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arXiv:0801.0014 · Physical Review B

Time reparametrization symmetry in spin-glass models

Horacio E. Castillo

Published Dec 29, 2008• 11 citations
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Abstract

We study the long-time aging dynamics of spin-glass models with two-spin interactions by performing a renormalization group (RG) transformation on the time variable in the nonequilibrium dynamical generating functional. We obtain the RG equations and find that the flow converges to an exact fixed point. We show that this fixed point is invariant under reparametrizations of the time variable. This continuous symmetry is broken, as evidenced by the fact that the observed correlations and responses are not invariant under it. We argue that this gives rise to the presence of Goldstone modes, and that those Goldstone modes shape the behavior of fluctuations in the nonequilibrium dynamics.

DOI
10.1103/physrevb.78.214430

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We study the long-time aging dynamics of spin-glass models with two-spin interactions by perform- ing a Renormalization Group transformation on the time variable in the non-equilibrium dynamical generating functional. We obtain the RG equations and find that the flow converges to an exact fixed point. We show that this fixed point is invariant under reparametrizations of the time variable.

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We study the long-time aging dynamics of spin-glass models with two-spin interactions by performing a renormalization group (RG) transformation on the time variable in the nonequilibrium dynamical generating functional. We obtain the RG equations and find that the flow converges to an exact fixed point. We show that this fixed point is invariant under reparametrizations of the time variable. This continuous symmetry is broken, as evidenced by the fact that the observed correlations and responses are not invariant under it. We argue that this gives rise to the presence of Goldstone modes, and that those Goldstone modes shape the behavior of fluctuations in the nonequilibrium dynamics.
Glassy materials are characterized by very slow dynamics, associated with a dramatic slowdown of molecular relaxation in structural glasses, and with a dramatic slowdown of spin relaxation in spin glasses. This slowdown of the dynamics has been captured in great part by the results obtained by dynamical mean field theories. In the case of supercooled liquids, the mean-field mode-coupling approach 1 has had some success in describing various features of the relaxation. In the case of spin glasses, a dynamical mean field theory based on a Langevin dynamics for the spins, examined within a functional integral formulation of the Martin-Siggia-Rose approach 2,3,4,5 , has been used to study the long-time relaxation. The dynamical mean field theory of spin glasses has successfully captured 3,4,5 some unusual properties of the spin dynamics, associated with the lack of equilibration, including the presence of physical aging and the breakdown of the equilibrium fluctuation-dissipation relations. However, mean field theories do not allow direct access to a description of the fluctuations in the dynamics. It turns out that fluctuations in the dynamics of glassy systems can in fact be rather strong, as it has been underlined by the discovery of dynamical heterogeneities 6,7 . Dynamical heterogeneities are nanometer-scale regions of molecules
We show that the generating functional describing the slow dynamics of spin-glass systems is invariant under reparametrizations of the time. This result is general and applies for both infinite and short-range models. It follows simply from the assumption that a separation between short time scales and long time scales exists in the system, and from the constraints of causality and unitarity. Global-time reparametrization invariance suggests that the low action excitations in a spin-glass may be smoothly spatially varying time reparametrizations. These Goldstone modes may provide the basis for an analytic dynamical theory of short-range spin glasses.
We study the long time dynamics of quantum spin glasses of rotors using the nonequilibrium Schwinger-Keldysh formalism. These models are known to have a quantum phase transition from a paramagnetic to a spin-glass phase, which we approach by looking at the divergence of the spin-relaxation rate at the transition point. In the aging regime, we determine the dynamical equations governing the time evolution of the spin response and correlation functions, and show that all terms in the equations that arise solely from quantum effects are irrelevant at long times under time reparametrization group (RPG) transformations. At long times, quantum effects enter only through the renormalization of the parameters in the dynamical equations for the classical counterpart of the rotor model. Consequently, quantum effects only modify the out-of-equilibrium fluctuation-dissipation relation (OEFDR), i.e. the ratio X between the temperature and the effective temperature, but not the form of the classical OEFDR.
We study the long time dynamics of a quantum version of the Sherrington-Kirkpatrick model. Time reparametrizations of the dynamical equations have a parallel with renormalization group transformations; in this language the long time behavior of this model is controlled by a reparametrization group ( R(p)G) fixed point of the classical dynamics. The irrelevance of quantum terms in the dynamical equations in the aging regime explains the classical nature of the out of equilibrium fluctuation-dissipation relation.
Abstract We recently showed that the two-dimensional Ising spin glass allows for a line of renormalization group fixed points which explains properties observed in numerical studies. We observe that this exact result corresponds to enhancement to a one-generator continuous internal symmetry. This finally explains why no finite temperature transition to a spin glass phase is observed in two dimensions. In more than two dimensions, instead, the continuous symmetry can be broken spontaneously and yields a spin glass order parameter which, for fixed temperature and disorder strength, takes continuous values in an interval. Such a feature is shared by the order parameter of the known mean field solution of the model with infinite-range interactions, which corresponds to infinitely many dimensions.
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.