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Physical Review

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

Arthur E. Ferdinand, Michael Ellis Fisher

Published Sep 10, 1969• 731 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.185.832

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

Helen Au-Yang, Michael Ellis Fisher

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Bounded and inhomogeneous Ising models. IV. Specific-heat amplitude for regular defects

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

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Critical two-dimensional Ising model with free, fixed ferromagnetic, fixed antiferromagnetic, and double antiferromagnetic boundaries

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Specific-Heat Scaling Functions for Ising and Heisenberg Models and Comparison with Experiments on Nickel

Richard Krasnow, Eugene Stanley

We calculate the scaled constant-magnetization and constant magnetic field specific heats $\frac{[{C}_{M}(\ensuremath{\epsilon},M)\ensuremath{-}{C}_{M}(\ensuremath{\epsilon},0)]}{{M}^{\ensuremath{-}\frac{\ensuremath{\alpha}}{\ensuremath{\beta}}}}$ and $\frac{[{C}_{H}(\ensuremath{\epsilon},H)\ensuremath{-}{C}_{H}(\ensuremath{\epsilon},0)]}{{H}^{\ensuremath{-}\frac{\ensuremath{\alpha}}{\ensuremath{\beta}\ensuremath{\delta}}}}$ as functions of the scaled variable $x\ensuremath{\equiv}\frac{\ensuremath{\epsilon}}{{M}^{\frac{1}{\ensuremath{\beta}}}}$ and $y\ensuremath{\equiv}\frac{\ensuremath{\epsilon}}{{H}^{\frac{1}{\ensuremath{\beta}\ensuremath{\delta}}}}$, respectively [where $\ensuremath{\epsilon}\ensuremath{\equiv}\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$ and ${T}_{c}$ is the critical temperature],for the spin-\textonehalf{} Ising model (bcc lattice) and the spin-\textonehalf{}, $\ensuremath{\infty}$ Heisenberg models (fcc lattice). Our calculations are based on previously calculated scaling functions for the $M(H,T)$ equation of state. We also calculate the amplitudes of the zero-field specific heat for $T>{T}_{c}$ and $T<{T}_{c}$. Thus, we obtain the functions ${C}_{M}(\ensuremath{\epsilon},M)$ and ${C}_{H}(\ensuremath{\epsilon},H)$, though for the Heisenberg models we cannot obtain the finite nonzero constant ${C}_{M}(0,0)={C}_{H}(0,0)$. We compare our calculated functions with the data of Connelly, Loomis, and Mapother on nickel.

