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

TLDR

The study analyzes the critical‑point anomaly of a plane square Ising lattice with periodic boundary conditions in the limit of large \(n\) while keeping the aspect ratio \(\xi=m/n\) fixed, and discusses heuristic arguments for finite‑size critical‑point effects in general systems. The authors present detailed graphs and numerical data, comparing their results with those for lattices with free edges. They find that the specific heat per spin of a large lattice follows a logarithmic scaling with corrections, that the finite‑lattice peak is rounded and shifted by \(O(1/n)\) relative to the critical temperature, with the shift direction depending on the aspect ratio \(\xi\) relative to a critical value \(\xi_{0}\).

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.