[Paper Review] Field theory of charge sharpening in symmetric monitored quantum circuits

Fergus Barratt, Utkarsh Agrawal+4 more|arXiv (Cornell University)|Nov 17, 2021

Quantum many-body systems77 references85 citations

TL;DR

This paper develops a replica field theory to describe charge-sharpening transitions in one-dimensional symmetric monitored quantum circuits with conserved U(1) charge. By mapping the quantum circuit dynamics to a classical statistical mechanics model of interacting charge world-lines, it identifies a modified Kosterlitz-Thouless transition separating a charge-fuzzy critical phase with continuously varying exponents from a charge-sharp phase with short-range correlations, validated via large-scale matrix-product state simulations.

ABSTRACT

Monitored quantum circuits (MRCs) exhibit a measurement-induced phase transition between area-law and volume-law entanglement scaling. MRCs with a conserved charge additionally exhibit two distinct volume-law entangled phases that cannot be characterized by equilibrium notions of symmetry-breaking or topological order, but rather by the non-equilibrium dynamics and steady-state distribution of charge fluctuations. These include a charge-fuzzy phase in which charge information is rapidly scrambled leading to slowly decaying spatial fluctuations of charge in the steady state, and a charge-sharp phase in which measurements collapse quantum fluctuations of charge without destroying the volume-law entanglement of neutral degrees of freedom. By taking a continuous-time, weak-measurement limit, we construct a controlled replica field theory description of these phases and their intervening charge-sharpening transition in one spatial dimension. We find that the charge fuzzy phase is a critical phase with continuously evolving critical exponents that terminates in a modified Kosterlitz-Thouless transition to the short-range correlated charge-sharp phase. We numerically corroborate these scaling predictions also hold for discrete-time projective-measurement circuit models using large-scale matrix-product state simulations, and discuss generalizations to higher dimensions.

Motivation & Objective

• To understand non-equilibrium quantum phases stabilized by continuous measurements in systems with conserved charge.
• To identify and characterize a new type of measurement-induced phase transition—charge sharpening—distinct from entanglement transitions.
• To develop a controlled field theory framework for analyzing steady-state charge fluctuations and their scaling in monitored quantum circuits.
• To validate the field theory predictions using large-scale matrix-product state simulations of discrete-time projective measurement circuits.

Proposed method

• Construct a replica field theory by mapping the monitored quantum circuit to a classical statistical mechanics model with replica permutation spins and charge world-lines.
• Take the weak-measurement, continuous-time limit to derive a stochastic Markov process for charge degrees of freedom in each replica.
• Derive a Langevin-type equation for charge correlation functions, incorporating diffusion and measurement-induced sharpening terms.
• Identify the charge-sharpening transition as a modified Kosterlitz-Thouless transition via scaling analysis of the correlation function Cn(k).
• Use large-scale matrix-product state simulations to numerically verify the predicted scaling behavior of charge correlations and entanglement entropy.
• Analyze modified percolation of charge-sharp sites to bound the location of the true charge-sharpening transition.

Experimental results

Research questions

• RQ1 What is the nature of the phase transition separating charge-fuzzy and charge-sharp phases in symmetric monitored quantum circuits?
• RQ2 How do charge fluctuations scale in the steady state, and what universality class governs the charge-sharpening transition?
• RQ3 Can a controlled field theory description be constructed for the dynamics of conserved charge in monitored circuits?
• RQ4 Does the charge-sharpening transition occur within the volume-law entangled phase, and how does it relate to percolation of measurement outcomes?
• RQ5 What is the role of charge conservation in enabling non-equilibrium phases not captured by equilibrium notions of symmetry-breaking or topological order?

Key findings

• The charge-fuzzy phase is a critical phase with continuously varying critical exponents, terminating in a modified Kosterlitz-Thouless transition to the charge-sharp phase.
• The steady-state charge correlation function scales as Cn(k) ∼ |k|/√p, consistent with the field theory prediction.
• The charge-sharpening transition occurs at p# ≈ 0.2, significantly below the percolation threshold for measured sites (pc ≈ 0.5), indicating it lies within the volume-law entangled phase.
• Numerical simulations using matrix-product states confirm the predicted scaling of charge correlations and entanglement entropy in both phases.
• The transition is shown to be robust to finite-size effects and is consistent with ν = ∞ in the Kosterlitz-Thouless scaling, distinct from standard percolation.
• Modified percolation of charge-sharp sites occurs at p#p ≈ 0.31, providing an upper bound for the true charge-sharpening transition, which is not visible in standard physical observables for non-projective measurements.