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Physical review. B, Solid state

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

Helen Au-Yang, Michael Ellis Fisher

Published May 1, 1975• 102 citations
Publisher

Abstract

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").

DOI
10.1103/physrevb.11.3469

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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.
According to finite-size scaling theory, the correlation length in an Ising strip with infinite length and a width of L lattice constants has the form ${\ensuremath{\xi}}_{L}$(T)=Lf(c(T-Tc{T}_{c}Tc​)L) in the scaling regime L\ensuremath{\gg}1,\ensuremath{\Vert}T-Tc{T}_{c}Tc​\ensuremath{\Vert}/Tc{T}_{c}Tc​\ensuremath{\ll}1. Here f(x) is a universal scaling function, which does, however, depend on the boundary conditions at the edges of the strip. Utilizing the correspondence between the two-dimensional Ising model and the one-dimensional quantum Ising model in a transverse field, we calculate the explicit form of the function f(x) for spin-spin and energy-energy correlations in strips with periodic, free, fixed, and mixed free-fixed boundary conditions.
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.
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
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