Fractal, Logarithmic, and Volume-Law Entangled Nonthermal Steady States via Spacetime Duality

The extension of many-body quantum dynamics to the nonunitary domain has led to a series of exciting developments, including new out-of-equilibrium entanglement phases and phase transitions. We show how a duality transformation between space and time, on one hand, and unitarity and nonunitarity, on the other, can be used to realize steady-state phases of nonunitary dynamics that exhibit a rich variety of behavior in their entanglement scaling with subsystem size—from logarithmic to extensive to fractal. We show how these outcomes in nonunitary circuits (that are “spacetime dual” to unitary circuits) relate to the growth of entanglement in time in the corresponding unitary circuits, and how they differ, through an exact mapping to a problem of unitary evolution with boundary decoherence, in which information gets “radiated away” from one edge of the system. In spacetime duals of chaotic unitary circuits, this mapping allows us to analytically derive a nonthermal volume-law entangled phase with a universal logarithmic correction to the entropy, previously observed in unitary-measurement dynamics. Notably, we also find robust steady-state phases with fractal entanglement scaling, S(ℓ)∼ℓ^{α} with tunable 0