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Journal of Physics C Solid State Physics

Regularly spaced defects in Ising models

Michael Ellis Fisher, Helen Au-Yang

Published Oct 21, 1975• 13 citations
Publisher

Abstract

The results of exact calculations of the critical behaviour of square-lattice Ising models with regularly spaced defects of finite density x=1/nm are reported and shown to satisfy certain scaling relations.
DOI
10.1088/0022-3719/8/20/003

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We address the problem of the definition of the finite-volume correlation length. First, we study the large-N limit of the N-vector model, and we show the existence of several constraints on the definition if regularity of the finite-size scaling functions and correct anomalous behaviour above the upper critical dimension are required. Then, we study in detail a model in which the zero mode is prohibited. Such. a model is a generalization of the fixed-magnetization Ising model which is equivalent to the lattice gas. Also in this case, we find that the finite-volume correlation length must satisfy appropriate constraints in order to obtain regular finite-size scaling functions, and, above the upper critical dimension, an anomalous scaling behaviour. The large-N results are confirmed by a one-loop calculation in the lattice phi (4) theory.
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.