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

• Published in: Physical review. X Vol. 11; no. 4; p. 041004
• Main Authors: Buchhold, M., Minoguchi, Y., Altland, A., Diehl, S.
• Format: Journal Article
• Language: English
• Published: College Park American Physical Society 01.10.2021
• Subjects: Background noise; Competition; Complexity; Decoupling; Degrees of freedom; Distillation; Eigenvectors; Entropy; Evolution; Fermions; Field theory; Magnets; Operators (mathematics); Phase transitions; Quantum entanglement; Quantum field theory; Quantum statistics; Quantum theory; Statistical analysis; Statistical mechanics; Wave functions
• ISSN: 2160-3308; 2160-3308
• DOI: 10.1103/PhysRevX.11.041004

A wave function subject to unitary time evolution and 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 this competition despite the randomness of single quantum trajectories, we construct ann-replica Keldysh field theory for the ensemble average of thenth 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, whilen−1others 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 measurement-induced pinning of the trajectory wave function into eigenstates of the measurement operators, which succeeds upon increasing the monitoring rate across a critical threshold.