The study of the equation of state of the two-dimensional Ising model is initiated by use of the recently calculated $n$ -point spin-correlation functions. In the scaling region where $T\ensuremath{\rightarrow}{T}_{c}$ and $h=B{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{15}{8}}(\frac{H}{k{T}_{c}})$ is of order 1 ( $B$ is a lattice-dependent constant) the free energy below ${T}_{c}$ is of the form $f(T, h)\ensuremath{-}f(T, 0)=\mathrm{const}{(1\ensuremath{-}\frac{{T}_{c}}{T})}^{2}$ $\ensuremath{\Sigma}{n=1}^{\ensuremath{\infty}}(\frac{1}{n!}){C}_{n}^{\ensuremath{-}}{h}^{n}$ where the lattice-dependent constant is chosen so that ${C}_{1}^{\ensuremath{-}}=1$. The constant ${C}_{2}^{\ensuremath{-}}$ has been studied previously. In this paper we study ${C}_{3}^{\ensuremath{-}}$ and ${C}_{4}^{\ensuremath{-}}$ and find as a first approximation $\frac{{C}_{3}^{\ensuremath{-}}}{3!}\ensuremath{\sim}\ensuremath{-}\frac{11}{72{\ensuremath{\pi}}^{2}}$ and $\frac{{C}_{4}^{\ensuremath{-}}}{4!}\ensuremath{\sim}\frac{5}{189\ensuremath{\pi}}+\frac{59}{162{\ensuremath{\pi}}^{3}}$. These are in close agreement with the low-temperature-series values of -0.01538 and 0.0195.The study of the equation of state of the two-dimensional Ising model is initiated by use of the recently calculated $n$-point spin-correlation functions. In the scaling region where $T\ensuremath{\rightarrow}{T}_{c}$ and $h=B{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{15}{8}}(\frac{H}{k{T}_{c}})$ is of order 1 ($B$ is a lattice-dependent constant) the free energy below ${T}_{c}$ is of the form $f(T, h)\ensuremath{-}f(T, 0)=\mathrm{const}{(1\ensuremath{-}\frac{{T}_{c}}{T})}^{2}$ $\ensuremath{\Sigma}{n=1}^{\ensuremath{\infty}}(\frac{1}{n!}){C}_{n}^{\ensuremath{-}}{h}^{n}$ where the lattice-dependent constant is chosen so that ${C}_{1}^{\ensuremath{-}}=1$. The constant ${C}_{2}^{\ensuremath{-}}$ has been studied previously. In this paper we study ${C}_{3}^{\ensuremath{-}}$ and ${C}_{4}^{\ensuremath{-}}$ and find as a first approximation $\frac{{C}_{3}^{\ensuremath{-}}}{3!}\ensuremath{\sim}\ensuremath{-}\frac{11}{72{\ensuremath{\pi}}^{2}}$ and $\frac{{C}_{4}^{\ensuremath{-}}}{4!}\ensuremath{\sim}\frac{5}{189\ensuremath{\pi}}+\frac{59}{162{\ensuremath{\pi}}^{3}}$. These are in close agreement with the low-temperature-series values of -0.01538 and 0.0195.

