Charge Sharpening via Tower of States in Replica SSB

In a previous work, we demonstrated that charge transport dynamics in $G$-symmetric random circuits can be understood as a consequence of strong to weak spontaneous $G$-symmetry breaking (SSB) in the steady states. For continuous $G$, the mechanism gives rise to gapless Nambu-Goldstone modes in an effective Hamiltonian description in a doubled Hilbert space, the dispersion of which determines how fast charges can transport. In this work, we extend our approach to study information-theoretic transitions in $G$-symmetric monitored random circuits, where a novel form of SSB in multiple copies of the Hilbert space provides a unifying framework for charge sharpening and purification transitions. In particular, the energy spacing in the Anderson tower of state (TOS) from this SSB in the replicated Hilbert space for higher moments of density matrix determines the timescale for charge sharpening and purification, providing a concrete connection between the information dynamics and SSB physics.