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arXiv:cond-mat/0105001 · Physical review. B, Condensed matter

Out-of-equilibrium dynamics of a quantum Heisenberg spin glass

Giulio Biroli, Olivier Parcollet

Published Feb 11, 2002• 32 citations
arXivHTMLPDF

Abstract

We study the out-of-equilibrium dynamics of the infinite range quantum Heisenberg spin glass model coupled to a thermal relaxation bath. The $\mathrm{SU}(2)$ spin algebra is generalized to $\mathrm{SU}(N)$ and we analyze the large-$N$ limit. The model displays a dynamical phase transition between a paramagnetic and a glassy phase. In the latter, the system remains out-of-equilibrium and displays an aging phenomenon, which we characterize using both analytical and numerical methods. In the aging regime, the quantum fluctuation-dissipation relation is violated and replaced over a very long time by its classical generalization, as in models involving simple spin algebras studied previously. We also discuss the effect of a finite coupling to the relaxation baths and their possible forms. This work completes and justifies previous studies on this model using a static approach.

DOI
10.1103/physrevb.65.094414

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We study with exact diagonalization techniques the Heisenberg model for a system of SU(2) spins with S=1/2S=1/2S=1/2 and random infinite-range exchange interactions. We calculate the critical temperature Tg{T}_{g}Tg​ for the spin-glass to paramagnetic transition. We obtain ${T}_{g}\ensuremath{\approx}0.13,$ in good agreement with previous quantum Monte Carlo and analytical estimates. We provide a detailed picture for the different kind of excitations which intervene in the dynamical response ${\ensuremath{\chi}}^{\ensuremath{''}}(\ensuremath{\omega},T)$ at T=0T=0T=0 and analyze their evolution as T increases. We also calculate the specific heat Cv(T).{C}_{v}(T).Cv​(T). We find that it displays a smooth maximum at ${T}_{M}\ensuremath{\approx}0.25,$ in good qualitative agreement with experiments. We argue that the fact that TM>Tg{T}_{M}>{T}_{g}TM​>Tg​ is due to a quantum disorder effect.
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
The infinite-range M-component models of quantum rotors and anharmonic oscillators coupled by random exchange interactions with Gaussian distributions are considered. Both models exhibit a quantum phase transition separating the spin glass and disordered phases where the quantum effects are regulated by a parameter \ensuremath{\Delta} (inverse momentum of inertia for rotor model and inverse mass for oscillator model) parametrizing the kinetic energy. The models allow, in the limit M\ensuremath{\rightarrow}\ensuremath{\infty}, for an exact solution in both the spin glass and disordered phases. Employing the quantum finite-temperature real-time description, we examine the static and dynamic properties of the systems without making use of the n-replica method. We find that the dynamical solution is marginally stable in the spin-glass phase with a constant static susceptibility below the spin-glass freezing temperature. A closed-form solution for the averaged free energy has been obtained and various thermodynamic functions are derived. In particular, the entropy is always positive in the spin-glass plane and goes to zero with a T3{\mathit{T}}^{3}T3 dependency in the low-temperature region. This is in contrast to the classical M-component Heisenberg model in the M\ensuremath{\rightarrow}\ensuremath{\infty} limit.