Dynamic linear response of the SK spin glass coupled microscopically to a bath

TL;DR

This paper develops a dynamic linear response theory for the SK spin glass coupled to a heat bath, relating susceptibility to static properties and analyzing finite size and temperature effects.

Contribution

It introduces a new relation between the internal field distribution and TAP solutions, enabling explicit dynamic response calculations for finite systems.

Findings

Frequency-dependent shift of the cusp temperature in susceptibility.

Explicit distribution functions from numerical solutions of TAP equations.

Finite size effects on dynamic response analyzed.

Abstract

The dynamic linear response theory of a general Ising model weakly coupled to a heat bath is derived employing the quantum statistical theory of Mori, treating the Hamiltonian of the spin bath coupling as a perturbation, and applying the Markovian approximation. Both the dynamic susceptibility and the relaxation function are expressed in terms of the static susceptibility and the static internal field distribution function. For the special case of the SK spin glass this internal field distribution can be related to the solutions of the TAP equations in the entire temperature region. Application of this new relation and the use of numerical solutions of the modified TAP equations leads for finite but large systems to explicit results for the distribution function and for dynamic linear response functions. A detailed discussion is presented which includes finite size effects. Due to the derived temperature dependence of the Onsager-Casimir coefficients a frequency-dependent shift of the cusp temperature of the real part of the dynamic susceptibility is found.