Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. B, Condensed matter

Exact results for the two-dimensional Ising model in a magnetic field: Tests of finite-size scaling theory

Borko Stošć, S. Milošević, Eugene Stanley

Published Jun 1, 1990• 34 citations
Publisher

Abstract

We use the transfer-matrix method to calculate density-of-states (DOS) functions for fully finite Ising-model systems. In this way, we obtain expressions for the corresponding partition functions for systems of sizes up to 12\ifmmode\times\else\texttimes\fi{}12 in zero field and up to 8\ifmmode\times\else\texttimes\fi{}8 in nonzero external magnetic field. We also establish a way of representing the DOS functions, in terms of the suitably scaled variables, that reveals a remarkably rapid convergence of these functions towards their thermodynamic limiting values as the system size increases. This rapid convergence implies that results obtained for systems of small sizes contain a surprisingly large amount of information about the behavior of infinite systems. In particular, we demonstrate that the finite-size-scaling-theory predictions are satisfied for small systems (treated exactly via this method) comparably well as for much larger systems (treated approximately via the Monte Carlo method). In addition, applying the method to the square Ising model in a nonzero field, we discover that the DOS functions for small systems display a regularity that one would expect to appear only for very large systems.

DOI
10.1103/physrevb.41.11466

Related papers

Correlation functions, free energies, and magnetizations in the two-dimensional random-field Ising model

S. L. A. de Queiroz, Robin B. Stinchcombe

Transfer-matrix methods are used to calculate spin-spin correlation functions (G), Helmholtz free energies (f) and magnetizations (m) in the two-dimensional random-field Ising model close to the zero-field bulk critical temperature T(c 0), on long strips of width L=3-18 sites, for binary field distributions. Analysis of the probability distributions of G for varying spin-spin distances R shows that describing the decay of their averaged values by effective correlation lengths is a valid procedure only for not very large R. Connections between field and correlation function distributions at high temperatures are established, yielding approximate analytical expressions for the latter, which are used for computation of the corresponding structure factor. It is shown that, for fixed R/L, the fractional widths of correlation-function distributions saturate asymptotically with L-2.2. Considering an added uniform applied field h, a connection between f(h), m(h), the Gibbs free energy g(m) and the distribution function for the uniform magnetization in a zero uniform field, P0(m), is derived and first illustrated for pure systems, and then applied for nonzero random field. From finite-size scaling and crossover arguments, coupled with numerical data, it is found that the width of P0(m) varies against (nonvanishing, but small) random-field intensity H0 as H(-3/7)(0).

arXiv:cond-mat/0105326•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Aug 28, 2001•5 citations

Transfer matrix and Monte Carlo tests of critical exponents in Ising model

J. Kaupužs

The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction ~1/L^0.25 of a very small magnitude also has been detected, as it can be expected from our recently developed GFD (grouping of Feynman diagrams) theory. Monte Carlo simulations of the squared magnetization of 3D Ising model have been performed by Wolff's algorithm in the range of the reduced temperatures t =< 0.000086 and system sizes L =< 410. The effective critical exponent beta_eff(t) tends to increase above the currently accepted numerical values. The critical coupling K_c=0.22165386(51) has been extracted from the Binder cumulant data within 96 =< L =< 384. The critical exponent 1/nu, estimated from the finite-size scaling of the derivatives of the Binder cumulant, tends to decrease slightly below the RG value 1.587 for the largest system sizes. The finite-size scaling of accurately simulated maximal values of the specific heat C_v in 3D Ising model confirms a logarithmic rather than power-like critical singularity of C_v.

arXiv:cond-mat/0405197•May 10, 2004•1 citation

Finite-size scaling of the density of zeros of the partition function in first- and second-order phase transitions

Richard J. Creswick, Seung‐Yeon Kim

The finite-size scaling form for the density of zeros of the partition function in first- and second-order phase transitions is derived. Using the finite-size scaling of the density of zeros, the order of a phase transition can be easily determined and the order parameter calculated from finite-size data. We illustrate the scaling theory using exact values for the zeros of the partition function of the two-dimensional Ising model in the complex magnetic-field plane.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Sep 1, 1997•35 citations

Two-Dimensional Ising Model in a Finite Magnetic Field

Michael Plischke, Daniel Charles Mattis

We present the results of numerical calculations giving accurate estimates of the magnetization of the two-dimensional Ising model on a square lattice. Moreover, we argue that these results are strict lower bounds to the correct magnetization $M(H,T)$. The estimates are obtained by dividing the infinite lattice into finite strips of width between two and nine spins and infinite length. The largest eigenvalue and the corresponding eigenvector of the transfer matrix are then obtained by an iterative process. The estimates of $M(H,T)$ converge to the correct answer for the infinite lattice everywhere except for a small region in the $T\ensuremath{-}H$ plane. We also compute isotherms and critical isobar for the corresponding lattice gas. Finally, we propose a new approximation to the transfer matrix, exactly solvable in two dimensions for $H=O$, which reproduces exactly the critical-point behavior of the full Ising model.

