Self-Correcting Quantum Memories Beyond the Percolation Threshold

We analyze several high dimensional generalizations of the toric code at nonzero temperature. We find that in large enough dimension, there can be a distinct separation between the critical temperature $T_c$, given by thermodynamic singularities, and the percolation temperature $T_p$, given by the percolation of defects. We argue that the regime $T_p<T<T_c$ is a range of temperatures where a self-correcting quantum memory can operate despite having percolating defects. We present analytic arguments and numerical evidence in support of this scenario, including a mean-field treatment and Monte Carlo simulations. Near $T_c$, simulations observe a large hysteretic behavior, which may have applications by allowing the self-correcting phase to survive in a "superheated" regime.