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Europhysics Letters (EPL)

On the Computation of Static Expectation Values from Dynamics in Spin Glasses

Silvio Franz, Jorge Kurchan B

Published Oct 1, 1992• 14 citations
Publisher

Abstract

We discuss the derivation of canonical correlation functions from the dynamic approach in long-range spin-glasses. We argue that such derivations are possible using the saddle-point approximation even in the presence of ergodicity breaking, within the following scenario: a ) the dynamical saddle-point solution is not unique at infinite times-scales; b ) a single dynamic solution does not yield the measurable correlation functions ( e.g. as observed at very long times in a computer simulation) but these have to be extracted from a sum over all such dynamic saddle-point processes. We propose a conjecture on how the different infinite-time solutions to the mean-field dynamical equations are related by a certain group of transformations. We then show that if all relevant saddle-points are obtained from Sompolinsky's solution through these transformations, the resulting P ( q ) is identical to that obtained with the Parisi scheme in replica space.

DOI
10.1209/0295-5075/20/3/002

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We discuss the derivation of canonical correlation functions from the dynamic approach in long-range spin-glasses. We argue that such derivations are possible using the saddle-point approximation even in the presence of ergodicity breaking, within the following scenario: a ) the dynamical saddle-point solution is not unique at infinite times-scales; b ) a single dynamic solution does not yield the measurable correlation functions ( e.g. as observed at very long times in a computer simulation) but these have to be extracted from a sum over all such dynamic saddle-point processes. We propose a conjecture on how the different infinite-time solutions to the mean-field dynamical equations are related by a certain group of transformations. We then show that if all relevant saddle-points are obtained from Sompolinsky's solution through these transformations, the resulting P ( q ) is identical to that obtained with the Parisi scheme in replica space.
For pt.I see ibid., vol.16, no.7, p.1219 (1983). The response of a simple infinite-range model of a spin glass to the imposition of a small external field is studied. The field is turned on at t=0, and the evolution of the spin correlations is followed as processes involving flips of successively larger and larger clusters of spins occur, until equilibrium is achieved at t= infinity . The evolution from one state to the other proceeds through a quasi-continuum of intermediate metastable states labelled by the fraction y of spins that flip simultaneously on that time scale. The time scale for such processes grows very long as the size N of the system increases (possibly like exp((Ny)12/Tg/T)), so in the infinite system, only an infinitesimal fraction of this evolution can be observed. The generalised equation of state describing the evolution of correlations has the form derived by Sompolinsky, but the physical interpretation is different. In particular, the author does not postulate any violation of the fluctuation-dissipation theorem in equilibrium; the terms in the equations which correspond to those which violate it in his theory have the form they do because a non-equilibrium situation is being described. The physical interpretation of the variable y also allows an additional relation between initial- and final-state correlation functions which leads to the Parisi equation for his order parameter q(x), with x=1-y.
We present a replica derivation for the claim that the averaged square of the local spin expectation value in the long-range spin glass model is equal to the minimum, q(0) of Parisi's order parameters q(x). This corresponds to the dynamic interpretation of q(x) due to Sompolinsky. We show that q(0) obeys a power law q(0) ∼ b 2/3 for low external fields b and all temperatures 0 < T < Tc. Furthermore we calculate the low field magnetization for all temperatures. The calculations are based on an exact relation for the renormalized local susceptibility which replaces the Parisi-Toulouse hypothesis.
This thesis aims at a description of the equilibrium dynamics of quantum spin glass systems. To this end a generic fermionic SU(2), spin 1/2 spin glass model with infinite-range interactions is defined in the first part. The model is treated in the framework of imaginary-time Grassmann field theory along with the replica formalism. A dynamical two-step decoupling procedure, which retains the full time dependence of the (replica-symmetric) saddle point, is presented. As a main result, a set of highly coupled self-consistency equations for the spin-spin correlations can be formulated. Beyond the so-called spin-static approximation two complementary systematic approximation schemes are developed in order to render the occurring integration problem feasible. One of these methods restricts the quantum-spin dynamics to a manageable number of bosonic Matsubara frequencies. A sequence of improved approximants to some quantity can be obtained by gradually extending the set of employed discrete frequencies. Extrapolation of such a sequence yields an estimate of the full dynamical solution. The other method is based on a perturbative expansion of the self-consistency equations in terms of the dynamical correlations. In the second part these techniques are applied to the isotropic Heisenberg spin glass both on the Fock space (HSGF) and, exploiting the Popov-Fedotov trick, on the spin space (HSGS). The critical temperatures of the paramagnet to spin glass phase transitions are determined accurately. Compared to the spin-static results, the dynamics causes slight increases of T_c by about 3% and 2%, respectively. For the HSGS the specific heat C(T) is investigated in the paramagnetic phase and, by way of a perturbative method, below but close to T_c. The exact C(T)-curve is shown to exhibit a pronounced non-analyticity at T_c and, contradictory to recent reports by other authors, there is no indication of maximum above T_c. In the last part of this thesis the spin glass model is augmented with a nearest-neighbor hopping term on an infinite-dimensional cubic lattice. An extended self-consistency structure can be derived by combining the decoupling procedure with the dynamical CPA method. For the itinerant Ising spin glass numerous solutions within the spin-static approximation are presented both at finite and zero temperature. Systematic dynamical corrections to the spin-static phase diagram in the plane of temperature and hopping strength are calculated, and the location of the quantum critical point is determined.
Abstract We present results of numerical simulations on a one-dimensional Ising spin glass with long-range interactions. Parameters of the model are chosen such that it is a proxy for a short-range spin glass above the upper critical dimension (i.e. in the mean-field regime). The system is quenched to a temperature well below the transition temperature and the growth of correlations is observed. The spatial decay of the correlations at distances less than the dynamic correlation length agrees quantitatively with the predictions of a static theory, the ‘metastate’, evaluated according to the replica symmetry breaking (RSB) theory. We also compute the dynamic exponent z ( T ) defined by and find that it is compatible with the mean-field value of the critical dynamical exponent for short range spin glasses.
Within the replica approach to mean-field spin-glasses the transition from ergodic high-temperature behaviour to the glassy low-temperature phase is marked by the instability of the replica-symmetric saddle-point. For general spin-glass models with non-Gaussian field distributions the corresponding Hessian is a 2n×2n2^n\times 2^n2n×2n matrix with the number nnn of replicas tending to zero eventually. We block-diagonalize this Hessian matrix using representation theory of the permutation group and identify the blocks related to the spin-glass susceptibility. Performing the limit n→0n\to 0n→0 within these blocks we derive expressions for the de~Almeida-Thouless line of general spin-glass models. Specifying these expressions to the cases of the Sherrington-Kirkpatrick, Viana-Bray, and the Lévy spin glass respectively we obtain results in agreement with previous findings using the cavity 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