Effective Theory for the Measurement-Induced Phase Transition of Dirac Fermions

TL;DR

This paper develops an effective theoretical framework for understanding measurement-induced phase transitions in Dirac fermions, revealing a critical phase with logarithmic entanglement growth and a gapped phase, separated by a BKT transition.

Contribution

It introduces an $n$-replica Keldysh field theory for measurement-induced transitions and applies it to Dirac fermions, uncovering a new phase diagram with critical and gapped phases.

Findings

Identifies a gapless critical phase with logarithmic entanglement growth.

Discovers a gapped area law phase separated by a BKT transition.

Provides a field-theoretic description of measurement-induced transitions in fermionic systems.

Abstract

A wave function exposed to measurements undergoes pure state dynamics, with deterministic unitary and probabilistic measurement induced state updates, defining a quantum trajectory. For many-particle systems, the competition of these different elements of dynamics can give rise to a scenario similar to quantum phase transitions. To access it despite the randomness of single quantum trajectories, we construct an $n$-replica Keldysh field theory for the ensemble average of the $n$-th moment of the trajectory projector. A key finding is that this field theory decouples into one set of degrees of freedom that heats up indefinitely, while $n-1$ others can be cast into the form of pure state evolutions generated by an effective non-Hermitian Hamiltonian. This decoupling is exact for free theories, and useful for interacting ones. In particular, we study locally measured Dirac fermions in $(1+1)$ dimensions, which can be bosonized to a monitored interacting Luttinger liquid at long wavelengths. For this model, the non-Hermitian Hamiltonian corresponds to a quantum Sine-Gordon model with complex coefficients. A renormalization group analysis reveals a gapless critical phase with logarithmic entanglement entropy growth, and a gapped area law phase, separated by a Berezinskii-Kosterlitz-Thouless transition. The physical picture emerging here is a pinning of the trajectory wave function into eigenstates of the measurement operators upon increasing the monitoring rate.