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 $$\mathsf {b}$$ . 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.