In this talk, I will explore the intriguing intersection of spontaneous symmetry breaking and open quantum dynamics.
First, I will introduce a family of Hamiltonians that circumvents the Hohenberg-Mermin-Wagner theorem, which prohibits the spontaneous breaking of continuous symmetries in spatial dimensions d>1 at finite temperatures. The classical/quantum correspondence suggests the absence of continuous symmetry breaking in one dimension at zero temperature. While the Heisenberg ferromagnet and Hamiltonians with coordinate-dependent symmetry charges are known exceptions, our Hamiltonian uniquely exhibits spontaneous breaking of U(1) “on-site” symmetry at zero temperature, despite the order parameter not commuting with the Hamiltonian.
In the second part, I will discuss the dynamics governed by a Brownian random circuit with strong U(1) symmetry. Interestingly, the emergent hydrodynamics of charge transport can be effectively described using the Hamiltonian from the first part. Moreover, the soft Goldstone modes associated with the spontaneous breaking of U(1) symmetry lead to diffusive or sub-diffusive charge dynamics. By constructing dispersive excited states of this effective Hamiltonian through a single-mode approximation, we gain a comprehensive understanding of many-body systems with conserved multipole moments and varying interaction ranges. Our approach further identifies exotic Krylov-space-resolved diffusive relaxation, verified numerically, even in the presence of dipole conservation.