Physical review. B, Solid state•Oct 1, 1970•11 citations

An empirical partition function for two-dimensional Ising Model in an external magnetic field

Rong Wei

There is no an accepted exact partition function (PF) for the two-dimensional (2D) Ising model with a non-zero external magnetic field to our knowledge. Here we infer an empirical PF for such an Ising model. We compare the PFs for two finite-size Ising lattices ($4\times 4$ and $4\times 6$) from this empirical PF with those from Wei (2018) (Wei, R.Q., 2018. An exact solution to the partition function of the finite-size Ising Model, arXiv: General Physics: 1805.01366.), and find that they are consistent very well. Based on this empirical PF, we further analyze and calculate the thermodynamic functions (heat capacity, magnetization, susceptibility) of this 2D Ising model and discuss the model's singularity semiquantitatively. Analysis and calculations from this PF show that they are coincident with those from other related studies; Especially the 2D Ising model in an external magnetic field has spontaneous magnetization (SM) calling the phenomenon at the critical temperature a phase transition, and the SM decreases with the increasing temperature. However, the decreasing variation of the SM here is different from that obtained by Yang (1952) from the 2D Ising model in a weak magnetic field.

arXiv:1905.04295•May 10, 2019•2 citations

A New Exact Method for Solving the Two-Dimensional Ising Model

S. Ranjbar, Gholamabbas Parsafar

We have used the two-dimensional Ising model with a limited number of rows, but with the coordination number of four for each site, to set up the transfer matrix for the model. From the solution of such a matrix, the exact thermodynamic properties have been obtained for the model with a definite number of rows, n . We have solved the matrix for n ≤ 7 and n ≤ 10 in the presence and absence of a magnetic field, respectively. On the basis of such solutions, we have proposed an analytical expression for the partition function of the model with any number of rows in the absence of a magnetic field. The proposed expression becomes more accurate when n is larger, in such a way that it becomes very accurate for n ≥ 8 and is exact for n → ∞. Our results show that the singularity of the specific heat occurs only for the model with infinite number of rows. In the presence of a magnetic field, the solution to the matrixes is too complicated to propose a general analytical expression for the partition function of the model with any number of rows. However, our exact solution for the model with n = 4, 5, 6, and 7 reveals an important point that such results are independent of n when the field−spin interaction energy is almost equal to or larger than that of the spin−spin interaction.

