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

• Published in: arXiv.org
• Main Authors: Leschke, Hajo, Rothlauf, Sebastian, Ruder, Rainer, Spitzer, Wolfgang
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 26.10.2021
• Subjects: Annealing; Beta; Commutativity; Free energy; Functionals; Magnetic fields; Mathematical functions; Mathematics - Mathematical Physics; Mathematics - Probability; Operators (mathematics); Physics - Disordered Systems and Neural Networks; Physics - Mathematical Physics; Physics - Statistical Mechanics; Spin glasses
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1912.06633

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.