Exact, Average, and Broken Symmetries in a Simple Adaptive Monitored Circuit

TL;DR

This paper explores how symmetry principles can classify phases and transitions in a simple adaptive monitored quantum circuit, revealing distinct symmetry behaviors and phase transitions with classical and percolation universality classes.

Contribution

It demonstrates the role of symmetry in non-equilibrium quantum circuits and maps the ordering transition to the Ising class, providing analytical and numerical insights.

Findings

Identification of symmetry-breaking, average symmetry, and exact symmetry phases.

Mapping of the ordering transition to the classical Ising universality class.

Observation of the entanglement transition in the percolation universality class.

Abstract

Symmetry is a powerful tool for understanding phases of matter in equilibrium. Quantum circuits with measurements have recently emerged as a platform for novel states of matter intrinsically out of equilibrium. Can symmetry be used as an organizing principle for these novel states, their phases and phase transitions? In this work, we give an affirmative answer to this question in a simple adaptive monitored circuit, which hosts an ordering transition in addition to a separate entanglement transition, upon tuning a single parameter. Starting from a symmetry-breaking initial state, depending on the tuning parameter, the steady state could (i) remain symmetry-broken, (ii) exhibit the average symmetry in the ensemble of trajectories, or (iii) exhibit the exact symmetry for each trajectory. The ordering transition is mapped to the transition in a classical majority vote model, described by the Ising universality class, while the entanglement transition lies in the percolation class. Numerical simulations are further presented to support the analytical understandings.