The Journal of Physical Chemistry B•Aug 12, 1999•9 citations
We use the transfer-matrix method to calculate density-of-states (DOS) functions for fully finite Ising-model systems. In this way, we obtain expressions for the corresponding partition functions for systems of sizes up to 12\ifmmode\times\else\texttimes\fi{}12 in zero field and up to 8\ifmmode\times\else\texttimes\fi{}8 in nonzero external magnetic field. We also establish a way of representing the DOS functions, in terms of the suitably scaled variables, that reveals a remarkably rapid convergence of these functions towards their thermodynamic limiting values as the system size increases. This rapid convergence implies that results obtained for systems of small sizes contain a surprisingly large amount of information about the behavior of infinite systems. In particular, we demonstrate that the finite-size-scaling-theory predictions are satisfied for small systems (treated exactly via this method) comparably well as for much larger systems (treated approximately via the Monte Carlo method). In addition, applying the method to the square Ising model in a nonzero field, we discover that the DOS functions for small systems display a regularity that one would expect to appear only for very large systems.
Transfer-matrix methods are used to calculate spin-spin correlation functions (G), Helmholtz free energies (f) and magnetizations (m) in the two-dimensional random-field Ising model close to the zero-field bulk critical temperature T(c 0), on long strips of width L=3-18 sites, for binary field distributions. Analysis of the probability distributions of G for varying spin-spin distances R shows that describing the decay of their averaged values by effective correlation lengths is a valid procedure only for not very large R. Connections between field and correlation function distributions at high temperatures are established, yielding approximate analytical expressions for the latter, which are used for computation of the corresponding structure factor. It is shown that, for fixed R/L, the fractional widths of correlation-function distributions saturate asymptotically with L-2.2. Considering an added uniform applied field h, a connection between f(h), m(h), the Gibbs free energy g(m) and the distribution function for the uniform magnetization in a zero uniform field, P0(m), is derived and first illustrated for pure systems, and then applied for nonzero random field. From finite-size scaling and crossover arguments, coupled with numerical data, it is found that the width of P0(m) varies against (nonvanishing, but small) random-field intensity H0 as H(-3/7)(0).
The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction ~1/L^0.25 of a very small magnitude also has been detected, as it can be expected from our recently developed GFD (grouping of Feynman diagrams) theory. Monte Carlo simulations of the squared magnetization of 3D Ising model have been performed by Wolff's algorithm in the range of the reduced temperatures t =< 0.000086 and system sizes L =< 410. The effective critical exponent beta_eff(t) tends to increase above the currently accepted numerical values. The critical coupling K_c=0.22165386(51) has been extracted from the Binder cumulant data within 96 =< L =< 384. The critical exponent 1/nu, estimated from the finite-size scaling of the derivatives of the Binder cumulant, tends to decrease slightly below the RG value 1.587 for the largest system sizes. The finite-size scaling of accurately simulated maximal values of the specific heat C_v in 3D Ising model confirms a logarithmic rather than power-like critical singularity of C_v.
We present the results of numerical calculations giving accurate estimates of the magnetization of the two-dimensional Ising model on a square lattice. Moreover, we argue that these results are strict lower bounds to the correct magnetization M(H,T)M(H,T)M(H,T). The estimates are obtained by dividing the infinite lattice into finite strips of width between two and nine spins and infinite length. The largest eigenvalue and the corresponding eigenvector of the transfer matrix are then obtained by an iterative process. The estimates of M(H,T)M(H,T)M(H,T) converge to the correct answer for the infinite lattice everywhere except for a small region in the $T\ensuremath{-}H$ plane. We also compute isotherms and critical isobar for the corresponding lattice gas. Finally, we propose a new approximation to the transfer matrix, exactly solvable in two dimensions for H=OH=OH=O, which reproduces exactly the critical-point behavior of the full Ising model.
There is no an accepted exact partition function (PF) for the two-dimensional (2D) Ising model with a non-zero external magnetic field to our knowledge. Here we infer an empirical PF for such an Ising model. We compare the PFs for two finite-size Ising lattices (4×44\times 44×4 and 4×64\times 64×6) from this empirical PF with those from Wei (2018) (Wei, R.Q., 2018. An exact solution to the partition function of the finite-size Ising Model, arXiv: General Physics: 1805.01366.), and find that they are consistent very well. Based on this empirical PF, we further analyze and calculate the thermodynamic functions (heat capacity, magnetization, susceptibility) of this 2D Ising model and discuss the model's singularity semiquantitatively. Analysis and calculations from this PF show that they are coincident with those from other related studies; Especially the 2D Ising model in an external magnetic field has spontaneous magnetization (SM) calling the phenomenon at the critical temperature a phase transition, and the SM decreases with the increasing temperature. However, the decreasing variation of the SM here is different from that obtained by Yang (1952) from the 2D Ising model in a weak magnetic field.
We have used the two-dimensional Ising model with a limited number of rows, but with the coordination number of four for each site, to set up the transfer matrix for the model. From the solution of such a matrix, the exact thermodynamic properties have been obtained for the model with a definite number of rows, n . We have solved the matrix for n ≤ 7 and n ≤ 10 in the presence and absence of a magnetic field, respectively. On the basis of such solutions, we have proposed an analytical expression for the partition function of the model with any number of rows in the absence of a magnetic field. The proposed expression becomes more accurate when n is larger, in such a way that it becomes very accurate for n ≥ 8 and is exact for n → ∞. Our results show that the singularity of the specific heat occurs only for the model with infinite number of rows. In the presence of a magnetic field, the solution to the matrixes is too complicated to propose a general analytical expression for the partition function of the model with any number of rows. However, our exact solution for the model with n = 4, 5, 6, and 7 reveals an important point that such results are independent of n when the field−spin interaction energy is almost equal to or larger than that of the spin−spin interaction.