Equilibrium dynamics of infinite-range quantum spin glasses in a field

We determine the low-energy spectrum and Parisi replica symmetry breaking function for the spin glass phase of the quantum Ising model with infinite-range random exchange interactions and transverse and longitudinal ($h$) fields. We show that, for all $h$, the spin glass state has full replica symmetry breaking, and the local spin spectrum is gapless with a spectral density which vanishes linearly with frequency. These results are obtained using an action functional$\text{unicode}x2014$argued to yield exact results at low frequencies$\text{unicode}x2014$that expands in powers of a spin glass order parameter, which is is bilocal in time, and a matrix in replica space. We also present the exact solution of the infinite-range spherical quantum $p$-rotor model at nonzero $h$: here, the spin glass state has one-step replica symmetry breaking, and gaplessness only appears after imposition of an additional marginal stability condition. Possible connections to experiments on random arrays of trapped Rydberg atoms are noted.