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Optical Memory and Neural Networks

Analytical expressions for a finite-size 2D Ising model

Iakov Karandashev, Boris Kryzhanovsky, M. Yu. Malsagov

Published Jul 1, 2017• 12 citations
Publisher

Abstract

Numerical methods are used to examine the thermodynamic characteristics of the twodimensional Ising model as a function of the number of spins N . Onsager’s solution is generalized to a finite-size lattice, and experimentally validated analytical expressions for the free energy and its derivatives are computed. The heat capacity at the critical point is shown to grow logarithmically with N . Due to the finite extent of the system the critical temperature can only be determined to some accuracy.

DOI
10.3103/s1060992x17030031

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Numerical methods are used to examine the thermodynamic characteristics of the twodimensional Ising model as a function of the number of spins N . Onsager’s solution is generalized to a finite-size lattice, and experimentally validated analytical expressions for the free energy and its derivatives are computed. The heat capacity at the critical point is shown to grow logarithmically with N . Due to the finite extent of the system the critical temperature can only be determined to some accuracy.
Numerical methods are used to examine the thermodynamic characteristics of the two-dimensional Ising model as a function of the number of spins N. Onsager's solution is generalized to a finite-size lattice, and experimentally validated analytical expressions for the free energy and its derivatives are computed. The heat capacity at the critical point is shown to grow logarithmically with N. Due to the finite extent of the system the critical temperature can only be determined to some accuracy.
For the 2D Ising model, we analyzed dependences of thermodynamic characteristics on number of spins by means of computer simulations. We compared experimental data obtained using the Fisher-Kasteleyn algorithm on a square lattice with N = l × l spins and the asymptotic Onsager solution ( N → ∞). We derived empirical expressions for critical parameters as functions of N and generalized the Onsager solution on the case of a finite-size lattice. Our analytical expressions for the free energy and its derivatives (the internal energy, the energy dispersion and the heat capacity) describe accurately the results of computer simulations. We showed that when N increased the heat capacity in the critical point increased as ln N . We specified restrictions on the accuracy of the critical temperature due to finite size of our system. Also in the finite-dimensional case, we obtained expressions describing temperature dependences of the magnetization and the correlation length. They are in a good qualitative agreement with the results of computer simulations by means of the dynamic Metropolis Monte Carlo method.
The analytic properties of the Onsager solution of the two-dimensional Ising lattice in zero external magnetic field are investigated. The free energy above the critical temperature Tc{T}_{c}Tc​ can be regarded as an analytic function in the uuu plane, where $u=\mathrm{exp}(\ensuremath{-}\frac{4J}{{k}_{B}T})$, with singularities at $u=0, \ensuremath{\infty}, \mathrm{and} \mathrm{exp}(\ifmmode\pm\else\textpm\fi{}\frac{4J}{{k}_{B}{T}_{c}})$. But its analytic continuation below the critical temperature does not describe a stable thermodynamic phase. Similar statements are true for the free energy below the critical temperature. The famous logarithmic singularity is briefly discussed. Finally, the knowledge of the analytic properties is shown, by elementary considerations, to explain the nonanalytic behavior at vanishing external field below the critical temperature---a result first enunciated by Yang and Lee for the free energy in the presence of an external magnetic field.
We compute the finite-size corrections to the free energy, internal energy and specific heat of the critical two-dimensional spin-1/2 Ising model on a triangular and hexagonal lattices wrapped on a torus. We find the general form of the finite-size corrections to these quantities, as well as explicit formulas for the first coefficients of each expansion. We analyze the implications of these findings on the renormalization-group description of the model. Key Words: Ising model; finite-size scaling; corrections to scaling; renormalization group; irrelevant operators; scaling functions. It is well-known that phase transitions in statistical-mechanical systems can occur only in the infinite-volume limit. In any finite system, all thermodynamic quantities (such as the magnetic susceptibility and the specific heat) are analytic functions of all parameters (such as the temperature and the magnetic field); but near a critical
The fact that the Ising model in higher dimensions than 1D features a phase transition at the critical temperature Tcdespite its apparent simplicity is one of the main reasons why it has lost none of its fascination and remains a central benchmark in modeling physical systems. Building on our previous work, where an approximative analytic free-energy expression for finite 2D-Ising lattices was introduced, we investigate different extrapolation strategies for estimating Tcof the infinite system from exactly solvable small lattices. Finite square lattices of linear dimension N with free and periodic boundary conditions were analyzed, exploiting their exactly accessible density of states to compute the heat capacity profiles C(T). Different approaches were compared, including scaling models for the peak temperature Tmax(N)and an envelope construction across the set of C(T)-profiles. We find that both approaches converge to the same asymptotic value and compare favorably to the established Binder cumulant method. Remarkably, a model for Tmaxwith a single model parameter following an N/(N+1)-law provides robust convergence, with a physical analogy motivating this proportionality. Our findings highlight that surprisingly few, but highly accurate, finite-size results are sufficient to obtain a precise extrapolation.