistent with the theoretical critical temperature of the two-dimensional square-lattice Ising model.Spin-configuration analysis further reveals enhanced fluctuations and domain reconstruction near the critical region.In addition, a thermalization criterion and an extended sampling test for L = 48 near the critical region are added to examine the influence of finite sampling and equilibration.These results indicate that Monte Carlo simulation of the two-dimensional Ising model can effectively reproduce continuous phase transition, critical fluctuation and finite-size effects, providing a basic computational case for understanding critical phenomena in complex systems and data-driven physical analysis.
Monte Carlo simulations are performed on the square lattice Ising system with both the nearest neighbor and the two-dimensional dipolar interactions in order to investigate the critical property of ferromagnetic phase transitions in this 2-dimensional system. Applying the finite size scaling theory to Monte Carlo data just near the transition point, the critical indices of this dipolar system are calculated and α=0.31±0.04, β =0.20±0.02, γ=1.08±0.20 and ν=0.72±0.10 are obtained. However, in the temperature region considerably far from the transition point, different values of α, β and γ seem to be adequate, which are rather close to those of the rigorous result of the 2-dimensional Ising model.
The nature of phase transitions occurring in the two-dimensional spin-1 lattice model with the first- and second-nearest-neighbor interactions is studied using Monte Carlo methods. This system models the adsorption of rigid diatomic molecules $(A\text{\ensuremath{-}}B)$ on (100) surfaces of crystals, assuming that each molecule is oriented perpendicularly to the crystal surface, and the binding energy depends on whether the AAA atom or BBB atom is adsorbed. It is shown that under the condition of a fully filled lattice, the order-disorder transition of the superantiferromagnetic (SAF) phase may occur via a continuous as well as a first-order phase transition, depending on the model parameters. The continuous order-disorder transition of the SAF phase is found to be nonuniversal. The nature of the various phase transitions is analyzed by the finite size scaling method. The application of this method is also demonstrated for the case of a second-order transition from one ordered phase to another ordered phase. We have also demonstrated that the phase diagram topology changes with the model parameters. While at low temperatures a first-order transition between the SAF phase and the $c(2\ifmmode\times\else\texttimes\fi{}2{)}_{\mathrm{AF}}$-ordered phase corresponding to a half-filled lattice occurs, this transition line at higher temperatures may terminate either by a triple point or a tricritical point. For the latter case a phase diagram with a different type of multicritical point is suggested.
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.
A two-dimensional Heisenberg model with random antiferromagnetic nearest-neighbor exchange is studied using quantum Monte Carlo techniques. As the strength of the randomness is increased, the system undergoes a transition from an antiferromagnetically ordered ground state to a gapless disordered state. The finite-size scaling of the staggered structure factor and susceptibility is consistent with a dynamic exponent z=2z=2z=2.