Complete replica solution for the transverse field Sherrington-Kirkpatrick spin glass model with continuous-time quantum Monte Carlo method

We construct a complete numerically exact solution of a mean-field quantum spin glass model-the transverse field Sherrington-Kirkpatrick model-by implementing a continuous-time quantum Monte Carlo method in the presence of full replica symmetry breaking. We extract the full numerically exact phase diagram, displaying a glassy phase with continuous replica symmetry breaking at small transverse fields and low temperatures. A paramagnetic phase emerges once thermal and quantum fluctuations melt the spin glass. We characterize both phases by extracting the order parameter as well as the static and dynamical local spin susceptibilities. The static susceptibility shows a plateau in the glassy phase, but it remains smooth across the phase boundary. For the imaginary part of the dynamical susceptibility, we find an Ohmic, i.e., linear in omega, scaling for small frequencies omega, with a slope independent of the transverse field. These results compare qualitatively well with ac susceptibility measurements on a dipole-coupled three-dimensional Ising magnet-the LiHoxY1-xF4 compound-in a transverse magnetic field. Our work provides a general framework for the exact numerical solution of mean-field quantum glass models, and it constitutes an important step towards understanding glassiness in realistic systems.