Charge fluctuation and charge-resolved entanglement in a monitored quantum circuit with $U(1)$ symmetry

TL;DR

This paper investigates a monitored quantum circuit with $U(1)$ symmetry, revealing a phase transition in charge fluctuations and entanglement properties, with critical behaviors similar to equilibrium Luttinger liquids.

Contribution

It uncovers a novel phase transition in charge fluctuations and entanglement in $U(1)$ symmetric circuits, with critical scaling behaviors analogous to Luttinger-liquid theory.

Findings

Identifies a phase transition characterized by charge fluctuation behavior.

Shows critical scaling of charge-resolved entanglement similar to Luttinger liquids.

Demonstrates that certain critical features do not persist below the transition.

Abstract

We study a (1+1)-dimensional quantum circuit consisting of Haar-random unitary gates and projective measurements that conserve a total $U(1)$ charge and thus have $U(1)$ symmetry. In addition to a measurement-induced entanglement transition between a volume-law and an area-law entangled phase, we find a phase transition between two phases characterized by bipartite charge fluctuation growing with the subsystem size or staying constant. At this charge-fluctuation transition, steady-state quantities obtained by evolving an initial state with a definitive total charge exhibit critical scaling behaviors akin to Tomonaga-Luttinger-liquid theory for equilibrium critical quantum systems with $U(1)$ symmetry, such as logarithmic scaling of bipartite charge fluctuation, power-law decay of charge correlation functions, and logarithmic scaling of charge-resolved entanglement whose coefficient becomes a universal quadratic function in a flux parameter. These critical features, however, do not persist below the transition in contrast to a recent prediction based on replica field theory and mapping to a classical statistical mechanical model.