Dynamics in presence of local symmetry-breaking impurities

Many examples of symmetries lead to universal slow relaxation in temporal correlation functions. We investigate the fate of these correlation functions in the presence of local symmetry-breaking impurities using noisy Brownian circuits. While correlation functions are expected to generically decay exponentially once the symmetries are broken, we find that they still exhibit a slow relaxation influenced by the approximately conserved quantities. This behavior can be analyzed using a superoperator formalism, namely the super Hamiltonian whose ground states correspond to conserved quantities, and low-lying excitations determine the late-time dynamics. The symmetry-breaking impurities only weakly perturb this super Hamiltonian, and preserve some approximate conserved quantities that determine the relaxation of correlation functions. For example, in one spatial dimension, the standard diffusive behavior associated with U(1) conserving systems becomes a different power law at long times when a symmetry-breaking impurity is included. We also consider the scenario of an impurity that destroys Hilbert space fragmentation, where we find that the finite saturation value that appears as a consequence of strong fragmentation persists for exponentially long times, as predicted from the exponentially small gap of the super Hamiltonian. Our approach provides a systematic method for understanding the effects of symmetry-breaking impurities on the relaxation dynamics of symmetric systems.