The Free Energy of a Quantum Sherrington\u2013Kirkpatrick Spin-Glass Model for Weak Disorder

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 mathsf {b}. More precisely, if the Gaussian disorder is weak in the sense that its standard deviation mathsf {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 mathsf {b}/mathsf {v}ge 0. The macroscopic annealed free energy turns out to be non-trivial and given, for any beta mathsf {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 mathsf {v}<1 we determine this minimum up to the order (beta mathsf {v})^{4} with the Taylor coefficients explicitly given as functions of beta mathsf {b} and with a remainder not exceeding (beta mathsf {v})^{6}/16. As a by-product we prove that the so-called static approximation to the minimization problem yields the wrong beta mathsf {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 mathsf {b}. Its essence dates back to Kac in 1956, but the formula was published only in 1989 by Gaveau and Schulman.