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

arXiv:cond-mat/0009024 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Exact amplitude ratio and finite-size corrections for the M×N square lattice Ising model

N. Sh. Izmailian, Chin‐Kun Hu

Published Feb 7, 2002• 42 citations
arXivHTMLPDF

Abstract

Let $f, U,$ and C represent, respectively, the free energy, the internal energy, and the specific heat of the critical Ising model on the $M\ifmmode\times\else\texttimes\fi{}N$ square lattice with periodic boundary conditions, and ${f}_{\ensuremath{\infty}}$ represents f for fixed $M/N$ and $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}.$ We find that $f, U,$ and C can be written as $N(f\ensuremath{-}{f}_{\ensuremath{\infty}})={\ensuremath{\sum}}_{i=1}^{\ensuremath{\infty}}{f}_{2i\ensuremath{-}1}{/N}^{2i\ensuremath{-}1}, U=\ensuremath{-}\sqrt{2}+{\ensuremath{\sum}}_{i=1}^{\ensuremath{\infty}}{u}_{2i\ensuremath{-}1}{/N}^{2i\ensuremath{-}1},$ and $C=8\mathrm{ln}N/\ensuremath{\pi}+{\ensuremath{\sum}}_{i=0}^{\ensuremath{\infty}}{c}_{i}{/N}^{i},$ i.e., $\mathrm{Nf}$ and U are odd functions of ${N}^{\ensuremath{-}1}.$ We also find that ${u}_{2i\ensuremath{-}1}{/c}_{2i\ensuremath{-}1}=1/\sqrt{2}$ and ${u}_{2i}{/c}_{2i}=0$ for $1<~i<\ensuremath{\infty}$ and obtain closed form expressions for $f, U,$ and C up to orders ${1/N}^{5}, {1/N}^{5},$ and ${1/N}^{3},$ respectively, which implies an analytic equation for ${c}_{5}.$

DOI
10.1103/physreve.65.036103

Related papers

Bounded and inhomogeneous Ising models. II. Specific-heat scaling function for a strip

Helen Au-Yang, Michael Ellis Fisher

The specific heat, energy, and free energy of an infinitely long, square-lattice ferromagnetic Ising strip consisting of $n$ parallel layers with free boundary conditions and a surface magnetic field ${H}_{1}={k}_{B}{\mathrm{Th}}_{1}$ imposed on the last layer, is analyzed for large $n$ in the light of finite-size scaling theory. It is shown rigorously that the free energy (and, similarly, the energy and specific heat) can be written asymptotically in the scaling form $\ensuremath{\Delta}f(n,t,{h}_{1})\ensuremath{\approx}{n}^{\ensuremath{-}2}(\mathrm{ln}n){W}_{1}(nt,{n}^{\frac{1}{2}}{h}_{1})+{n}^{\ensuremath{-}2}{W}_{2}(nt,{n}^{\frac{1}{2}}{h}_{1})$, where $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$. The scaling functions ${W}_{i}(x,y)$ are computed in explicit closed form and shown to verify all the analyticity and asymptotic requirements anticipated by scaling theory. Furthermore, in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ at fixed $t\ensuremath{\ne}0$, the bulk and surface contributions to the thermodynamic properties are found to account for all except a correction of order ${e}^{\frac{\mathrm{pna}}{\ensuremath{\xi}(T)}}$, where $a$ is the lattice spacing and $\ensuremath{\xi}(T)$ is the bulk correlation length; the value of the small rational constant $p$ is interpreted in terms of interference effects between the two opposite boundaries (or "surfaces").

