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Physical review. B, Condensed matter

Spin anisotropy and crossover in the Potts model

D. J. Wallace, A. P. Young

Published Mar 1, 1978• 91 citations
Publisher

Abstract

A group-theoretical analysis is presented of the perturbation's in the $n+1$-state Potts model, which arise when the equivalence between the states is broken. In the continuum version of this model, described by $n$ real fields, all possible quadratic perturbations transform according to two irreducible representations of the underlying symmetry group, so there are two associated cross-over exponents. One of these exponents is equal to the order-parameter exponent $\ensuremath{\beta}$ for any $n$. In the limit $n\ensuremath{\rightarrow}0$ the other exponent $\ensuremath{\varphi}$ describes crossover between percolation and random Ising behavior and is also related to an exponent describing the conductivity of a random resistor network close to the percolation threshold. We show that $\ensuremath{\varphi}=1$, in this limit, to all orders in perturbation theory. Consequently, there are no new exponents associated with crossover to anisotropic behavior for $n\ensuremath{\rightarrow}0$.

DOI
10.1103/physrevb.17.2384

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Physical review. B, Solid state•Sep 1, 1974•256 citations
A group-theoretical analysis is presented of the perturbation's in the n+1n+1n+1-state Potts model, which arise when the equivalence between the states is broken. In the continuum version of this model, described by nnn real fields, all possible quadratic perturbations transform according to two irreducible representations of the underlying symmetry group, so there are two associated cross-over exponents. One of these exponents is equal to the order-parameter exponent \ensuremathβ\ensuremath{\beta}\ensuremathβ for any nnn. In the limit n\ensuremath→0n\ensuremath{\rightarrow}0n\ensuremath→0 the other exponent \ensuremathφ\ensuremath{\varphi}\ensuremathφ describes crossover between percolation and random Ising behavior and is also related to an exponent describing the conductivity of a random resistor network close to the percolation threshold. We show that \ensuremathφ=1\ensuremath{\varphi}=1\ensuremathφ=1, in this limit, to all orders in perturbation theory. Consequently, there are no new exponents associated with crossover to anisotropic behavior for n\ensuremath→0n\ensuremath{\rightarrow}0n\ensuremath→0.
A continuous ${\ensuremath{\varphi}}^{3}$ field theory with quadratic symmetry breaking and isotropic trilinear interaction ${g}_{0} \ensuremath{\Sigma}{d}_{\mathrm{ijk}}{\ensuremath{\varphi}}_{i}{\ensuremath{\varphi}}_{j}{\ensuremath{\varphi}}_{k}$ is studied, to one-loop order in dimension $d=6\ensuremath{-}\ensuremath{\epsilon}$, for an anisotropic (n+1n+1n+1)-state Potts model. Effective critical exponents for the crossover induced by quadratic symmetry breaking are calculated, and the results, in the limit n=0n=0n=0, are applicable to the thermally driven crossover in random bond-diluted ferromagnets near the percolation threshold. The limitation of a single renormalized ggg is explicitly pointed out.
The effect of quadratic symmetry breaking (QSB) on two representations for the Potts vectors in a continuum-field model are studied to two-loop order in renormalized perturbation theory in $d=6\ensuremath{-}\ensuremath{\epsilon}$ dimensions, in extension of an earlier group-theoretical analysis by Wallace and Young. The explicit dependence of the crossover exponent $\ensuremath{\varphi}$ that corresponds to QSB that destroys the equivalence between pairs of Potts vectors is obtained as a function of ddd and nnn for the ppp-state model with p=n+1p=n+1p=n+1. It is shown that this exponent follows from the calculation of vertex functions in a representation due to Wallace and Young, whereas a second crossover exponent $\overline{\ensuremath{\varphi}}$, that can be identified with the critical exponent $\ensuremath{\beta}$, and which corresponds to QSB that favors a single Potts vector against the others, follows from a calculation using the representation of the Potts vectors due to Priest and Lubensky.
