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

TL;DR

This work constructs a replica field theory for monitored quantum many-body dynamics under the quantum jumps protocol, providing a Keldysh action for $R$ replicated degrees of freedom in which the non-Hermitian evolution is replica-diagonal and the quantum jumps couple replicas. Focusing on the monitored Ising chain with density monitoring, the authors identify a replica-symmetric saddle point that matches the unconditional Lindblad dynamics for the averaged state, and derive the replica-off-diagonal sector as a Non-Linear Sigma Model whose symmetry class is DIII (or D in special limits). The analysis reveals how the replica structure encodes the entangling behavior seen in simulations and shows the weak-monitoring phase is perturbatively stable for small monitoring strength, with the universality governed by the associated NLSM. The framework unifies and extends previous field-theoretic treatments of monitored free fermions and Majorana systems, and opens routes to exploring more general monitors, interactions, and disorder in the context of measurement-induced entanglement transitions.

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.

Figures (3)

• Figure 1: Cartoon of the setup. A monitored fermionic chain evolving under quantum jump protocol, characterised by a deterministic evolution driven by a non-Hermitian Hamiltonian $H_{\rm nH}$$a)$ and stochastic Quantum Jumps $b)$.
• Figure 2: Scheme of the considered contour $\mathcal{C}$
• Figure 3: Scheme of the considered contour $C$, which runs as usual in a counterclockwise manner.