Physical review. B, Solid state•May 1, 1975•102 citations

Bounded and Inhomogeneous Ising Models. I. Specific-Heat Anomaly of a Finite Lattice

Arthur E. Ferdinand, Michael Ellis Fisher

The critical-point anomaly of a plane square $m\ifmmode\times\else\texttimes\fi{}n$ Ising lattice with periodic boundary conditions (a torus) is analyzed asymptotically in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ with $\ensuremath{\xi}=\frac{m}{n}$ fixed. Among other results, it is shown that for fixed $\ensuremath{\tau}=\frac{n(T\ensuremath{-}{T}_{c})}{{T}_{c}}$, the specific heat per spin of a large lattice is given by $\frac{{C}_{\mathrm{mn}}(T)}{{k}_{\mathrm{B}}\mathrm{mn}}={A}_{0}\mathrm{ln}n+B(\ensuremath{\tau}, \ensuremath{\xi})+{B}_{1}(\ensuremath{\tau})\frac{(\mathrm{ln}n)}{n}+\frac{{B}_{2}(\ensuremath{\tau}, \ensuremath{\xi})}{n}+O[\frac{{(\mathrm{ln}n)}^{3}}{{n}^{2}}],$ where explicit expressions can be given for ${A}_{0}$ and for the functions $B$, ${B}_{1}$, and ${B}_{2}$. It follows that the specific-heat peak of the finite lattice is rounded on a scale $\ensuremath{\delta}=\frac{\ensuremath{\Delta}T}{{T}_{c}}\ensuremath{\sim}\frac{1}{n}$, while the maximum in ${C}_{\mathrm{mn}}(T)$ is displaced from ${T}_{c}$ by $\ensuremath{\epsilon}=\frac{({T}_{c}\ensuremath{-}{T}_{max})}{{T}_{c}}\ensuremath{\sim}\frac{1}{n}$. For ${\ensuremath{\xi}}_{0}>\ensuremath{\xi}>{{\ensuremath{\xi}}_{0}}^{\ensuremath{-}1}$, where ${\ensuremath{\xi}}_{0}=3.13927\ensuremath{\cdots}$, the maximum lies above ${T}_{c}$; but for $\ensuremath{\xi}>{\ensuremath{\xi}}_{0}$ or $\ensuremath{\xi}<{{\ensuremath{\xi}}_{0}}^{\ensuremath{-}1}$, the maximum is depressed below ${T}_{c}$; when $\ensuremath{\xi}=\ensuremath{\infty}, {\ensuremath{\xi}}_{0}, or {{\ensuremath{\xi}}_{0}}^{\ensuremath{-}1}$, the relative shift in the maximum from ${T}_{c}$ is only of order $\frac{(\mathrm{ln}n)}{{n}^{2}}$. Detailed graphs and numerical data are presented, and the results are compared with some for lattices with free edges. Some heuristic arguments are developed which indicate the possible nature of finite-size critical-point effects in more general systems.

Physical Review•Sep 10, 1969•731 citations

Exact finite-size corrections for the square-lattice Ising model with Brascamp-Kunz boundary conditions

N. Sh. Izmailian, Koryun B. Oganesyan, Chin‐Kun Hu

Finite-size scaling, finite-size corrections, and boundary effects for critical systems have attracted much attention in recent years. Here we derive exact finite-size corrections for the free energy F and the specific heat C of the critical ferromagnetic Ising model on the Mu x 2 Nu square lattice with Brascamp-Kunz (BK) boundary conditions [J. Math. Phys. 15, 66 (1974)] and compare such results with those under toroidal boundary conditions. When the ratio xi/2=(Mu+1)/2 Nu is smaller than 1 the behaviors of finite-size corrections for C are quite different for BK and toroidal boundary conditions; when ln(xi/2) is larger than 3, finite-size corrections for C in two boundary conditions approach the same values. In the limit Nu-->infinity we obtain the expansion of the free energy for infinitely long strip with BK boundary conditions. Our results are consistent with the conformal field theory prediction for the mixed boundary conditions by Cardy [Nucl. Phys. B 275, 200 (1986)] although the definitions of boundary conditions in two cases are different in one side of the long strip.

arXiv:cond-mat/0202282•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•May 22, 2002•48 citations

Universal finite-size scaling functions with exact nonuniversal metric factors

Ming‐Chya Wu, Chin‐Kun Hu, N. Sh. Izmailian

Using exact partition functions and finite-size corrections for the Ising model on finite square, plane triangular, and honeycomb lattices and extending a method [J. Phys. 19, L1215 (1986)] to subtract leading singular terms from the free energy, we obtain universal finite-size scaling functions for the specific heat, internal energy, and free energy of the Ising model on these lattices with exact nonuniversal metric factors.

