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arXiv:cond-mat/0009059 · Physical Review Letters

Critical Behavior of the Two-Dimensional Ising Susceptibility

William P. Orrick, Bernie G. Nickel, Anthony John Guttmann, Jacques H. H. Perk

Published Apr 30, 2001• 52 citations
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Abstract

We report computations of the short- and long-distance (scaling) contributions to the square-lattice Ising susceptibility. Both computations rely on summation of correlation functions, obtained using nonlinear partial difference equations. In terms of a temperature variable $\ensuremath{\tau}$, linear in ${T/T}_{c}\ensuremath{-}1$, the short-distance terms have the form ${\ensuremath{\tau}}^{p}(\mathrm{ln}|\ensuremath{\tau}|{)}^{q}$ with $p\ensuremath{\ge}{q}^{2}$. A high- and low-temperature series of $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}323$ terms, generated using an algorithm of complexity $\mathrm{O}({N}^{6})$, are analyzed to obtain the scaling part, which when divided by the leading $|\ensuremath{\tau}{|}^{\ensuremath{-}7/4}$ singularity contains only integer powers of $\ensuremath{\tau}$. Contributions of distinct irrelevant variables are identified and quantified at leading orders $|\ensuremath{\tau}{|}^{9/4}$ and $|\ensuremath{\tau}{|}^{17/4}$.

DOI
10.1103/physrevlett.86.4120

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We have made substantial advances in elucidating the properties of the susceptibility of the square lattice Ising model. We discuss its analyticity properties, certain closed form expressions for subsets of the coefficients, and give an algorithm of complexity O(N6) to determine its first N coefficients. As a result, we have generated and analyzed series with more than 300 terms in both the high- and low-temperature regime. We quantify the effect of irrelevant variables to the scaling-amplitude functions. In particular, we find and quantify the breakdown of simple scaling, in the absence of irrelevant 1 scaling fields, arising first at order |T −Tc | 9/4, though high-low temperature symmetry is still preserved. At terms of order |T −Tc | 17/4 and beyond, this symmetry is no longer present. The short-distance terms are shown to have the form (T − Tc) p (log |T − Tc|) q with p ≥ q2. Conjectured exact expressions for some correlation functions and series coefficients in terms of elliptic theta functions also foreshadow future developments.
Four terms in the expansion for the zero-field susceptibility of the quadratic Ising lattice at criticality are known exactly. We have computed three additional terms in this expansion by analyzing Nickel's high-temperature series for this lattice with second-order homogeneous differential approximants and Pad\'e techniques. These three terms are found to vary as \ensuremath{\Vert}t${\ensuremath{\Vert}}^{1/4}$,t, and \ensuremath{\Vert}t${\ensuremath{\Vert}}^{5/4}$ with t=1-Tc{T}_{c}Tc​/T, and their respective coefficients are obtained to within 0.01% accuracy. It is shown that, if terms of the form \ensuremath{\Vert}t${\ensuremath{\Vert}}^{1/4}$ln\ensuremath{\Vert}t\ensuremath{\Vert} and \ensuremath{\Vert}t${\ensuremath{\Vert}}^{5/4}$ln\ensuremath{\Vert}t\ensuremath{\Vert} were present, their amplitudes would be less than ${10}^{\mathrm{\ensuremath{-}}5}$ of the amplitudes of the \ensuremath{\Vert}t${\ensuremath{\Vert}}^{1/4}$ and \ensuremath{\Vert}t${\ensuremath{\Vert}}^{5/4}$ terms, respectively. The results are in accord with the predictions of the renormalization group if irrelevant variables are neglected and indicate that, if singularities due to irrelevant variables are present, their associated exponents must exceed 2. The results are also used to predict the coefficients of the low-temperature series for this system. Agreement to as much as one part in 105{10}^{5}105 with the known first eleven coefficients of this series is obtained.
We investigate the complex-temperature singularities of the susceptibility of the 2D Ising model on a square lattice. From an analysis of low-temperature series expansions, we find evidence that, as one approaches the point u=u s =-1 (where u=e '4K ) from within the complex extensions of the FM or AFM phases, the susceptibility has a divergent singularity of the form X approximately A s '(1+u)(- gamma s ') with exponent gamma s '=3/2. The critical amplitude A s ' is calculated. Other critical exponents are found to be alpha s '= sigma s =0 and beta s = 1/4 , so that the scaling relation alpha s '+2 beta s + gamma s ' is satisfied. However, using exact results for beta s on the square, triangular, and honeycomb lattices, we show that universality is violated at this singularity: beta s is lattice-dependent. Finally, from an analysis of spin-spin correlation functions, we demonstrate that the correlation length and hence susceptibility are finite as one approaches the point u=-1 from within the symmetric phase. This is confirmed by an explicit study of high-temperature series expansions.
The susceptibility of simple-cubic-lattice ferromagnetic Ising films of nnn two-dimensional lattice layers is studied by extrapolation of high-temperature series expansions (to eleventh and twelfth orders) for n=3,4,…,14n=3,4,\dots{},14n=3,4,…,14 layers with periodic boundary conditions, and n=3,4,…,10n=3,4,\dots{},10n=3,4,…,10 layers with free-surface conditions. The corresponding surface susceptibility series for sc, bcc, and fcc lattices are analyzed in the light of universality hypotheses. On the basis of finite-size scaling theory explicit scaling functions are constructed describing the crossover from two-dimensional to three-dimensional critical behavior in terms of the ratio $\frac{n}{\ensuremath{\xi}(T)}$, where $\ensuremath{\xi}\ensuremath{\sim}\ensuremath{\Delta}{T}^{\ensuremath{-}\ensuremath{\nu}}$ is the bulk correlation length.
The zero·field susceptibility x ( T) of the antiferromagnet in the Ising square lattice with the symmetric interaction was evaluated by carrying out the lattice sum of the short · range correlation function X ( m, n) (O~m, n~4), which is exactly and explicitly calculated on the basis of a "generalized Wronskian " form with dimension m. It is naturally derived from the singularity of X(m, n) that the highest singularity of x ( T) (per site) near the critical temperature Tc is logarithmic as kB Tz ( T) =A + B(l- T/Tc)lnI1- T/Tcl, with A =0.1596264029 and B=0.2009359779 in the best approximation of the present calculation. Both A and B are completely equal below and above the critical temperature.
If scaling variables and scaling expressions are chosen judiciously, ``critical'' scaling analyses do not need to be restricted to a narrow range of temperatures near the critical point. The standard scaling variable $(T\ensuremath{-}{T}_{c})/{T}_{c}$ is inadapted to wide ranges of temperature because it diverges at high temperatures. With the variable $\ensuremath{\tau}=(T\ensuremath{-}{T}_{c})/T$, in the high-dimension (mean-field) limit, the reduced susceptibility and the second-moment correlation length of the Ising ferromagnet depend on temperature as $\ensuremath{\chi}(T)={\ensuremath{\tau}}^{\ensuremath{-}1}$ and $\ensuremath{\xi}(T)={T}^{\ensuremath{-}1/2}{\ensuremath{\tau}}^{\ensuremath{-}1/2}$ exactly over the entire temperature range above the critical temperature Tc{T}_{c}Tc​. For the canonical two-dimensional square lattice near-neighbor Ising ferromagnet, it is shown that compact ``extended scaling'' expressions analogous to the high-dimensional limit form but with the appropriate exponents give accurate approximations to the true temperature dependencies, again over the entire temperature range from Tc{T}_{c}Tc​ to infinity. Within this approach, for near-neighbor interaction systems, there is no crossover temperature above which mean-field-like behavior sets in.