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Chinese Journal of Physics

Calculation of Numerical Microcanonical Ensemble Method on 2D Statistical Models

Jone-Zen Wang, Tzong‐Jer Yang

Published Jun 1, 1998• 0 citations
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Abstract

In our earlier work, we proposed a new microcanonical ensemble method and showed that the numerical results of this method are correct in 1D models. In this paper, the numerical data in 2D models are presented. One of our calculations is concentrated on fermionic models, because fermionic models behave quite differently from bosonic models when the dimension is larger than two (there may be boson-fermion duality in 1D). Our results come out to be consistent with exact values. To our knowledge, there is no other numerical methods which can present reliable calculation on fermionic models with large lattice, especially when the temperature is low. In addition to the test on fermionic models, we also show our results on quantum 2D XY model. We calculate longitudinal spin-spin correlation, specific heat, vortex density and vortex pair density. These results agree with the calculations by other methods. The data of vortex density and vortex pair density seem to display directly the unbinding of vortices- and antivortices. PACS. 05.30.Ch- Quantum ensemble theory. PACS. 65.50.+m- Thermodynamic properties and entropy.

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In our earlier work, we proposed a new microcanonical ensemble method and showed that the numerical results of this method are correct in 1D models. In this paper, the numerical data in 2D models are presented. One of our calculations is concentrated on fermionic models, because fermionic models behave quite differently from bosonic models when the dimension is larger than two (there may be boson-fermion duality in 1D). Our results come out to be consistent with exact values. To our knowledge, there is no other numerical methods which can present reliable calculation on fermionic models with large lattice, especially when the temperature is low. In addition to the test on fermionic models, we also show our results on quantum 2D XY model. We calculate longitudinal spin-spin correlation, specific heat, vortex density and vortex pair density. These results agree with the calculations by other methods. The data of vortex density and vortex pair density seem to display directly the unbinding of vortices- and antivortices. PACS. 05.30.Ch- Quantum ensemble theory. PACS. 65.50.+m- Thermodynamic properties and entropy.
Based on the microcanonical ensemble theory in statistical mechanics, we devise a method that can be used to enhance the capability of numerical calculation on statistical models with large lattice sizes. In our method, we take the expectation value of the energy, as defined in quantum mechanics, instead of the eigenvalue as the energy of a physical system. We show mathematically that the relevant physical quantities obtained in this way are unchanged in the thermodynamical limit and we apply this method to numerical calculations. In this paper, we present our numerical results with the one-dimensional spinless fermionic model as a first test of our method. The numerical calculations are done to a 4096 lattice size using a computer with a speed of about 40 mflops. Our numerical data agree quite well with the exact values. Also, the fluctuation of data is small, in contrast to that obtained using the quantum Monte Carlo method. \textcopyright{} 1996 The American Physical Society.
The authors have carried out microcanonical Monte Carlo (MC) simulations for the two-dimensional (2D) classical XY model on a square lattice having 900 spins with periodic boundary conditions. The temperature dependence of the energy, specific heat, mean-square magnetization and vortex density show good agreement with the existing canonical MC simulation results. They thus demonstrate ergodicity in the microcanonical MC simulations for the 2D classical XY model. The maximum of the temperature derivative of the vortex density is found to occur at the temperature corresponding to the maximum of the specific heat.
We use numerical linked cluster expansions (NLCEs) to study correlated lattice systems in one and two dimensions.For fermions in the honeycomb lattice, we study its finite-temperature properties and short-range spin correlations using NLCEs and determinantal quantum Monte Carlo (DQMC) simulations.For the homogeneous system, we analyze a number of thermodynamic quantities, including the entropy, the specific heat, uniform and staggered spin susceptibilities, short-range spin correlations, and the double occupancy at and away from half filling.Employing a local density approximation (LDA), we examine the viability of adiabatic cooling by increasing the interaction strength for homogeneous as well as for trapped systems.We also use NLCEs to study thermodynamic properties of the two-dimensional spin-1/2 Ising, XY, and Heisenberg models with bimodal random-bond disorder on the square and honeycomb lattices.In all cases, the nearest-neighbor coupling between the spins takes values ±J with equal probability.We obtain the disorder averaged (over all disorder configurations) energy, entropy, specific heat, and uniform magnetic susceptibility in each case.These results are compared with the corresponding ones in the clean models.Analytic expressions are obtained for low orders in the expansion of these thermodynamic quantities in inverse temperature.For many-body localization in disordered isolated systems, we study quantum quenches in the thermodynamic limit.By a quantum quench it is meant that the initial state is stationary with respect iii to an initial Hamiltonian, which is suddenly changed to a new (time-independent) Hamiltonian.The latter then drives the (unitary) dynamics of the system.We are interested in the time average of observables after relaxation following the quench, which can be computed under the so-called diagonal ensemble.
We show how to generalize the zero-temperature Lanczos method for calculating dynamical correlation functions to finite temperatures. The key is the microcanonical ensemble which allows us to replace the involved canonical ensemble with a single appropriately chosen state; in the thermodynamic limit it provides the same physics as the canonical ensemble but with the evaluation of a single expectation value. We can employ the same system sizes as for zero temperature but, whereas the statistical fluctuations present in small systems are prohibitive, the spectra of the largest system sizes are surprisingly smooth. We investigate, as a test case, the spin conductivity of the spin-1/2 anisotropic Heisenberg model and in particular we present a comparison of spectra obtained by the canonical and microcanonical ensemble methods.
We present results for the equation of state of the two-dimensional Hubbard model on an isotropic square lattice as obtained from a controlled and numerically exact large-cluster dynamical mean field simulation. Our results are obtained for large but finite systems and are extrapolated to infinite system size using a known finite-size scaling relation, and are supplemented by reliable error bars accounting for all sources of errors. We establish the importance of examining the decay of spatial spin correlations to determine whether a sufficiently large cluster has been used and with this in mind we present the energy, entropy, double occupancy, and nearest-neighbor spin correlations extrapolated to the thermodynamic limit. We discuss the implications of these calculations on pseudogap physics of the 2D Hubbard model away from half filling, where we find a strong behavioral shift in energy below a temperature T∗{T}^{*}T∗ which becomes more pronounced for larger clusters. Finally, we provide reference calculations and tables for the equation of state for values of doping away from half filling which are of interest to cold-atom experiments.