The free energy of a quantum Sherrington-Kirkpatrick 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 $b$. 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 $b/v\ge 0$. 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.

v1: 36 pages, 1 figure; v2: 38 pages, 1 figure, high-field limit added, 3 references added, some misprints corrected, minor stylistic improvements, back references included; v3: 41 pages, 3 figures, information added, references added or updated, misprints corrected, stylistic improvements; v4: final version after minor editing