arXiv:cond-mat/0303166•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jun 26, 2003•38 citations

Domain walls and shape dependence of finite-size correction terms in the two-dimensional Ising-model critical region

Peter H. Kleban, Gulden Akinci

Ferdinand and Fisher have analyzed the finite-size behavior of the two-dimensional Ising model with periodic boundary conditions for $T$ near ${T}_{c}$. In the thermodynamic limit, the leading correction term to the specific heat has an unusual and unexpected behavior as the shape of the $m\ifmmode\times\else\texttimes\fi{}n$ lattice is changed. For shape parameter $s=\frac{m}{n}=1$, the peak in the specific-heat correction term occurs at a reduced temperature ${\ensuremath{\tau}}_{max}=2{K}_{c}n\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}>0$. With increasing $s$ the peak moves to negative $\ensuremath{\tau}$ values and then increases to 0. We show that this effect may be understood by keeping only the two largest eigenvalues of the transfer matrix. To this approximation, which is good for $s=1$ and improves exponentially with increasing $s$, all the shape dependence of the free energy is due to a factor that can be expressed entirely in terms of a domain-wall energy. Furthermore, the functional form of this factor is the same as that of the finite-length correction to the one-dimensional Ising model. Thus the dependence of ${\ensuremath{\tau}}_{max}$ on $s$ is qualitatively similar to the dependence of ${T}_{max}^{1}$ the location of the one-dimensional Ising-model specific-heat peak on $N$, the number of spins. We also argue that the two-dimensional partition function is expressible at least for $s>1$ as a factor due to domains, and another due to a one-dimensional Ising array of domain walls. Approximate expressions are obtained for the effective number of spins and the effective coupling of the array. We also point out that there is reason to believe the one-dimensional Ising array of domain walls at fixed $\ensuremath{\tau}$ connects smoothly with a similar array for $T<{T}_{c}$.

Physical review. B, Condensed matter•Aug 1, 1983•21 citations

Exact Finite-Size-Scaling Corrections to the Critical Two-Dimensional Ising Model on a Torus. II. Triangular and hexagonal lattices

