We present a critical analysis of the Sompolinsky theory of equilibrium dynamics. By using the spherical
2+p spin-glass model we test the asymptotic static limit of the Sompolinsky solution showing that it fails to yield a thermodynamically stable solution. We then present an alternative formulation, based on the Crisanti, H\"orner, and Sommers [Z. Phys. B: Condens. Matter 92, 257 (1993)] dynamical solution of the spherical
p-spin spin-glass model, reproducing a stable static limit that coincides, in the case of a one step replica symmetry breaking ansatz, with the solution at the dynamic free energy threshold at which the relaxing system gets stuck off equilibrium. We formally extend our analysis to any number of replica symmetry breakings
R. In the limit $R\ensuremath{\rightarrow}\ensuremath{\infty}$, both formulations lead to the Parisi antiparabolic differential equation. This is the special case, though, where no dynamic blocking threshold occurs. The formulation does not contain the additional order parameter $\ensuremath{\Delta}$ of the Sompolinsky theory.
A generalization of the usual spherical model of spin-glasses is introduced, and its static and dynamic behavior is fully studied. The generalized model is given the name quantum spherical model to contrast it with the usual model which we call the classical spherical model. The results of the quantum model reduce to those of the classical model in appropriate limits. However, the quantum model shows some new and interesting features. The equilibrium entropy of spin-glasses which is negative at sufficiently low temperatures in the classical model is found positive in the quantum version, and it vanishes at zero temperature. The dynamic properties of quantum and classical spherical models are similar except in the infinite time limit when different equilibrium behavior is obtained as expected. The magnetization, order-parameter, and on-site correlations are found to decay exponentially above the transition temperature, and algebraically at and below it in the long-time limit. A closed-form expression for the dynamic susceptibility valid at all temperatures is also obtained. The algebraic decay predicted by our model may serve as a model form of behavior in the interpretation of experimental relaxation data on spin-glasses.
We study the out-of-equilibrium dynamics of the infinite range quantum Heisenberg spin glass model coupled to a thermal relaxation bath. The
SU(2) spin algebra is generalized to
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.
We determine the low-energy spectrum and Parisi replica-symmetry-breaking function for the spin-glass phase of the quantum Ising model with infinite-range random exchange interactions and transverse and longitudinal ( h ) fields. We show that, for all h , the spin-glass state has full replica symmetry breaking and the local spin spectrum is gapless, with a spectral density that vanishes linearly with frequency. These results are obtained using an action functional—argued to yield exact results at low frequencies—that expands in powers of a spin-glass order parameter, which is bilocal in time, and a matrix in replica space. We also present the exact solution of the infinite-range spherical quantum p -rotor model at nonzero h : here, the spin-glass state has one-step replica symmetry breaking and gaplessness only appears after imposition of an additional marginal stability condition. Possible connections to experiments on random arrays of trapped Rydberg atoms are noted. Published by the American Physical Society 2024
Langevin equations for the relaxation of spin fluctuations in a soft-spin version of the Edwards-Anderson model are used as a starting point for the study of the dynamic and static properties of spin-glasses. An exact uniform Lagrangian for the average dynamic correlation and response functions is derived for arbitrary range of random exchange, using a functional-integral method proposed by De Dominicis. The properties of the Lagrangian are studied in the mean-field limit which is realized by considering an infinite-ranged random exchange. In this limit, the dynamics are represented by a stochastic equation of motion of a single spin with self-consistent (bare) propagator and Gaussian noise. The low-frequency and the static properties of this equation are studied both above and below
Tc. Approaching
Tc from above, spin fluctuations slow down with a relaxation time proportional to ${|T\ensuremath{-}{T}_{c}|}^{\ensuremath{-}1}$ whereas at
Tc the damping function vanishes as ${\ensuremath{\omega}}^{\frac{1}{2}}$. We derive a criterion for dynamic stability below
Tc. It is shown that a stable solution necessarily violates the fluctuation-dissipation theorem below
Tc. Consequently, the spin-glass order parameters are the time-persistent terms which appear in both the spin correlations and the local response. This is shown to invalidate the treatment of the spin-glass order parameters as purely static quantities. Instead, one has to specify the manner in which they relax in a finite system, along time scales which diverge in the thermodynamic limit. We show that the finite-time correlations decay algebraically with time as ${t}^{\ensuremath{-}\ensuremath{\nu}}$ at all temperatures below
Tc, with a temperature-dependent exponent $\ensuremath{\nu}$. Near
Tc, $\ensuremath{\nu}$ is given (in the Ising case) as $\ensuremath{\nu}(T)\ensuremath{\sim}\frac{1}{2}\ensuremath{-}{\ensuremath{\pi}}^{\ensuremath{-}1}(\frac{1\ensuremath{-}T}{{T}_{c}})+\ensuremath{\sigma}{(\frac{1\ensuremath{-}T}{{T}_{c}})}^{2}$. A tentative calculation of $\ensuremath{\nu}$ at
T=0 K is presented. We briefly discuss the physical origin of the violation of the fluctuation-dissipation theorem.