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arXiv:1305.3638

The Quantum Spherical Spin Glass Model: A Limitation to Static Approximation

Junior, Vilarbo da Silva, Alexsandro Marian Carvalho

Published May 15, 2013• 0 citations
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Abstract

In this work, we confront the static approximate with a exact solution in the quantum spherical p-spin interaction model (p->oo and p=2). On the one hand, this study indicates that the static approximate corresponds to exact solution in the cases p->oo and p = 2 in the classic regime. On the other, the static approximate differs from the exact solution for p=2 in the quantum regime.

DOI
10.48550/arxiv.1305.3638

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In this work, we confront the static approximate with a exact solution in the quantum spherical p-spin interaction model (p->oo and p=2). On the one hand, this study indicates that the static approximate corresponds to exact solution in the cases p->oo and p = 2 in the classic regime. On the other, the static approximate differs from the exact solution for p=2 in the quantum regime.
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 present an exact analytic solution of the short-range interaction disordered quantum spherical model with symmetric binary $\ifmmode\pm\else\textpm\fi{}J$ bond distribution on the Bethe lattice (with coordination number zzz). We find the zero temperature quantum spin glass phase transition for coordinations $z\ensuremath{\ge}3$ but not for the one-dimensional system (z=2)(z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2)(z=2). Quantum dynamics is examined via various correlation functions on the infinite tree which are evaluated in a closed form. The relationship between the results for a fully connected lattice $(z\ensuremath{\rightarrow}\ensuremath{\infty})$ and the usual mean-field approach is established.
We present a critical analysis of the Sompolinsky theory of equilibrium dynamics. By using the spherical 2+p2+p2+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 ppp-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 RRR. 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.
We study a quantum extension of the spherical p-spin-glass model using the imaginary-time replica formalism. We solve the model numerically and we discuss two analytical approximation schemes that capture most of the features of the solution. The phase diagram and the physical properties of the system are determined in thermodynamic equilibrium and under conditions of marginal stability. In both cases, the phase diagram consists of two qualitatively different regions. If the transition temperature is higher than a certain value T∗,{T}^{*},T∗, quantum effects are qualitatively irrelevant and the phase transition is second order, as in the classical case. When quantum fluctuations depress the transition temperature below T∗,{T}^{*},T∗, the transition becomes first order. The susceptibility is discontinuous and shows hysteresis across the first-order line. This behavior is reminiscent of that observed in the dipolar Ising spin-glass ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}$ in an external transverse magnetic field. We discuss the thermodynamics and the stationary dynamics of both states. The spectrum of magnetic excitations of the equilibrium spin-glass state is gapped, leading to an exponentially small specific heat at low temperatures. That of the marginally stable state is gapless and its specific heat varies linearly with temperature, as generally observed in glasses at low temperature. We show that the properties of the marginally stable state are closely related to those obtained in studies of the real-time dynamics of the system weakly coupled to a quantum thermal bath. Finally, we discuss a possible application of our results to the problem of polymers in random media.
Abstract We study a large- N bosonic quantum mechanical sigma-model with a spherical target space subject to disordered interactions, more colloquially known as the p -spin spherical model. Replica symmetry is broken at low temperatures and for sufficiently weak quantum fluctuations, which drives the system into a spin glass (SG) phase. The first half of this paper is dedicated to a discussion of this model’s thermodynamics, with particular emphasis on the marginally stable SG . This phase exhibits an emergent conformal symmetry in the strong coupling regime, which dictates its thermodynamic properties. It is associated with an extensive number of nearby states in the free energy landscape. We discuss in detail an elegant approximate solution to the SG equations, which interpolates between the conformal regime and an ultraviolet-complete short distance solution. In the second half of this paper we explore the real-time dynamics of the model and uncover quantum chaos as measured by out-of-time-order four-point functions, both numerically and analytically. We find exponential Lyapunov growth, which intricately depends on the model’s couplings and becomes strongest in the quantum critical regime. We emphasize that the SG phase also exhibits quantum chaos, albeit with parametrically smaller Lyapunov exponent than in the replica symmetric phase. An analytical calculation in the marginal SG phase suggests that this Lyapunov exponent vanishes in a particular infinite coupling limit. We comment on the potential meaning of these observations from the perspective of holography.