Physical review. B, Condensed matterNov 1, 197816 citations

On Ising model in magnetic field on the lattice

Raghav G. Jha

We study the free energy of the classical square-lattice Ising model in a uniform real magnetic field $h$ , for which no exact solution is known. We first conjecture a closed-form expression built from the one-dimensional Ising chain and Onsager's hypergeometric solution. Comparing with high-precision tensor renormalization group (TRG) data, we find that it deviates by up to $2.6\%$ , has no spontaneous magnetization, and gives an effective critical-isotherm exponent $δ\to 1$ . However, using the Lee--Yang circle theorem, we show that the exact free energy is a superposition of one-dimensional Ising chains in the same field, weighted by a temperature-dependent (normalized, but not always positive) distribution of squared Lee--Yang gaps, of which our first expression is a single-chain approximation. Exact results (Onsager, Yang, the strong-field expansion and the Lee--Yang solution at $βh = iπ/2$ ) become sum rules for this distribution. We propose a simple form for it, fixed by Onsager's free energy and the exact strong-field limit without any fit to numerical data. This \emph{improved} conjecture agrees with TRG to $0.15\%$ for all $β$ and $h$ , reproduces Yang's spontaneous magnetization exactly, has a positive specific heat, and yields the critical isotherm $M\propto h^{1/δ}$ with $δ= 15.2$ and amplitude $1.055$ , compared with the exact $δ=15$ and $1.059$ . A refined form, which builds in additional input from Ising field theory (including the exact critical isotherm, $δ=15$ ) and a few constants adjusted to transfer-matrix and TRG data, matches the TRG free energy to $4\times 10^{-5}$ .We study the free energy of the classical square-lattice Ising model in a uniform real magnetic field $h$, for which no exact solution is known. We first conjecture a closed-form expression built from the one-dimensional Ising chain and Onsager's hypergeometric solution. Comparing with high-precision tensor renormalization group (TRG) data, we find that it deviates by up to $2.6\%$, has no spontaneous magnetization, and gives an effective critical-isotherm exponent $δ\to 1$. However, using the Lee--Yang circle theorem, we show that the exact free energy is a superposition of one-dimensional Ising chains in the same field, weighted by a temperature-dependent (normalized, but not always positive) distribution of squared Lee--Yang gaps, of which our first expression is a single-chain approximation. Exact results (Onsager, Yang, the strong-field expansion and the Lee--Yang solution at $βh = iπ/2$) become sum rules for this distribution. We propose a simple form for it, fixed by Onsager's free energy and the exact strong-field limit without any fit to numerical data. This \emph{improved} conjecture agrees with TRG to $0.15\%$ for all $β$ and $h$, reproduces Yang's spontaneous magnetization exactly, has a positive specific heat, and yields the critical isotherm $M\propto h^{1/δ}$ with $δ= 15.2$ and amplitude $1.055$, compared with the exact $δ=15$ and $1.059$. A refined form, which builds in additional input from Ising field theory (including the exact critical isotherm, $δ=15$) and a few constants adjusted to transfer-matrix and TRG data, matches the TRG free energy to $4\times 10^{-5}$.

arXiv:2504.18744Apr 25, 20250 citations

Applicability of n -vicinity method for calculation of free energy of Ising model

Boris Kryzhanovsky, Leonid B. Litinskii

arXiv:1701.07332Physica A Statistical Mechanics and its ApplicationsNov 3, 201619 citations

Some remarks on Kadanoff's variational approximations for renormalization-group transformations

Michel Droz, Sheldon L. Katz, J. D. Gunton

The recursion equations corresponding to the one-hypercube approximation of Kadanoff are studied in more detail for the Wilson-Fisher $n$ -vector model in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions and the $d$ -dimensional Ising model. The free energy for the one-dimensional Ising model is solved exactly in this approximation and is shown to give the correct singular behavior near zero temperature and the exact free energy in zero field. The corrections to scaling exponents are shown to be exact to first order in $\ensuremath{\epsilon}$ for the Wilson-Fisher model and are given for the $d=2,3, \mathrm{and} 4$ dimensional Ising models. The amplitude of the specific-heat divergence is calculated for the two-dimensional Ising model and is shown to be in reasonable agreement with the exact result.The recursion equations corresponding to the one-hypercube approximation of Kadanoff are studied in more detail for the Wilson-Fisher $n$-vector model in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions and the $d$-dimensional Ising model. The free energy for the one-dimensional Ising model is solved exactly in this approximation and is shown to give the correct singular behavior near zero temperature and the exact free energy in zero field. The corrections to scaling exponents are shown to be exact to first order in $\ensuremath{\epsilon}$ for the Wilson-Fisher model and are given for the $d=2,3, \mathrm{and} 4$ dimensional Ising models. The amplitude of the specific-heat divergence is calculated for the two-dimensional Ising model and is shown to be in reasonable agreement with the exact result.

