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Physical Review A

Entropy of the two-dimensional Ising model

Daniel R. Stump

Published Nov 1, 1987• 4 citations
Publisher

Abstract

A numerical calculation of the entropy of the two-dimensional Ising model is described, for nonzero external field. The calculation makes use of the Monte Carlo method to simulate a kind of microcanonical ensemble.
DOI
10.1103/physreva.36.4439

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We generalize the microcanonical algorithm developed by Creutz et al. (1986), and make a detailed comparison with the exact solution in the case of a two-dimensional Ising model at finite volume. We present a new numerical method to compute the temperature in the microcanonical ensemble. This allows us to define a `thermalization` criterion to estimate the point where the differences between canonical and microcanonical results are the smallest. This criterion is shown to work well in the case of the two-dimensional Ising system.
We have provided a concise introduction to the Ising model as one of the most important models in statistical mechanics and in studying the phenomenon of phase transition. The required theoretical background and derivation of the Hamiltonian of the model have also been presented. We finally have discussed the computational method and details to numerically solve the two- and three-dimensional Ising problems using Monte Carlo simulations. The related computer codes in both Python and Fortran, as well as a simulation trick to visualize the spin lattice, have also been provided.
We report the results of a microcanonical simulation of the two- and three-dimensional Ising models at criticality. We also present a microcanonical algorithm that allows a simultaneous simulation of a lattice spin system at different energies, in our case the number of different energies is 32. The critical behavior of the systems was studied via a recently proposed microcanonical renormalization-group technique that yields independent estimates for the critical energy and the critical temperature, as well as for three critical exponents providing a direct test of hyperscaling. Our results in two dimensions are in good agreement with exact results. In the three dimensions our quoted values are consistent with Monte Carlo estimates recently reported in the literature. We obtain uc{\mathit{u}}_{\mathit{c}}uc​=-0.991(1), Tc{\mathit{T}}_{\mathit{c}}Tc​=4.5112(3), \ensuremath{\beta}/\ensuremath{\nu}=0.541(1), \ensuremath{\nu}=0.630(3), and \ensuremath{\alpha}=0.109(1).
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.