Jesús Salas

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

•Jan 1, 2001•20 citations
Let f,U,f, U,f,U, and C represent, respectively, the free energy, the internal energy, and the specific heat of the critical Ising model on the M\ifmmode×\else\texttimes\fiNM\ifmmode\times\else\texttimes\fi{}NM\ifmmode×\else\texttimes\fiN square lattice with periodic boundary conditions, and f\ensuremath∞{f}_{\ensuremath{\infty}}f\ensuremath∞​ represents f for fixed M/NM/NM/N and N\ensuremath→\ensuremath∞.\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}.N\ensuremath→​\ensuremath∞. We find that f,U,f, U,f,U, and C can be written as N(f\ensuremath−f\ensuremath∞)=\ensuremath∑i=1\ensuremath∞f2i\ensuremath−1/N2i\ensuremath−1,U=\ensuremath−2+\ensuremath∑i=1\ensuremath∞u2i\ensuremath−1/N2i\ensuremath−1,N(f\ensuremath{-}{f}_{\ensuremath{\infty}})={\ensuremath{\sum}}_{i=1}^{\ensuremath{\infty}}{f}_{2i\ensuremath{-}1}{/N}^{2i\ensuremath{-}1}, U=\ensuremath{-}\sqrt{2}+{\ensuremath{\sum}}_{i=1}^{\ensuremath{\infty}}{u}_{2i\ensuremath{-}1}{/N}^{2i\ensuremath{-}1},N(f\ensuremath−f\ensuremath∞​)=\ensuremath∑i=1\ensuremath∞​f2i\ensuremath−1​/N2i\ensuremath−1,U=\ensuremath−2​+\ensuremath∑i=1\ensuremath∞​u2i\ensuremath−1​/N2i\ensuremath−1, and C=8lnN/\ensuremathπ+\ensuremath∑i=0\ensuremath∞ci/Ni,C=8\mathrm{ln}N/\ensuremath{\pi}+{\ensuremath{\sum}}_{i=0}^{\ensuremath{\infty}}{c}_{i}{/N}^{i},C=8lnN/\ensuremathπ+\ensuremath∑i=0\ensuremath∞​ci​/Ni, i.e., Nf\mathrm{Nf}Nf and U are odd functions of N\ensuremath−1.{N}^{\ensuremath{-}1}.N\ensuremath−1. We also find that u2i\ensuremath−1/c2i\ensuremath−1=1/2{u}_{2i\ensuremath{-}1}{/c}_{2i\ensuremath{-}1}=1/\sqrt{2}u2i\ensuremath−1​/c2i\ensuremath−1​=1/2​ and u2i/c2i=0{u}_{2i}{/c}_{2i}=0u2i​/c2i​=0 for 1< i<\ensuremath∞1<~i<\ensuremath{\infty}1< i<\ensuremath∞ and obtain closed form expressions for f,U,f, U,f,U, and C up to orders 1/N5,1/N5,{1/N}^{5}, {1/N}^{5},1/N5,1/N5, and 1/N3,{1/N}^{3},1/N3, respectively, which implies an analytic equation for c5.{c}_{5}.c5​.
The specific heat, energy, and free energy of an infinitely long, square-lattice ferromagnetic Ising strip consisting of nnn parallel layers with free boundary conditions and a surface magnetic field H1=kBTh1{H}_{1}={k}_{B}{\mathrm{Th}}_{1}H1​=kB​Th1​ imposed on the last layer, is analyzed for large nnn in the light of finite-size scaling theory. It is shown rigorously that the free energy (and, similarly, the energy and specific heat) can be written asymptotically in the scaling form $\ensuremath{\Delta}f(n,t,{h}_{1})\ensuremath{\approx}{n}^{\ensuremath{-}2}(\mathrm{ln}n){W}_{1}(nt,{n}^{\frac{1}{2}}{h}_{1})+{n}^{\ensuremath{-}2}{W}_{2}(nt,{n}^{\frac{1}{2}}{h}_{1})$, where $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$. The scaling functions Wi(x,y){W}_{i}(x,y)Wi​(x,y) are computed in explicit closed form and shown to verify all the analyticity and asymptotic requirements anticipated by scaling theory. Furthermore, in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ at fixed $t\ensuremath{\ne}0$, the bulk and surface contributions to the thermodynamic properties are found to account for all except a correction of order ${e}^{\frac{\mathrm{pna}}{\ensuremath{\xi}(T)}}$, where aaa is the lattice spacing and $\ensuremath{\xi}(T)$ is the bulk correlation length; the value of the small rational constant ppp is interpreted in terms of interference effects between the two opposite boundaries (or "surfaces").
The critical-point anomaly of a plane square $m\ifmmode\times\else\texttimes\fi{}n$ Ising lattice with periodic boundary conditions (a torus) is analyzed asymptotically in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ with $\ensuremath{\xi}=\frac{m}{n}$ fixed. Among other results, it is shown that for fixed $\ensuremath{\tau}=\frac{n(T\ensuremath{-}{T}_{c})}{{T}_{c}}$, the specific heat per spin of a large lattice is given by $\frac{{C}_{\mathrm{mn}}(T)}{{k}_{\mathrm{B}}\mathrm{mn}}={A}_{0}\mathrm{ln}n+B(\ensuremath{\tau}, \ensuremath{\xi})+{B}_{1}(\ensuremath{\tau})\frac{(\mathrm{ln}n)}{n}+\frac{{B}_{2}(\ensuremath{\tau}, \ensuremath{\xi})}{n}+O[\frac{{(\mathrm{ln}n)}^{3}}{{n}^{2}}],$ where explicit expressions can be given for A0{A}_{0}A0​ and for the functions