Strong-to-weak symmetry breaking in monitored dipole conserving quantum circuits

Abstract

We explore the information-theoretic phases of monitored quantum circuits subject to dynamics that conserves both charge and dipole moment, as well as measurements of the local charge density. Explicitly, both charge and dipole-moment conservation are strong symmetries, but under the dynamics they can be spontaneously broken to weak symmetries: this spontaneous symmetry breaking has an information-theoretic interpretation in terms of whether one can learn global charges from local measurements. We find a rich phase diagram: in one spatial dimension, charge is always easy to learn, while dipole moment can be either easy or hard. In two dimensions, we find three phases: for frequent measurements, both charge and dipole moment are easy to learn; as the measurement rate is decreased, first dipole moment and then charge become hard. In two dimensions, the low-measurement phase is an exotic critical phase with anisotropic spacetime scaling, analogous to a smectic liquid crystal.

Figures (2)

• Figure 1: Non-equilibrium phases in monitored dipole-moment conserving circuits. a) Illustration of hierarchical strong-to-weak spontaneous symmetry breaking (SW-SSB): (left) both charge and dipole sectors are mixed and hence weakly symmetric; (middle) charge is strongly symmetric while dipole is weakly symmetric; (right) both charge and dipole are strongly symmetric. b) The hierarchy of non-equilibrium phases in monitored quantum circuits realized by tuning the measurement rate $\gamma$ (top: 1+1D, bottom: 2+1D). We characterize the non-equilibrium phases by SW-SSB, survival time (i.e., sharpening time) of a weakly-symmetric initial state in a finite system of linear extent $L$, and the dynamical exponent $z$. Here, ✓ indicates the weakly symmetric (SW-SSB) phase, ✗ the strongly symmetric phase, and quasi long-range order (QLRO) power-law decaying correlations. For the gapped degrees of freedom, $z$ is left blank.
• Figure 2: Strongly symmetric quantum channel. We consider a brickwork circuit composed of gates conserving the global charge and dipole moment. Each gate connects configurations within the same symmetry sector via Haar-random unitaries. The unitary evolution is followed by a completely dephasing channel, and local measurements are performed at each site with a certain rate $\gamma$.