Interpolations for a quantum Parisi formula in transverse field mean-field spin glass models

A quantum Parisi formula for the transverse field Sherrington-Kirkpatrick (SK) model is proven with an elementary mathematical method. First, a self-overlap corrected quantum model of the transverse field SK model is represented in terms of the Hamiltonian with annealed random interactions. The interpolation given by Guerra and Toninelli is extended to the self-overlap corrected quantum model. It is proven that the infinite-volume limit of the free energy density exists in the operator formalism. Next, another interpolation developed by Guerra and Talagrand is applied to obtain a finite step replica-symmetry breaking (RSB) bound on the free energy density in the transverse field SK model. The interpolation enables us to show that the deviation of the RSB solution from the exact solution vanishes in the self-overlap corrected quantum model in a functional representation of the quantum spin operators. Finally, the corrected terms are removed by the Hopf-Lax formula for a nonlinear partial differential equation to show the quantum Parisi formula for the original transverse field SK model. The formula is extended to that for the transverse field mean-field $p$-spin glass model.