BBB, B1{B}_{1}B1​, and B2{B}_{2}B2​. It follows that the specific-heat peak of the finite lattice is rounded on a scale $\ensuremath{\delta}=\frac{\ensuremath{\Delta}T}{{T}_{c}}\ensuremath{\sim}\frac{1}{n}$, while the maximum in Cmn(T){C}_{\mathrm{mn}}(T)Cmn​(T) is displaced from Tc{T}_{c}Tc​ by $\ensuremath{\epsilon}=\frac{({T}_{c}\ensuremath{-}{T}_{max})}{{T}_{c}}\ensuremath{\sim}\frac{1}{n}$. For ${\ensuremath{\xi}}_{0}>\ensuremath{\xi}>{{\ensuremath{\xi}}_{0}}^{\ensuremath{-}1}$, where ${\ensuremath{\xi}}_{0}=3.13927\ensuremath{\cdots}$, the maximum lies above Tc{T}_{c}Tc​; but for $\ensuremath{\xi}>{\ensuremath{\xi}}_{0}$ or $\ensuremath{\xi}<{{\ensuremath{\xi}}_{0}}^{\ensuremath{-}1}$, the maximum is depressed below Tc{T}_{c}Tc​; when $\ensuremath{\xi}=\ensuremath{\infty}, {\ensuremath{\xi}}_{0}, or {{\ensuremath{\xi}}_{0}}^{\ensuremath{-}1}$, the relative shift in the maximum from Tc{T}_{c}Tc​ is only of order (lnn)n2\frac{(\mathrm{ln}n)}{{n}^{2}}n2(lnn)​. Detailed graphs and numerical data are presented, and the results are compared with some for lattices with free edges. Some heuristic arguments are developed which indicate the possible nature of finite-size critical-point effects in more general systems.
Finite-size scaling, finite-size corrections, and boundary effects for critical systems have attracted much attention in recent years. Here we derive exact finite-size corrections for the free energy F and the specific heat C of the critical ferromagnetic Ising model on the Mu x 2 Nu square lattice with Brascamp-Kunz (BK) boundary conditions [J. Math. Phys. 15, 66 (1974)] and compare such results with those under toroidal boundary conditions. When the ratio xi/2=(Mu+1)/2 Nu is smaller than 1 the behaviors of finite-size corrections for C are quite different for BK and toroidal boundary conditions; when ln(xi/2) is larger than 3, finite-size corrections for C in two boundary conditions approach the same values. In the limit Nu-->infinity we obtain the expansion of the free energy for infinitely long strip with BK boundary conditions. Our results are consistent with the conformal field theory prediction for the mixed boundary conditions by Cardy [Nucl. Phys. B 275, 200 (1986)] although the definitions of boundary conditions in two cases are different in one side of the long strip.
Ferdinand and Fisher have analyzed the finite-size behavior of the two-dimensional Ising model with periodic boundary conditions for TTT near Tc{T}_{c}Tc​. In the thermodynamic limit, the leading correction term to the specific heat has an unusual and unexpected behavior as the shape of the $m\ifmmode\times\else\texttimes\fi{}n$ lattice is changed. For shape parameter s=mn=1s=\frac{m}{n}=1s=nm​=1, the peak in the specific-heat correction term occurs at a reduced temperature ${\ensuremath{\tau}}_{max}=2{K}_{c}n\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}>0$. With increasing sss the peak moves to negative $\ensuremath{\tau}$ values and then increases to 0. We show that this effect may be understood by keeping only the two largest eigenvalues of the transfer matrix. To this approximation, which is good for s=1s=1s=1 and improves exponentially with increasing sss, all the shape dependence of the free energy is due to a factor that can be expressed entirely in terms of a domain-wall energy. Furthermore, the functional form of this factor is the same as that of the finite-length correction to the one-dimensional Ising model. Thus the dependence of ${\ensuremath{\tau}}_{max}$ on sss is qualitatively similar to the dependence of Tmax1{T}_{max}^{1}Tmax1​ the location of the one-dimensional Ising-model specific-heat peak on NNN, the number of spins. We also argue that the two-dimensional partition function is expressible at least for s>1s>1s>1 as a factor due to domains, and another due to a one-dimensional Ising array of domain walls. Approximate expressions are obtained for the effective number of spins and the effective coupling of the array. We also point out that there is reason to believe the one-dimensional Ising array of domain walls at fixed $\ensuremath{\tau}$ connects smoothly with a similar array for T<TcT<{T}_{c}T<Tc​.
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