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.