# The Free Energy of a Quantum Sherrington–Kirkpatrick Spin-Glass Model for Weak Disorder

Leschke H, Rothlauf S, Ruder R, Spitzer W (2021)

**Publication Type:** Journal article

**Publication year:** 2021

### Journal

**Book Volume:** 182

**Article Number:** 55

**Journal Issue:** 3

**DOI:** 10.1007/s10955-020-02689-8

### 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 / β, 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≥ 0. The macroscopic annealed free energy turns out to be non-trivial and given, for any βv> 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 βv< 1 we determine this minimum up to the order (βv) ^{4} with the Taylor coefficients explicitly given as functions of βb and with a remainder not exceeding (βv) ^{6}/ 16. As a by-product we prove that the so-called static approximation to the minimization problem yields the wrong βb-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 βb. Its essence dates back to Kac in 1956, but the formula was published only in 1989 by Gaveau and Schulman.

### Authors with CRIS profile

### Involved external institutions

### How to cite

**APA:**

Leschke, H., Rothlauf, S., Ruder, R., & Spitzer, W. (2021). The Free Energy of a Quantum Sherrington–Kirkpatrick Spin-Glass Model for Weak Disorder. *Journal of Statistical Physics*, *182*(3). https://dx.doi.org/10.1007/s10955-020-02689-8

**MLA:**

Leschke, Hajo, et al. "The Free Energy of a Quantum Sherrington–Kirkpatrick Spin-Glass Model for Weak Disorder." *Journal of Statistical Physics* 182.3 (2021).

**BibTeX:** Download