Exponentially slow thermalization and the robustness of Hilbert space fragmentation

The phenomenon of Hilbert space fragmentation, whereby dynamical constraints fragment Hilbert space into many disconnected sectors, provides a simple mechanism by which thermalization can be arrested. However, little is known about how thermalization occurs in situations where the constraints are not exact. To study this, we consider a situation in which a fragmented 1d chain with pair-flip constraints is coupled to an ergodicity-restoring thermal bath at its boundary. We numerically observe an exponentially long thermalization time in Hamiltonian dynamics, manifested in both entanglement dynamics and the relaxation of local observables. To understand this, we study an analogous model of random unitary circuit dynamics, whose thermalization time we prove scales exponentially with system size. Slow thermalization in this model is shown to be a consequence of strong bottlenecks in configuration space, which restrict how the system can explore Hilbert space, and demonstrate a new way of producing anomalously slow thermalization dynamics.