A cutoff-dependent formulation of renormalized perturbation theory is extended to study the crossover of the parallel susceptibility in systems with quadratic symmetry breaking near a bicritical point, to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$. In terms of the true temperature variable $t=T\ensuremath{-}{T}_{c}(g)$, for anisotropy ggg, and crossover variable $x=\frac{g}{{t}^{\ensuremath{\varphi}}}$, the effective susceptibility exponent ${\ensuremath{\gamma}}_{\mathrm{eff}}(t,x)$ is calculated for two ${\ensuremath{\varphi}}^{4}$-field models, one with a sharp momentum cutoff $\ensuremath{\Lambda}$ and the other with a smooth term ${\ensuremath{\Lambda}}^{\ensuremath{-}2}{({\ensuremath{\nabla}}^{2}\ensuremath{\varphi})}^{2}$. Corrections to the first model due to insertions of composite operators of canonical dimension six, inside the scaling regime, are explicitly calculated and shown to yield small but not negligible contributions to crossover behavior. The role of new, finite renormalizations is discussed and it is shown that either a fairly monotonic ${\ensuremath{\gamma}}_{\mathrm{eff}}(t,x)$ is obtained or a displaced maximum may appear depending on the outcome of a higher-order renormalization-group study, which is not done here.
The randomly diluted resistor network is formulated in terms of an n-replicated s-state Potts model with a spin-spin coupling constant J in the limit when first n, then s, and finally 1/J go to zero. This limit is discussed and to leading order in 1/J the generalized susceptibility is shown to reproduce the results of the accompanying paper where the resistor network is treated using the xy model. This Potts Hamiltonian is converted into a field theory by the usual Hubbard-Stratonovich transformation and thereby a renormalization-group treatment is developed to obtain the corrections to the critical exponents to first order in \ensuremath{\epsilon}=6-d, where d is the spatial dimensionality. The recursion relations are shown to be the same as for the xy model. Their detailed analysis (given in the accompanying paper) gives the resistance crossover exponent as ${\ensuremath{\varphi}}_{1}$=1+\ensuremath{\epsilon}/42, and determines the critical exponent, t for the conductivity of the randomly diluted resistor network at concentrations, p, just above the percolation threshold: t=(d-2)\ensuremath{\nu}+${\ensuremath{\varphi}}_{1}$, where \ensuremath{\nu} is the critical exponent for the correlation length at the percolation threshold. These results correct previously accepted results giving \ensuremath{\varphi}=1 to all orders in \ensuremath{\epsilon}. The new result for ${\ensuremath{\varphi}}_{1}$ removes the paradox associated with the numerical result that t>1 for d=2, and also shows that the Alexander-Orbach conjecture, while numerically quite accurate, is not exact, since it disagrees with the \ensuremath{\epsilon} expansion.
We present results of a numerical simulation of the qqq-state random bond Potts model in two dimensions and for large qqq. In particular, care is taken to study the crossover from the pure model to the random model, as well as the crossover from the percolation to the random model. We show how to determine precisely the random fixed point and measure critical exponents at this point.
The crossover behavior of the susceptibilities ${\ensuremath{\chi}}^{\ensuremath{\alpha}\ensuremath{\alpha}}$ of spin systems with anisotropic exchange coupling is discussed on the basis of an "extended" scaling hypothesis which gives ${\ensuremath{\chi}}^{\ensuremath{\alpha}\ensuremath{\alpha}}(T,g)\ensuremath{\approx}A{t}^{\ensuremath{-}\ensuremath{\gamma}}{X}_{\ensuremath{\alpha}}(\frac{Bg}{{t}^{\ensuremath{\varphi}}})$, where ggg is the anisotropy parameter, $t=\frac{(T\ensuremath{-}{T}_{c0})}{{T}_{c0}}$, with Tc0{T}_{c0}Tc0​ and $\ensuremath{\gamma}$ being the iso tropic critical temperature, and susceptibility exponent, while AAA and BBB are model-dependent amplitudes. Analysis of high-temperature series expansions for ${\ensuremath{\chi}}^{\ensuremath{\alpha}\ensuremath{\alpha}}(T,g)$ as polynomials in ggg for rhombic and axial anisotropy in the fcc, bcc, and simple cubic classical Heisenberg (n=3n=3n=3) and classical XYXYXY and planar-spin models (N=2N=2N=2), verifies the scaling with crossover exponents $\ensuremath{\varphi}=1.25\ifmmode\pm\else\textpm\fi{}0.015$ (n=3n=3n=3) and $\ensuremath{\varphi}=1.175\ifmmode\pm\else\textpm\fi{}0.015$ (n=2n=2n=2). The universality of the scaling functions ${X}_{\ensuremath{\alpha}}(x)$ is demonstrated both for small xxx and in the anisotropic limit $T\ensuremath{\rightarrow}{T}_{c}(g)$, where $x\ensuremath{\rightarrow}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{x}$; for n=3n=3n=3, accurate representations are constructed in the form ${X}_{\ensuremath{\alpha}}(x)\ensuremath{\simeq}{P}_{\ensuremath{\alpha}}(\frac{x}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{x}}){(1\ensuremath{-}\frac{x}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{x}})}^{\ensuremath{-}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\gamma}}}$, where $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\gamma}}$ is the anisotropic susceptibility exponent, while ${P}_{\ensuremath{\alpha}}(z)$ varies smoothly from ${P}_{\ensuremath{\alpha}}(0)=1$ to ${P}_{\ensuremath{\alpha}}(1)\ensuremath{\simeq}1.10$.