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arXiv:2302.12820 · Physical Review X

Nonlinear Sigma Models for Monitored Dynamics of Free Fermions

Michele Fava, Lorenzo Piroli, Tobias Swann, Denis Bernard, Adam Nahum

Published Dec 8, 2023• 92 citations
arXivHTMLPDF

Abstract

We derive field theory descriptions for measurement-induced phase transitions in free fermion systems.We focus on a multiflavor Majorana chain, undergoing Hamiltonian evolution with continuous monitoring of local fermion parity operators.Using the replica trick, we map the dynamics to the imaginary time evolution of an effective spin chain and use the number of flavors as a large parameter for a controlled derivation of the effective field theory.This is a nonlinear sigma model for an orthogonal N × N matrix, in the replica limit N → 1. (On a boundary of the phase diagram, another sigma model with higher symmetry applies.)Together with known results for the renormalization-group beta function, this derivation establishes the existence of stable phases-nontrivially entangled and disentangled, respectively-in the physically relevant replica limit N → 1.In the nontrivial phase, an asymptotically exact calculation shows that the bipartite entanglement entropy for a system of size L scales as ðln LÞ 2 , in contrast to findings in previously studied models.Varying the relative strength of Hamiltonian evolution and monitoring, as well as a dimerization parameter, the model's phase diagram contains transitions out of the nontrivial phase, which we map to vortex-unbinding transitions in the sigma model, and also contains separate critical points on the measurement-only axis.We highlight the close analogies as well as the differences with the replica approach to Anderson transitions in disordered systems.

