The free energy of a quantum Sherrington-Kirkpatrick spin-glass model for weak disorder

TL;DR

This paper extends classical results on the Sherrington-Kirkpatrick spin-glass model to the quantum case with a transverse field, showing that under weak disorder the free energy simplifies and no spin-glass phase exists.

Contribution

It rigorously proves that for weak disorder, the quantum model's free energy matches the annealed free energy and identifies the minimizer of the associated variational problem.

Findings

Free energy equals annealed free energy under weak disorder.

No spin-glass phase for any ratio of transverse field to disorder strength.

Explicit characterization of the variational minimizer for the free energy.

Abstract

We extend two rigorous results of Aizenman, Lebowitz, and Ruelle in their pioneering paper of 1987 on the Sherrington-Kirkpatrick spin-glass model without external magnetic field to the quantum case with a "transverse field" of strength $b$. More precisely, if the Gaussian disorder is weak in the sense that its standard deviation $v>0$ is smaller than the temperature $1/\beta$, then the (random) free energy almost surely equals the annealed free energy in the macroscopic limit and there is no spin-glass phase for any $b/v\geq0$. The macroscopic annealed free energy (times $\beta$) turns out to be non-trivial and given, for any $\beta v>0$, by the global minimum of a certain functional of square-integrable functions on the unit square according to a Varadhan large-deviation principle. For $\beta v<1$ we determine this minimum up to the order $(\beta v)^4$ with the Taylor coefficients explicitly given as functions of $\beta b$ and with a remainder not exceeding $(\beta v)^6/16$. As a by-product we prove that the so-called static approximation to the minimization problem yields the wrong $\beta b$-dependence even to lowest order. Our main tool for dealing with the non-commutativity of the spin-operator components is a probabilistic representation of the Boltzmann-Gibbs operator by a Feynman-Kac (path-integral) formula based on an independent collection of Poisson processes in the positive half-line with common rate $\beta b$. Its essence dates back to Kac in 1956, but the formula was published only in 1989 by Gaveau and Schulman.