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arXiv:cond-mat/0204615 · Physical review. B, Condensed matter

Spin-glass transition in a magnetic field: A renormalization group study

Iveta R. Pimentel, T. Temesvári, C. De Dominicis

Published Jun 6, 2002• 50 citations
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Abstract

We study the transition of short-range Ising spin glasses in a magnetic field, within a general replica symmetric field theory, which contains three masses and eight cubic couplings, that is defined in terms of the fields representing the replicon, anomalous, and longitudinal modes. We discuss the symmetry of the theory in the limit of replica number $\stackrel{\ensuremath{\rightarrow}}{n}0,$ and consider the regular case where the longitudinal and anomalous masses remain degenerate. The spin-glass transitions in zero and nonzero fields are analyzed in a common framework. The mean-field treatment shows the usual results, that is a transition in zero field, where all the modes become critical, and a transition in nonzero field, at the de Almeida--Thouless (AT) line, with only the replicon mode critical. Renormalization group methods are used to study the critical behavior, to order $\ensuremath{\varepsilon}=6\ensuremath{-}d.$ In the general theory we find a stable fixed point associated with a spin-glass transition in a zero field. This fixed point becomes unstable in the presence of a small magnetic field, and we calculate crossover exponents, which we relate to zero-field critical exponents. In a finite magnetic field, we find no physical stable fixed point to describe the AT transition, in agreement with previous results of other authors.

DOI
10.1103/physrevb.65.224420

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We study the transition of short-range Ising spin glasses in a magnetic field, within a general replica symmetric field theory, which contains three masses and eight cubic couplings, that is defined in terms of the fields representing the replicon, anomalous, and longitudinal modes. We discuss the symmetry of the theory in the limit of replica number n\ensuremath→0,\stackrel{\ensuremath{\rightarrow}}{n}0,n\ensuremath→0, and consider the regular case where the longitudinal and anomalous masses remain degenerate. The spin-glass transitions in zero and nonzero fields are analyzed in a common framework. The mean-field treatment shows the usual results, that is a transition in zero field, where all the modes become critical, and a transition in nonzero field, at the de Almeida--Thouless (AT) line, with only the replicon mode critical. Renormalization group methods are used to study the critical behavior, to order \ensuremathε=6\ensuremath−d.\ensuremath{\varepsilon}=6\ensuremath{-}d.\ensuremathε=6\ensuremath−d. In the general theory we find a stable fixed point associated with a spin-glass transition in a zero field. This fixed point becomes unstable in the presence of a small magnetic field, and we calculate crossover exponents, which we relate to zero-field critical exponents. In a finite magnetic field, we find no physical stable fixed point to describe the AT transition, in agreement with previous results of other authors.
We use the generic replica symmetric cubic field theory to study the transition of short-range Ising spin glasses in a magnetic field around the upper critical dimension. A novel fixed point is found from the application of the renormalization group. In the spin-glass limit, this fixed point governs the critical behavior of a class of systems characterized by a single cubic parameter. For this universality class, the spin-glass susceptibility diverges at criticality, whereas the longitudinal mode remains massive. The third mode, however, behaves unusually. The physical consequences of this unusual behavior are discussed, and a comparison with the conventional de Almeida-Thouless scenario is presented.
The asymptotical behavior of physical quantities, like the order parameter, the replicon, and longitudinal masses, is studied around the zero-field spin-glass transition point when a small external magnetic field is applied. An effective field theory to model this asymptotics contains a small perturbation in its Lagrangian which breaks the zero-field symmetry. A first-order renormalization group supplemented by perturbational results provides the scaling functions. The perturbative zero of the scaling function for the replicon mass defines a generic Almeida-Thouless surface stemming from the zero-field fixed point.
Critical point scaling in a field HHH applies for the limits $t\ensuremath{\rightarrow}0$ (where $t=T/{T}_{c}\ensuremath{-}1)$ and $H\ensuremath{\rightarrow}0$ but with the ratio $R=t/{H}^{2/\ensuremath{\Delta}}$ finite. $\ensuremath{\Delta}$ is a critical exponent of the zero-field transition. We study the replicon correlation length $\ensuremath{\xi}$ and from it the crossover scaling function f(R)f(R)f(R) defined via $1/(\ensuremath{\xi}{H}^{4/(d+2\ensuremath{-}\ensuremath{\eta})})\ensuremath{\sim}f(R)$. We have calculated analytically f(R)f(R)f(R) for the mean-field limit of the Sherrington-Kirkpatrick model. In dimension d=3d=3d=3, we have determined the exponents and the critical scaling function f(R)f(R)f(R) within two versions of the Migdal-Kadanoff (MK) renormalization group procedure. One of the MK versions gives results for f(R)f(R)f(R) in d=3d=3d=3 in reasonable agreement with those of the Monte Carlo simulations at the values of RRR for which they can be compared. If there were a de Almeida-Thouless (AT) line for $d\ensuremath{\le}6$, it would appear as a zero of the function f(R)f(R)f(R) at some negative value of RRR, but there is no evidence for such behavior. This is consistent with the arguments that there should be no AT line for $d\ensuremath{\le}6$, which we review.
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.
We study in Ising spin glasses the finite-size effects near the spin-glass transition in zero field and at the de Almeida-Thouless transition in a field by Monte Carlo methods and by analytical approximations. In zero field, the finite-size scaling function associated with the spin-glass susceptibility of the Sherrington-Kirkpatrick mean-field spin-glass model is of the same form as that of one-dimensional spin-glass models with power-law long-range interactions in the regime where they can be a proxy for the Edwards-Anderson short-range spin-glass model above the upper critical dimension. We also calculate a simple analytical approximation for the spin-glass susceptibility crossover function. The behavior of the spin-glass susceptibility near the de Almeida-Thouless transition line has also been studied, but here we have only been able to obtain analytically its behavior in the asymptotic limit above and below the transition. We have also simulated the one-dimensional system in a field in the non-mean-field regime to illustrate that when the Imry-Ma droplet length scale exceeds the system size one can then be erroneously lead to conclude that there is a de Almeida-Thouless transition even though it is absent.
By using real-space renormalization group (RG) methods, we show that spin glasses in a field display a new kind of transition in high dimensions. The corresponding critical properties and the spin-glass phase are governed by two nonperturbative zero-temperature fixed points of the RG flow. We compute the critical exponents and discuss the RG flow and its relevance for three-dimensional systems. The new spin-glass phase we discovered has unusual properties, which are intermediate between the ones conjectured by droplet and full replica symmetry-breaking theories. These results provide a new perspective on the long-standing debate about the behavior of spin glasses in a field.