Physical review. B, Solid state•Jul 1, 1973•20 citations
The critical-point anomaly of a plane square m\ifmmode×\else\texttimes\finm\ifmmode\times\else\texttimes\fi{}nm\ifmmode×\else\texttimes\fin Ising lattice with periodic boundary conditions (a torus) is analyzed asymptotically in the limit n\ensuremath→\ensuremath∞n\ensuremath{\rightarrow}\ensuremath{\infty}n\ensuremath→\ensuremath∞ with \ensuremathξ=mn\ensuremath{\xi}=\frac{m}{n}\ensuremathξ=nm​ fixed. Among other results, it is shown that for fixed \ensuremathτ=n(T\ensuremath−Tc)Tc\ensuremath{\tau}=\frac{n(T\ensuremath{-}{T}_{c})}{{T}_{c}}\ensuremathτ=Tc​n(T\ensuremath−Tc​)​, the specific heat per spin of a large lattice is given by Cmn(T)kBmn=A0lnn+B(\ensuremathτ,\ensuremathξ)+B1(\ensuremathτ)(lnn)n+B2(\ensuremathτ,\ensuremathξ)n+O[(lnn)3n2],\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}}],kB​mnCmn​(T)​=A0​lnn+B(\ensuremathτ,\ensuremathξ)+B1​(\ensuremathτ)n(lnn)​+nB2​(\ensuremathτ,\ensuremathξ)​+O[n2(lnn)3​], 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 \ensuremathδ=\ensuremathΔTTc\ensuremath∼1n\ensuremath{\delta}=\frac{\ensuremath{\Delta}T}{{T}_{c}}\ensuremath{\sim}\frac{1}{n}\ensuremathδ=Tc​\ensuremathΔT​\ensuremath∼n1​, while the maximum in Cmn(T){C}_{\mathrm{mn}}(T)Cmn​(T) is displaced from Tc{T}_{c}Tc​ by \ensuremathϵ=(Tc\ensuremath−Tmax)Tc\ensuremath∼1n\ensuremath{\epsilon}=\frac{({T}_{c}\ensuremath{-}{T}_{max})}{{T}_{c}}\ensuremath{\sim}\frac{1}{n}\ensuremathϵ=Tc​(Tc​\ensuremath−Tmax​)​\ensuremath∼n1​. For \ensuremathξ0>\ensuremathξ>\ensuremathξ0\ensuremath−1{\ensuremath{\xi}}_{0}>\ensuremath{\xi}>{{\ensuremath{\xi}}_{0}}^{\ensuremath{-}1}\ensuremathξ0​>\ensuremathξ>\ensuremathξ0​\ensuremath−1, where \ensuremathξ0=3.13927\ensuremath⋯{\ensuremath{\xi}}_{0}=3.13927\ensuremath{\cdots}\ensuremathξ0​=3.13927\ensuremath⋯, the maximum lies above Tc{T}_{c}Tc​; but for \ensuremathξ>\ensuremathξ0\ensuremath{\xi}>{\ensuremath{\xi}}_{0}\ensuremathξ>\ensuremathξ0​ or \ensuremathξ<\ensuremathξ0\ensuremath−1\ensuremath{\xi}<{{\ensuremath{\xi}}_{0}}^{\ensuremath{-}1}\ensuremathξ<\ensuremathξ0​\ensuremath−1, the maximum is depressed below Tc{T}_{c}Tc​; when \ensuremathξ=\ensuremath∞,\ensuremathξ0,or\ensuremathξ0\ensuremath−1\ensuremath{\xi}=\ensuremath{\infty}, {\ensuremath{\xi}}_{0}, or {{\ensuremath{\xi}}_{0}}^{\ensuremath{-}1}\ensuremathξ=\ensuremath∞,\ensuremathξ0​,or\ensuremathξ0​\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.
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").
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 $M\ifmmode\times\else\texttimes\fi{}N$ square lattice with periodic boundary conditions, and ${f}_{\ensuremath{\infty}}$ represents f for fixed M/NM/NM/N and $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}.$ We find that f,U,f, U,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., Nf\mathrm{Nf}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 u2i/c2i=0{u}_{2i}{/c}_{2i}=0u2i​/c2i​=0 for $1<~i<\ensuremath{\infty}$ 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 amplitude, $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{A}(x,\ensuremath{\tau})$, of the logarithmic specific-heat singularity of a square Ising lattice with bond defects regularly spaced on an $m\ifmmode\times\else\texttimes\fi{}n$ grid (with x=1mnx=\frac{1}{\mathrm{mn}}x=mn1​ and $\ensuremath{\tau}=\frac{n}{m}$) is calculated asymptotically with the result $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{A}(x,\ensuremath{\tau})={A}_{0}\ensuremath{-}{A}_{1}x\mathrm{ln}{x}^{\ensuremath{-}1}\ensuremath{-}|{A}_{2}(\ensuremath{\tau})x\ensuremath{-}{A}_{3}{x}^{2}{\mathrm{ln}}^{2}x+O({x}^{2}\mathrm{ln}x)$, as $x\ensuremath{\rightarrow}0$. The coefficient A1{A}_{1}A1​ verifies the scaling prediction A1=C1{A}_{1}={C}_{1}A1​=C1​, where C1{C}_{1}C1​ is the amplitude of the (ln∣t∣)2{(\mathrm{ln}|t|)}^{2}(ln∣t∣)2 term in the incremental specific heat due to a single, isolated defect.
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​.
The critical two-dimensional Ising model is studied with four types boundary conditions: free, fixed ferromagnetic, fixed antiferromagnetic, and fixed double antiferromagnetic. Using bond propagation algorithms with surface fields, we obtain the free energy, internal energy, and specific heat numerically on square lattices with a square shape and various combinations of the four types of boundary conditions. The calculations are carried out on the square lattices with size N×N and 30<N<1000. The numerical data are analyzed with finite-size scaling. The bulk, edge, and corner terms are extracted very accurately. The exact results are conjectured for the corner logarithmic term in the free energy, the edge logarithmic term in the internal energy, and the corner logarithmic term in the specific heat. The corner logarithmic terms in the free energy agree with the conformal field theory very well.
We calculate the scaled constant-magnetization and constant magnetic field specific heats $\frac{[{C}_{M}(\ensuremath{\epsilon},M)\ensuremath{-}{C}_{M}(\ensuremath{\epsilon},0)]}{{M}^{\ensuremath{-}\frac{\ensuremath{\alpha}}{\ensuremath{\beta}}}}$ and $\frac{[{C}_{H}(\ensuremath{\epsilon},H)\ensuremath{-}{C}_{H}(\ensuremath{\epsilon},0)]}{{H}^{\ensuremath{-}\frac{\ensuremath{\alpha}}{\ensuremath{\beta}\ensuremath{\delta}}}}$ as functions of the scaled variable $x\ensuremath{\equiv}\frac{\ensuremath{\epsilon}}{{M}^{\frac{1}{\ensuremath{\beta}}}}$ and $y\ensuremath{\equiv}\frac{\ensuremath{\epsilon}}{{H}^{\frac{1}{\ensuremath{\beta}\ensuremath{\delta}}}}$, respectively [where $\ensuremath{\epsilon}\ensuremath{\equiv}\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$ and Tc{T}_{c}Tc​ is the critical temperature],for the spin-\textonehalf{} Ising model (bcc lattice) and the spin-\textonehalf{}, $\ensuremath{\infty}$ Heisenberg models (fcc lattice). Our calculations are based on previously calculated scaling functions for the M(H,T)M(H,T)M(H,T) equation of state. We also calculate the amplitudes of the zero-field specific heat for T>TcT>{T}_{c}T>Tc​ and T<TcT<{T}_{c}T<Tc​. Thus, we obtain the functions ${C}_{M}(\ensuremath{\epsilon},M)$ and ${C}_{H}(\ensuremath{\epsilon},H)$, though for the Heisenberg models we cannot obtain the finite nonzero constant CM(0,0)=CH(0,0){C}_{M}(0,0)={C}_{H}(0,0)CM​(0,0)=CH​(0,0). We compare our calculated functions with the data of Connelly, Loomis, and Mapother on nickel.