Physical review. B, Solid stateJan 1, 19777 citations

Calculation of free energy of a three-dimensional Ising-like system in an external field with the use of the model

Ihor V. Pylyuk, Mykhailo P. Kozlovskii

Physica A Statistical Mechanics and its ApplicationsAug 22, 20104 citations

The study of the equation of state of the two-dimensional Ising model is initiated by use of the recently calculated

n

-point spin-correlation functions. In the scaling region where $T\ensuremath{\rightarrow}{T}_{c}$ and $h=B{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{15}{8}}(\frac{H}{k{T}_{c}})$ is of order 1 (

B

is a lattice-dependent constant) the free energy below

{T}_{c}

is of the form $f(T, h)\ensuremath{-}f(T, 0)=\mathrm{const}{(1\ensuremath{-}\frac{{T}_{c}}{T})}^{2}$ $\ensuremath{\Sigma}{n=1}^{\ensuremath{\infty}}(\frac{1}{n!}){C}_{n}^{\ensuremath{-}}{h}^{n}$ where the lattice-dependent constant is chosen so that ${C}_{1}^{\ensuremath{-}}=1$. The constant ${C}_{2}^{\ensuremath{-}}$ has been studied previously. In this paper we study ${C}_{3}^{\ensuremath{-}}$ and ${C}_{4}^{\ensuremath{-}}$ and find as a first approximation $\frac{{C}_{3}^{\ensuremath{-}}}{3!}\ensuremath{\sim}\ensuremath{-}\frac{11}{72{\ensuremath{\pi}}^{2}}$ and $\frac{{C}_{4}^{\ensuremath{-}}}{4!}\ensuremath{\sim}\frac{5}{189\ensuremath{\pi}}+\frac{59}{162{\ensuremath{\pi}}^{3}}$. These are in close agreement with the low-temperature-series values of -0.01538 and 0.0195.

We study the free energy of the classical square-lattice Ising model in a uniform real magnetic field

h

, for which no exact solution is known. We first conjecture a closed-form expression built from the one-dimensional Ising chain and Onsager's hypergeometric solution. Comparing with high-precision tensor renormalization group (TRG) data, we find that it deviates by up to

2.6\%

, has no spontaneous magnetization, and gives an effective critical-isotherm exponent

δ\to 1

. However, using the Lee--Yang circle theorem, we show that the exact free energy is a superposition of one-dimensional Ising chains in the same field, weighted by a temperature-dependent (normalized, but not always positive) distribution of squared Lee--Yang gaps, of which our first expression is a single-chain approximation. Exact results (Onsager, Yang, the strong-field expansion and the Lee--Yang solution at

βh = iπ/2

) become sum rules for this distribution. We propose a simple form for it, fixed by Onsager's free energy and the exact strong-field limit without any fit to numerical data. This \emph{improved} conjecture agrees with TRG to

0.15\%

for all

β

and

h

, reproduces Yang's spontaneous magnetization exactly, has a positive specific heat, and yields the critical isotherm

M\propto h^{1/δ}

with

δ= 15.2

and amplitude

1.055

, compared with the exact

δ=15

and

1.059

. A refined form, which builds in additional input from Ising field theory (including the exact critical isotherm,

δ=15

) and a few constants adjusted to transfer-matrix and TRG data, matches the TRG free energy to

4\times 10^{-5}

The recursion equations corresponding to the one-hypercube approximation of Kadanoff are studied in more detail for the Wilson-Fisher

n

-vector model in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions and the

d

-dimensional Ising model. The free energy for the one-dimensional Ising model is solved exactly in this approximation and is shown to give the correct singular behavior near zero temperature and the exact free energy in zero field. The corrections to scaling exponents are shown to be exact to first order in $\ensuremath{\epsilon}$ for the Wilson-Fisher model and are given for the

d=2,3, \mathrm{and} 4

dimensional Ising models. The amplitude of the specific-heat divergence is calculated for the two-dimensional Ising model and is shown to be in reasonable agreement with the exact result.