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Physical review. B, Condensed matter

Spin-glass effects on critical behavior in systems with extended quenched disorder

L. De Cesare, Maria Teresa Mercaldo

Published Jan 1, 1999• 8 citations
Publisher

Abstract

A renormalization-group approach is used for investigating the effects of the Parisi replica symmetry breaking (``spin-glass effects'') on critical properties of d-dimensional systems with extended quenched disorder along ${\ensuremath{\varepsilon}}_{d}$ directions. Within a double expansion scheme to first order in $(\ensuremath{\varepsilon}=4\ensuremath{-}d, {\ensuremath{\varepsilon}}_{d})$ it is found that for $d<~4+{\ensuremath{\varepsilon}}_{d}$ and order-parameter dimensionalities $n<~4,$ a full fixed-point instability with a consequent runaway towards a strong-coupling regime occurs in the parameter space while, for $d<4$ and $n>4,$ compatibly with the modified Harris criterion, an additional marginally stable fixed point exists which might control a new anisotropic random critical behavior. In any case, the traditional replica-symmetric Dorogovtsev-like solution for classical and quantum systems is unstable against the replica symmetry-breaking modes.

DOI
10.1103/physrevb.59.855

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A renormalization-group approach is used for investigating the effects of the Parisi replica symmetry breaking (``spin-glass effects'') on critical properties of d-dimensional systems with extended quenched disorder along \ensuremathεd{\ensuremath{\varepsilon}}_{d}\ensuremathεd​ directions. Within a double expansion scheme to first order in (\ensuremathε=4\ensuremath−d,\ensuremathεd)(\ensuremath{\varepsilon}=4\ensuremath{-}d, {\ensuremath{\varepsilon}}_{d})(\ensuremathε=4\ensuremath−d,\ensuremathεd​) it is found that for d< 4+\ensuremathεdd<~4+{\ensuremath{\varepsilon}}_{d}d< 4+\ensuremathεd​ and order-parameter dimensionalities n< 4,n<~4,n< 4, a full fixed-point instability with a consequent runaway towards a strong-coupling regime occurs in the parameter space while, for d<4d<4d<4 and n>4,n>4,n>4, compatibly with the modified Harris criterion, an additional marginally stable fixed point exists which might control a new anisotropic random critical behavior. In any case, the traditional replica-symmetric Dorogovtsev-like solution for classical and quantum systems is unstable against the replica symmetry-breaking modes.
We study the critical properties of the weakly disordered p-component ferromagnet in terms of the renormalization group (RG) theory generalized to take into account the replica symmetry breaking (RSB) effects coming from the multiple local minima solutions of the mean-field equations. Recently it has been shown that for p<4 the traditional RG flows at dimensions D=4- epsilon , which are usually considered as describing the disorder-induced universal critical behaviour, are unstable with respect to the RSB potentials as found in spin glasses, and a new type of stable one-step RSB fixed point has been discovered. Here it is demonstrated that for a general type of the Parisi RSB structures there exist no stable fixed points, and the RG flows lead to the strong-coupling regime at the finite scale R * approximately exp(1/u), where u is the small parameter describing the disorder. The physical consequences of the obtained RG solutions are discussed. In particular, we argue that discovered RSB strong-coupling phenomena indicate the onset of a new spin-glass-type critical behaviour in the temperature interval tau < tau * approximately exp(-1/u) near Tr. The possible relevance of the considered RSB effects for the Griffith phase is also discussed.
A renormalization-group treatment is used for investigating the critical properties of d-dimensional systems with anisotropic quenched disorder totally correlated in ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{d}}$ dimensions and randomly distributed in the remaining d-${\mathrm{\ensuremath{\epsilon}}}_{\mathit{d}}$ space directions (d=4-\ensuremath{\epsilon}) with long-range correlation which decays with the distance as a power law involving a characteristic parameter \ensuremath{\sigma}. Within a triple (\ensuremath{\epsilon},${\mathrm{\ensuremath{\epsilon}}}_{\mathit{d}}$,\ensuremath{\sigma}) expansion it is shown that the recursion relations possess a novel focus-type random fixed point leading to new anisotropic critical properties. Under certain conditions, different critical regimes and crossovers are found to occur, consistent with the modified version of the Harris criterion obtained for the random problem under study.
We study the critical properties of the weakly disordered two-dimensional Ising and Baxter models in terms of the renormalization group (RG) theory generalized to take into account replica symmetry breaking (RSB) effects. Recently it has been shown that the traditional replica-symmetric RG flows in the dimension are unstable with respect to the RSB potentials and a new spin-glass type critical phenomena has been discovered (Dotsenko Vik S, Harris B, Sherrington D and Stinchbombe R 1995 J. Phys. A: Math. Gen. 28 3093; Dotsenko Vik S and Feldman D E 1995 J. Phys. A: Math. Gen. 28 5183). In contrast, here it is demonstrated that in the considered two-dimensional systems the renormalization-group flows are stable with respect to the RSB modes. It is shown that the solution of the renormalization group equations with arbitrary starting RSB coupling matrix exhibits asymptotic approach to the traditional replica-symmetric ones. Thus, to leading order the non-perturbative RSB degrees of freedom do not affect the critical phenomena in the two-dimensional weakly disordered Ising and Baxter models studied earlier.
A field-theory approach is used to investigate the 'spin-glass effects' on the critical behaviour of systems with weak temperature-like quenched disorder. The renormalization group (RG) analysis of the effective Hamiltonian of a model with replica symmetry breaking (RSB) potentials of a general type is carried out in the two-loop approximation. The fixed point (FP) stability, recently found within the one-step RSB RG treatment, is further explored in terms of replicon eigenvalues. We find that the traditional FPs, which are usually considered to describe the disorder-induced universal critical behaviour, remain stable when the continuous RSB modes are taken into account.
We study critical behaviour of disordered magnets near four dimensions. We consider the system with explicit cubic anisotropy and scalar disorder and that with random direction of anisotropy axis. The quenched disorder is taken into account by replica method. Using the method of operator product expansion, we derive in the first order to εεε approximation the renormalization group equations taking into account possible replica symmetry breaking.
We consider quantum rotors or Ising spins in a transverse field on a d-dimensional lattice, with random, frustrating, short-range, exchange interactions. The quantum dynamics are associated with a finite moment of inertia for the rotors, and with the transverse field for the Ising spins. For a suitable distribution of exchange constants, these models display spin-glass and quantum paramagnet phases and a zero-temperature (T) quantum transition between them. An earlier exact solution for the critical properties of a model with infinite-range interactions cna be reproduced by minimization of a Landau effective-action functional for the model in finite d with short-range interactions. The functional is expressed in terms of a composite spin field which is bilocal in time. The mean-field phase diagram near the T=0 critical point is mapped out as a function of T, strength of the quantum coupling, and applied fields. The spin-glass phase has replica symmetry breaking; but, as in the classical Ising spin glass, the order parameter becomes replica symmetric as T\ensuremath{\rightarrow}0. Next we examine the consequences of fluctuations about the mean field for the critical properties. Above d=8, and with certain restrictions on the values of the Landau couplings, we find that the transition is controlled by a Gaussian fixed point with mean-field critical exponents. For couplings not attracted by the Gaussian fixed point above d=8, and for all physical couplings below d=8, we find runaway renormalization-group flows to strong coupling. General scaling relations that should be valid even at the strong-coupling fixed point are proposed and compared with Monte Carlo simulations.