Boundary Perturbation Effects in Quantum Systems with Conserved Energy and Continuous Symmetry

We investigate one-dimensional systems with both energy conservation and a continuous symmetry, focusing on the impact of a boundary perturbation that breaks the continuous symmetry. Our study reveals two distinct dynamical phases: one in which the corresponding charge exhibits extensive fluctuations, and another where the charge remains conserved. These phenomena are observed in both free and interacting models. We provide an explanation using the effective Hamiltonian derived from degenerate perturbation theory. In the Floquet setting, we show that the frozen phase can survive at high driving frequencies but vanishes at low frequencies. The emergence of the charge-frozen phase is attributed to effective energy conservation, and we demonstrate that this phase disappears when effective energy conservation is broken or replaced by other symmetries.