The free energy of a quantum Sherrington-Kirkpatrick spin-glass model for weak disorder
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 we...
Saved in:
Published in | arXiv.org |
---|---|
Main Authors | , , , |
Format | Paper Journal Article |
Language | English |
Published |
Ithaca
Cornell University Library, arXiv.org
26.10.2021
|
Subjects | |
Online Access | Get full text |
Cover
Loading…
Summary: | 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. |
---|---|
ISSN: | 2331-8422 |
DOI: | 10.48550/arxiv.1912.06633 |