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Physical review. B, Condensed matter

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

Peter H. Kleban, Gulden Akinci

Published Aug 1, 1983• 21 citations
Publisher

Abstract

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 $T<{T}_{c}$.

DOI
10.1103/physrevb.28.1466

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Physical Review•Sep 10, 1969•731 citations
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×\else\texttimes\finm\ifmmode\times\else\texttimes\fi{}nm\ifmmode×\else\texttimes\fin 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 \ensuremathτmax=2Kcn(T\ensuremath−Tc)Tc>0{\ensuremath{\tau}}_{max}=2{K}_{c}n\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}>0\ensuremathτmax​=2Kc​nTc​(T\ensuremath−Tc​)​>0. With increasing sss the peak moves to negative \ensuremathτ\ensuremath{\tau}\ensuremathτ 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 \ensuremathτmax{\ensuremath{\tau}}_{max}\ensuremathτmax​ 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 \ensuremathτ\ensuremath{\tau}\ensuremathτ connects smoothly with a similar array for T<TcT<{T}_{c}T<Tc​.
Ferdinand and Fisher have discovered some unusual lattice-shape effects on the finite-size-induced specific-heat peak shift in the two-dimensional Ising-model critical region. It is shown that they are well approximated by keeping only the two largest eigenvalues of the transfer matrix, and may be expressed entirely in terms of a one-dimensional Ising array of domain walls. Approximate expressions for the effective coupling and number of spins are given. It is also pointed out that in general domain walls should be thermodynamically important for such finite-size effects.
The finite-size effect is studied in the Kasteleyn model of dimers on the brick lattice. This model is isomorphic to an anisotropic domain-wall model. Asymptotic analysis of the exact Pfaffian solution for the specific heat establishes that finite-size-scaling theory is valid near the critical point of this model. The finite-size-scaling function is a function of a scaled temperature variable \ensuremath{\tau} and a shape factor scrr=N2{N}^{2}N2/M, where 2N is the number of lattice points in the direction perpendicular to the preferred axis for the domain walls and 2M is the number of lattice points parallel to the preferred axis. The scaled temperature variable \ensuremath{\tau} is given by MN2{\mathrm{MN}}^{2}MN2t/(M+N2{N}^{2}N2), where t is the reduced temperature. As a function of \ensuremath{\tau} the scaling function scrP(\ensuremath{\tau},scrr) is a sequence of \ensuremath{\delta} functions in the limit scrr=0 and a smooth single-peaked function in the limit scrr=\ensuremath{\infty}. In the latter case the specific heat per lattice site can be written as ,\ensuremath{\infty}), where \ensuremath{\alpha} is the bulk specific-heat exponent with the known value of (1/2) and ${\ensuremath{\nu}}_{M}$ is found to have the value of 1. In the case scrr=0, the specific heat per lattice site can be written in an equivalent form by replacing M by N and ${\ensuremath{\nu}}_{m}$ by ${\ensuremath{\nu}}_{N}$ which takes the value ${\ensuremath{\nu}}_{N}$=(1/2). According to finite-size-scaling theory ${\ensuremath{\nu}}_{M}$ and ${\ensuremath{\nu}}_{N}$ may be interpreted to be the critical exponents ${\ensuremath{\nu}}_{\mathrm{y}}$ and ${\ensuremath{\nu}}_{\mathrm{x}}$, respectively, of the divergent length scales in the two principal directions. Our exact values of ${\ensuremath{\nu}}_{m}$ and ${\ensuremath{\nu}}_{N}$ are in agreement with the values of ${\ensuremath{\nu}}_{\mathrm{y}}$ and ${\ensuremath{\nu}}_{\mathrm{x}}$ predicted for general anisotropic domain-wall models.
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
Exact ground states of two-dimensional Ising spin glasses with Gaussian and bimodal $(\ifmmode\pm\else\textpm\fi{}J)$ distributions of the disorder are calculated using a ``matching''algorithm, which allows large system sizes of up to N=4802{N=480}^{2}N=4802 spins to be investigated. We study domain walls induced by two rather different types of boundary condition changes, and, in each case, analyze the system-size dependence of an appropriately defined ``defect energy,'' which we denote by $\ensuremath{\Delta}E.$ For Gaussian disorder, we find a power-law behavior $\ensuremath{\Delta}E\ensuremath{\sim}{L}^{\ensuremath{\theta}},$ with $\ensuremath{\theta}=\ensuremath{-}0.266(2)$ and $\ensuremath{\theta}=\ensuremath{-}0.282(2)$ for the two types of boundary condition changes. These results are in reasonable agreement with each other, allowing for small systematic effects. They also agree well with earlier work on smaller sizes. The negative value indicates that two dimensions is below the lower critical dimension dc.{d}_{c}.dc​. For the $\ifmmode\pm\else\textpm\fi{}J$ model, we obtain a different result, namely that the domain-wall energy saturates at a nonzero value for $\stackrel{\ensuremath{\rightarrow}}{L}\ensuremath{\infty},$ so $\ensuremath{\theta}=0,$ indicating that the lower critical dimension for the $\ifmmode\pm\else\textpm\fi{}J$ model is exactly dc=2.{d}_{c}=2.dc​=2.
In order to investigate the effects of connectivity and proximity in the specific heat, a special class of exactly solvable planar layered Ising models has been studied in the thermodynamic limit. The Ising models consist of repeated uniform horizontal strips of width mmm connected by sequences of vertical strings of length nnn mutually separated by distance NNN, with N=1,2N=1,2N=1,2 and 333. We find that the critical temperature Tc(N,m,n)T_c(N,m,n)Tc​(N,m,n), arising from the collective effects, decreases as nnn and NNN increase, and increases as mmm increases, as it should be. The amplitude A(N,m,n)A(N,m,n)A(N,m,n) of the logarithmic divergence at the bulk critical temperature Tc(N,m,n)T_c(N,m,n)Tc​(N,m,n) becomes smaller as nnn and mmm increase. A rounded peak, with size of order ln⁡m\ln mlnm and signifying the one-dimensional behavior of strips of finite width mmm, appears when nnn is large enough. The appearance of these rounded peaks does not depend on mmm as much, but depends rather more on NNN and nnn, which is rather perplexing. Moreover, for fixed mmm and nnn, the specific heats are not much different for different NNN. This is a most surprising result. For N=1N=1N=1, the spin-spin correlation in the center row of each strip can be written as a Toeplitz determinant with a generating function which is much more complicated than in Onsager's Ising model. The spontaneous magnetization in that row can be calculated numerically and the spin-spin correlation is shown to have two-dimensional Ising behavior.
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