Thermodynamics and Universality for Mean Field Quantum Spin Glasses

Nicholas Crawford

Published Jan 1, 2006• 10 citations

Abstract

ABSTRACT. We study aspects of the thermodynamics of quantum versions of spin glasses. By means of the Lie-Trotter formula for exponential sums of operators, we adapt methods used to analyze classical spin glass models to answer analogous questions about quantum models.

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Nicholas Crawford

Published Jan 1, 2006• 10 citations

OpenAlex

Abstract

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Thermodynamics and Universality for Mean Field Quantum Spin Glasses

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arXiv:1907.01609Jul 2, 20197 citations

We introduce the transverse Ising model as a prototype for discussing quantum phase transition. Next we introduce Suzuki-Trotter formalism to show the correspondence between

d

-dimensional quantum system with a

(d+1)

-dimensional classical system. We then discuss transverse Ising spin glass models, namely S-K model, E-A model, and the

\pm J

model with Ising spin in transverse field. We briefly discuss the mean field, exact diagonalization, and quantum Monte Carlo results for their phase diagrams. Next we discuss the question of replica symmetry restoration in quantum spin glasses (due to tunnelling possibility through the barriers). Then we discuss the quantum annealing technique and indicate its relationship with replica symmetry restoration in quantum spin glasses. We have also breifly discussed the possibility of Quantum Annealing in context of kinetically constrained systems. Mean-field calculation for BCS superconductivity, Real-space RG calculation for one dimensional transverse Ising system, scattering amplitude calculation for tunneling through asymmetric barrier (useful for quantum kinetically constrained system) are given in the appendices.

x

and

z

). Those of

z

-type are non-ergodic extended eigenstates (NEE) in the basis of

\{σ_z\}

operators that inherit the structure of the classical spin glass with exponentially long decay times of Edwards Anderson order parameter at any finite value of transverse field

B_{\perp}

. Those of

x

-type form narrow bands of NEEs that conserve the integer-valued

x

-magnetization. Quantum evolution within a given band of each type is described by a Hamiltonian that belongs to either the ensemble of Preferred Basis Levi matrices (

z

-type) or Gaussian Orthogonal ensemble (

x

-type). We characterize the non-equilibrium dynamics using fractal dimension

D

that depends on energy density (temperature) and plays a role of thermodynamic potential:

D=0

in MBL phase, $0