DOI
10.1103/physrevx.13.041045

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We derive field theory descriptions for measurement-induced phase transitions in free fermion systems.We focus on a multiflavor Majorana chain, undergoing Hamiltonian evolution with continuous monitoring of local fermion parity operators.Using the replica trick, we map the dynamics to the imaginary time evolution of an effective spin chain and use the number of flavors as a large parameter for a controlled derivation of the effective field theory.This is a nonlinear sigma model for an orthogonal N × N matrix, in the replica limit N → 1. (On a boundary of the phase diagram, another sigma model with higher symmetry applies.)Together with known results for the renormalization-group beta function, this derivation establishes the existence of stable phases-nontrivially entangled and disentangled, respectively-in the physically relevant replica limit N → 1.In the nontrivial phase, an asymptotically exact calculation shows that the bipartite entanglement entropy for a system of size L scales as ðln LÞ 2 , in contrast to findings in previously studied models.Varying the relative strength of Hamiltonian evolution and monitoring, as well as a dimerization parameter, the model's phase diagram contains transitions out of the nontrivial phase, which we map to vortex-unbinding transitions in the sigma model, and also contains separate critical points on the measurement-only axis.We highlight the close analogies as well as the differences with the replica approach to Anderson transitions in disordered systems.
We study measurement-induced phases of free fermion systems with U ( 1 ) symmetry. Following a recent approach developed for Majorana chains, we derive a field theory description for the purity and bipartite entanglement at large space and timescales. We focus on a multiflavor one-dimensional chain with random complex hoppings and continuous monitoring of the local fermion density. By means of the replica trick, and using the number of flavors as a large parameter controlling our approximations, we derive an effective field theory made up of a SU(N) nonlinear sigma model ( NL σ M ) coupled to fluctuating hydrodynamics. Contrary to the case of noninteracting Majorana fermions, displaying no U ( 1 ) symmetry, we find that the bipartite entanglement entropy satisfies an area law for all monitoring rates but with a nontrivial scaling of entanglement when the correlation length is large. We provide numerical evidence supporting our claims. We briefly show how imposing a reality condition on the hoppings can change the NL σ M and also discuss higher-dimensional generalizations. Published by the American Physical Society 2024
Measurement-induced phase transitions (MIPTs) in monitored quantum dynamics are nonequilibrium phase transitions between quantum-chaotic (volume-law entangled) and entanglement-suppressed, area-law phases. Here we reveal how monitored dynamics are situated within the framework of general far-from-equilibrium, quantum condensed-matter physics. Measurement-induced heating effects scramble the distribution function in generic (interacting) monitored fermion systems, and this enables a simplified symmetry-based description of the dynamics. We demonstrate the equivalence of the Keldysh technique with the conventional Statistical-Mechanics Model for circuits, resulting from a doubled Hilbert-space (Choi-Jamio\l{}kowski) mapping. We illustrate this using the monitored dynamics of interacting fermions with a conserved charge, deriving a unified effective-field theory that captures all phases and phase transitions. The noninteracting counterpart in one-dimensional space only has an area-law phase, with no MIPT. This was explained via an effective nonlinear sigma model replica field theory possessing a very large symmetry. We show that other phases and phase transitions emerge when the replica symmetry is reduced by interactions. The reduced symmetry combines a replica permutation symmetry and charge-conservation within each replica. The former and its spontaneous breaking govern the MIPT, which can be recognized via a separatrix in the renormalization-group flow. The replica-resolved charge conservation dictates the ``charge-sharpening'' transition between two kinds of dynamics, where the global charge information is either hidden or reconstructible from the measurements. The field theory explains why the charge-sharpening transition should occur only in the volume-law phase. Our framework provides a template for other classes of MIPTs and situates these within the arena of nonequilibrium condensed-matter physics.
We explore, both analytically and numerically, the quantum dynamics of a many-body free-fermion system subjected to local density measurements. We begin by extending the mapping to the nonlinear sigma-model (NLSM) field theory for the case of finite evolution time T and different classes of initial states, which lead to different NLSM boundary conditions. The analytical formalism is then used to study how quantum correlations gradually develop, with increasing T , from those determined by the initial state towards their steady-state form. The analytical results are confirmed by numerical simulations for several types of initial states. We further consider the long-time limit, when the system in d + 1 space-time dimensions becomes quasi-one-dimensional, and analyze the scaling of the “localization” time (which is simultaneously the purification time and the charge-sharpening time for this class of problems). The analytical predictions for scaling properties are fully confirmed by numerical simulations in a d = 2 model around the measurement-induced phase transition. We use this dynamical approach to determine numerically the measurement-induced transition point and the associated correlation-length critical exponent.
We develop an analytical approach to the study of one-dimensional free fermions subject to random projective measurements of local site occupation numbers, based on the Keldysh path-integral formalism and replica trick.In the limit of rare measurements, γ=J ≪ 1 (where γ is measurement rate per site and J is hopping constant in the tight-binding model), we derive a nonlinear sigma model (NLSM) as an effective field theory of the problem.Its replica-symmetric sector is described by a Uð2Þ=Uð1Þ × Uð1Þ ≃ S 2 sigma model with diffusive behavior, and the replica-asymmetric sector is a two-dimensional NLSM defined on SUðRÞ manifold with the replica limit R → 1.On the Gaussian level, valid in the limit γ=J → 0, this model predicts a logarithmic behavior for the second cumulant of number of particles in a subsystem and for the entanglement entropy.However, the one-loop renormalization group analysis allows us to demonstrate that this logarithmic growth saturates at a finite value ∼ðJ=γÞ 2 even for rare measurements, which corresponds to the area-law phase.This implies the absence of a measurement-induced entanglement phase transition for free fermions.The crossover between logarithmic growth and saturation, however, happens at exponentially large scale, ln l corr ∼ J=γ.This makes this crossover very sharp as a function of the measurement frequency γ=J, which can be easily confused with a transition from the logarithmic to area law in finite-size numerical calculations.We have performed a careful numerical analysis, which supports our analytical predictions.
A theory of the measurement-induced entanglement phase transition for free-fermion models in d>1d>1d>1 dimensions is developed. The critical point separates a gapless phase with ${\ensuremath{\ell}}^{d\ensuremath{-}1}\mathrm{ln}\ensuremath{\ell}$ scaling of the second cumulant of the particle number and of the entanglement entropy and an area-law phase with ${\ensuremath{\ell}}^{d\ensuremath{-}1}$ scaling, where $\ensuremath{\ell}$ is a size of the subsystem. The problem is mapped onto an SU(R)\mathrm{SU}(R)SU(R) replica nonlinear sigma model in d+1d+1d+1 dimensions, with $R\ensuremath{\rightarrow}1$. Using renormalization-group analysis, we calculate critical indices in one-loop approximation justified for $d=1+\ensuremath{\epsilon}$ with $\ensuremath{\epsilon}\ensuremath{\ll}1$. Further, we carry out a numerical study of the transition for a d=2d=2d=2 model on a square lattice, determine numerically the critical point, and estimate the critical index of the correlation length, $\ensuremath{\nu}\ensuremath{\approx}1.4$.
Measurement-induced phase transitions are nonequilibrium transitions between phases characterized by distinct entanglement scaling behaviors, driven by the competition between unitary dynamics and measurements. Despite recent numerical efforts, how quenched disorder affects these transitions remains unclear. In this work, we study a ddd-dimensional noninteracting fermionic system subject to both quenched disorder and continuous monitoring of the local particle density, and derive an effective field theory describing its long-time universal behaviors. We find that the system is governed by the same nonlinear sigma model as in the case of clean monitored fermions, with disorder entering only through a modification of model parameters. This result suggests that the presence or absence of a measurement-induced phase transition is unaffected by the introduction of disorder. In spatial dimension d=1d=1d=1, the system exhibits only an area law phase and no transition, whereas in d>1d>1d>1, a transition persists between an area×\times×log law phase and an area law phase, with the critical measurement rate suppressed by disorder. Numerical simulation confirms the absence of MIPT in d=1d=1d=1, and the linear shift of the critical measurement rate by disorder scattering rate in higher dimensions predicted by analytical theory.