Replica Field Theory of Quantum Jumps Monitoring: Application to the Ising Chain

TL;DR

This paper develops a replica field theory for monitored quantum many-body systems under quantum jumps, applying it to the Ising chain to understand entanglement phases and symmetry classes.

Contribution

It introduces a novel replica field theory framework for monitored quantum systems and derives an effective field theory for the Ising chain with quantum jump monitoring.

Findings

Derives the replica-diagonal saddle point describing the average state.

Formulates a Non-Linear Sigma Model for the off-diagonal sector.

Identifies symmetry classes DIII or D depending on parameters.

Abstract

In this work we derive the replica field theory for monitored quantum many-body systems evolving under the quantum jumps protocol, corresponding to a non-Hermitian evolution interspersed with random quantum jumps whose distribution is state-dependent. We show that the density matrix of $R$ replicas evolves according to a master equation where the non-Hermitian term is replica-diagonal while coupling among replicas are due to quantum jumps. We write down the associated Keldysh action and study its behavior for the specific case of the Ising Chain with monitoring of particle density and tunable anisotropy, interpolating between free fermions with strong U(1) symmetry and the Ising chain with Z$_2$ symmetry. We derive the effective field theory in terms of slowly varying fields and obtain the replica-diagonal saddle point, which we show to describe the average state. We then go beyond saddle point and derive the effective field theory describing the replica off-diagonal sector, which takes the form of a Non-Linear Sigma Model. The symmetry class is either DIII or D, depending on the parameters of the Ising chain, except at a special symmetric point, where we recover the results for free fermions. We discuss the implications of these findings for the entangling phase observed numerically for the monitored Ising chain.