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Acta Physica Sinica

THREE-DIMENSIONAL QUANTUM SPIN GLASS THEORY (Ⅱ)——STABILITY ANALYSIS

Yi Lin, K.L. Yao, (1)华中理工大学物理系和激光技术国家实验室,武汉430074; (2)中国高等科学技术中心(世界实验室)

Published Jan 1, 1993• 1 citation
Publisher

Abstract

Based on our previous work, introducing an extended Goldbart ansatz, all eigenvalues of the Hessian matrix in our three-dimensional spin glass model are obtained. It is found that the vanishing smallest eigenvalue λ3 has two solutions, which correspond to the Gabay-Toulouse (GT) upper and Almeida-Thouless (AT) lower critical lines respectively. Finally, the relation between our theory and other method is discussed.

DOI
10.7498/aps.42.992

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Based on our previous work, introducing an extended Goldbart ansatz, all eigenvalues of the Hessian matrix in our three-dimensional spin glass model are obtained. It is found that the vanishing smallest eigenvalue λ3 has two solutions, which correspond to the Gabay-Toulouse (GT) upper and Almeida-Thouless (AT) lower critical lines respectively. Finally, the relation between our theory and other method is discussed.
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.
Using generalized Parisi's replica symmetry breaking scheme, the quantum spin glass theory proposed by our previous work is extended to the first-step replica symmetry breaking. A set of new self-consistent equations of spin self-interact ions and spin glass order parameters are obtained. The corresponding local susceptibility is reduced. It is found that the replica symmetry solution for vector spin glass model at low temperature must be broken through calculating the free-energy of the replica symmetry breaking.
Using generalized Parisi's replica symmetry breaking scheme, the quantum spin glasstheory proposed by our previous work is extended to the first-step replica symmetry breaking. A set of new self-consistent equations of spin self-interact ions and spin glass order pa rameters are obtained. The corresponding local susceptibility is reduced. It is found that thereplica symmetry solution for vector spin glass model at low temperature must be brokenthrough calculating the free-energy of the replica symmetry breaking.
We study, near Tc{T}_{c}Tc​, the stability of Parisi's solution for the long-range spin-glass. In addition to the discrete, longitudinal spectrum found by Thouless, de Almeida, and Kosterlitz, we find transverse bands depending on one or two continuous parameters, and a host of zero modes occupying most of the parameter space. All eigenvalues are non-negative, proving that Parisi's solution is marginally stable.
We study, near Tc{T}_{c}Tc​, the stability of Parisi's solution for the long-range spin-glass. In addition to the discrete, "longitudinal" spectrum found by Thouless, de Almeida, and Kosterlitz, we find "transverse" bands depending on one or two continuous parameters, and a host of zero modes occupying most of the parameter space. All eigenvalues are non-negative, proving that Parisi's solution is marginally stable.