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Physical review. B, Condensed matter

Crossover exponents for the Potts model with quadratic symmetry breaking

W. K. Theumann, M. A. Gusmão

Published Sep 1, 1984• 5 citations
Publisher

Abstract

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 $d$ and $n$ for the $p$-state model with $p=n+1$. 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.

DOI
10.1103/physrevb.30.2800

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Physical review. B, Condensed matter•Jun 1, 1983•5 citations
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−\ensuremathϵd=6\ensuremath{-}\ensuremath{\epsilon}d=6\ensuremath−\ensuremathϵ dimensions, in extension of an earlier group-theoretical analysis by Wallace and Young. The explicit dependence of the crossover exponent \ensuremathφ\ensuremath{\varphi}\ensuremathφ 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 \ensuremathφ‾\overline{\ensuremath{\varphi}}\ensuremathφ​, that can be identified with the critical exponent \ensuremathβ\ensuremath{\beta}\ensuremathβ, 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.
Renormalized perturbation theory with generalized minimal subtraction is used as an appropriate renormalization-group procedure for the study of crossover behavior in the continuum version of the p-state Potts model with quadratic and trilinear symmetry breaking, within the representation of Priest and Lubensky, by means of a two-loop-order calculation in d=6-\ensuremath{\epsilon} dimensions. The boundary between first- and second-order phase transitions is studied for longitudinal and transverse ordering as a function of p. A fixed-point runaway for longitudinal ordering is consistent with a mean-field interpretation of a first-order transition for p>p∗{p}^{\mathrm{*}}p∗, where p∗{p}^{\mathrm{*}}p∗\ensuremath{\lesssim}2 but not with a second-order transition for pp∗{p}^{\mathrm{*}}p∗. Finite and stable fixed points are obtained for transverse ordering, one that follows by crossover from the symmetric fixed point for 2(13/3), in consistency with the usual mean-field interpretation of a second-order transition for 2p3.
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 critical exponents for two ${\ensuremath{\varphi}}^{3}$-field theories with long-range (LR) interactions decaying as 1/${R}^{d+\ensuremath{\sigma}}$, \ensuremath{\sigma}>0, are calculated to two-loop order in renormalized perturbation theory in d=3\ensuremath{\sigma}-\ensuremath{\epsilon}' dimensions. One is the continuum version of the p-state Potts model and the other is the scalar field theory with imaginary coupling that describes the Yang-Lee edge-singularity problem. The two crossover exponents for quadratic symmetry breaking discussed by Wallace and Young and in recent work by the present authors are also calculated in the first case. By means of renormalization-group recursion relations to one-loop order, it is shown that the LR fixed point is stable for all \ensuremath{\sigma}2 whenever ${\ensuremath{\eta}}_{\mathrm{SR}}$, the critical correlation-function exponent for short-range (SR) interactions, is negative, implying a discontinuity of critical exponents at \ensuremath{\sigma}=2. This is the case for the (p2)-state Potts model and for the Yang-Lee edge-singularity problem, and is in agreement with recent results by Chang and Sak for the Ising spin-glass problem. For the (p>2)-state Potts model there is an indication of a continuous crossover to SR behavior at \ensuremath{\sigma}=2-${\ensuremath{\eta}}_{\mathrm{SR}}$, with ${\ensuremath{\eta}}_{\mathrm{SR}}$>0. It is pointed out that a number of exact results [\ensuremath{\beta}=(1-\ensuremath{\epsilon}'/2\ensuremath{\sigma})\ensuremath{\sigma}\ensuremath{\nu}, \ensuremath{\sigma}^=(d-\ensuremath{\sigma})/(d+\ensuremath{\sigma}), in which \ensuremath{\sigma}^ is the Yang-Lee edge-singularity exponent, and ${\ensuremath{\nu}}^{\mathrm{\ensuremath{-}}1}$=(d-\ensuremath{\sigma})/2 for a scalar theory] may apply within the LR expansion depending on \ensuremath{\eta}=2-\ensuremath{\sigma} (shown here to hold at least to two-loop order) being exact, to all orders.
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 $\ensuremath{\beta}$ for any nnn. 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 $n\ensuremath{\rightarrow}0$.
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.
It is shown that a trilinear symmetry breaking which destroys the equivalence of the states in a continuum generalization of the ppp-state Potts model yields first-order phase transitions for all p>1p>1p>1, in contrast to the results of the symmetric theory where there is a second-order transition for p<2p<2p<2 and a first-order transition for p>2p>2p>2, in $d=6\ensuremath{-}\ensuremath{\epsilon}$ dimensions, and in spite of mean-field---theory predictions. A calculation in renormalized perturbation theory, to one-loop order, on a three-trilinear-coupling theory yields two new accessible and partly stable asymmetric fixed points beyond the symmetric one, except for the three-state Potts model, one for ${2.2}^{\ensuremath{-}}\ensuremath{\le}p<\ensuremath{\infty}$ and the other for 1<p<1331<p<\frac{13}{3}1<p<313​. There is a fixed-point runaway for the first one when $p<{2.2}^{\ensuremath{-}}$ and for the second when p>133p>\frac{13}{3}p>313​, which are interpreted as the usual first-order transitions. For the indicated ranges of ppp the transitions are of first order near a spinodal point, with uniaxial ordering in the first case and transverse ordering in the second. Critical exponents that could describe the approach to the spinodal points are explicitly calculated.