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Physical review. B, Condensed matter

Two-loop renormalization of Potts spin-glass

Mau-chung Chang

Published Apr 1, 1985• 0 citations
Publisher

Abstract

The q-state Potts spin-glass (SG) with short-ranged random-exchange interaction is formulated field theoretically. The one-loop calculation [M.-c. Chang, Phys. Rev. B 28, 6592 (1983)] shows that the Potts SG transition is of first order when q\ensuremath{\ge}4 and for q=3 it is undetermined because the Potts SG fixed point is only marginally stable. The calculations are carried out to two-loop order with use of renormalized perturbation theory by means of minimal subtraction procedure. The results show that the three-state Potts SG fixed point is infrared stable, and we conclude that it is second-order transition. The critical exponents to order ${\ensuremath{\epsilon}}^{2}$ (\ensuremath{\epsilon}=6-d) are given.

DOI
10.1103/physrevb.31.4329

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The q-state Potts spin-glass (SG) with short-ranged random-exchange interaction is formulated field theoretically. The one-loop calculation [M.-c. Chang, Phys. Rev. B 28, 6592 (1983)] shows that the Potts SG transition is of first order when q\ensuremath{\ge}4 and for q=3 it is undetermined because the Potts SG fixed point is only marginally stable. The calculations are carried out to two-loop order with use of renormalized perturbation theory by means of minimal subtraction procedure. The results show that the three-state Potts SG fixed point is infrared stable, and we conclude that it is second-order transition. The critical exponents to order \ensuremathϵ2{\ensuremath{\epsilon}}^{2}\ensuremathϵ2 (\ensuremath{\epsilon}=6-d) are given.
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.
We consider the spin-glass (SG) with long-range (LR) random exchange interaction varying with distance RRR as ${R}^{\ensuremath{-}\frac{d}{2}\ensuremath{-}\frac{\ensuremath{\sigma}}{2}}$, where ddd is the space dimensionality and $\ensuremath{\sigma}>0$. It is shown that the LR fixed point is stable if $\ensuremath{\sigma}<2$. The nonclassical ($d<3\ensuremath{\sigma}$) LR critical exponents in powers of $3\ensuremath{\sigma}\ensuremath{-}d$ are not continuous at $\ensuremath{\sigma}=2$ when the transition to short-range (SR) exchange is made.
In the present paper we analyze the critical properties of a quantum spherical spin-glass model with short-range random interactions. Since the model allows for rigorous detailed calculations, we can show how the effective partition function calculated with the help of the replica method for the spin-glass fluctuating fields ${Q}_{\ensuremath{\alpha}\ensuremath{\gamma}}(\stackrel{P\vec}{k}{\ensuremath{\omega}}_{1}{\ensuremath{\omega}}_{2})$ separates into a mean-field contribution for ${Q}_{\ensuremath{\alpha}\ensuremath{\alpha}}(0;\ensuremath{\omega};\ensuremath{-}\ensuremath{\omega})$ and a strictly short-range partition function for the fields ${Q}_{\ensuremath{\alpha}\ensuremath{\ne}\ensuremath{\gamma}}(\stackrel{P\vec}{k}{\ensuremath{\omega}}_{1}{\ensuremath{\omega}}_{2})$. Here $\ensuremath{\alpha},\ensuremath{\gamma}=1,\dots{},n$ are replica indices. The mean-field part WMF{W}_{\text{MF}}WMF​ coincides with previous results. The short-range part WSR{W}_{\text{SR}}WSR​ describes a phase transition in a Q3{Q}^{3}Q3-field theory, where the fluctuating fields depend on a space variable $\stackrel{P\vec}{r}$ and times ${\ensuremath{\tau}}_{1}$ and ${\ensuremath{\tau}}_{2}$. This we analyze using the renormalization group with dimensional regularization and minimal subtraction of dimensional poles. By generalizing standard field theory methods to our particular situation, we can identify the critical dimensionality as dc=5{d}_{c}=5dc​=5 at very low temperatures due to the dimensionality shift Dc=dc+1=6{D}_{c}={d}_{c}+1=6Dc​=dc​+1=6. We then perform an ${ϵ}^{\ensuremath{'}}$ expansion to order one loop to calculate the critical exponents by solving the renormalization-group equations.
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 model of the quantum (transverse) Ising spin glass with a long-range random interaction is proposed and studied here. In this model the random interaction between two spins at a distance rij{r}_{\mathrm{ij}}rij​ apart falls algebraically as ${1/r}_{\mathrm{ij}}^{(d+\ensuremath{\sigma})/2}.$ Here, apart from the strength of the quantum fluctuations, the interaction range $\ensuremath{\sigma}$ is also tunable. We have studied the ground-state properties of the model, extending the ``Droplet model'' of the short-range quantum spin glass. The model is also studied using a modified form of the effective Landau action describing the transition of the short-range quantum spin glass. The important features which are due to the long-range interaction are clearly mentioned. Field theoretical renormalization group calculations fail to locate any stable weak-coupling fixed point in the non-mean-field region. The simplest conjecture is that beyond the mean-field region, the critical behavior is governed by the infinite randomness fixed point. We extend the phenomenological scaling relation for the short-range quantum spin glass (based on the assumption of the existence of a dangerously irrelevant operator) to the present long-range interacting case. Most of the possible interesting aspects associated with the quantum transition in the present model are elaborately discussed.
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