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 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