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

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

M. Buchhold, Y. Minoguchi, A. Altland, and S. Diehl

Phys. Rev. X 11, 041004 – Published 7 October 2021

Abstract

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 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 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.

Received 15 March 2021
Revised 1 July 2021
Accepted 9 August 2021

DOI:https://doi.org/10.1103/PhysRevX.11.041004

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Nonequilibrium statistical mechanics Quantum entanglement Quantum measurements Quantum simulation Quantum stochastic processes

1-dimensional spin chains Quantum wires

Luttinger liquid model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

M. Buchhold ¹, Y. Minoguchi², A. Altland¹, and S. Diehl¹

¹Institut für Theoretische Physik, Universität zu Köln, D-50937 Cologne, Germany
²Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1040 Vienna, Austria

Popular Summary

The recent discovery of a novel type of phase transition—driven by the competition between a quantum system’s internal evolution and its external observation—has sparked significant research into its nature and origin. Induced by measurements, the transition is marked by a qualitative change of the behavior of the entanglement entropy. Here, we provide the missing bridge from micro- to macrophysics by developing a new theory that describes the transition from the perspective of nonequilibrium quantum statistical mechanics.

The decisive step in understanding the change from microscopic simplicity to macroscopic complexity lies in the identification of the relevant degrees of freedom. Our work takes that step for this novel class of measurement-induced phase transitions. Such transitions then appear as a natural consequence of an underlying quantum field theory, distilling the quantum physics of the monitored many-body wave function from the noisy background due to random measurement outcomes. To make this more concrete, we consider a system of free Dirac fermions and show that, by observation alone, the Dirac fermions are made to behave like strongly interacting electrons or planar magnets—they undergo what is known as a Berezinskii-Kosterlitz-Thouless quantum phase transition.

Our approach connects the novel class of measurement-induced transitions more firmly to quantum phase transitions. We demonstrate that measurement-induced transitions reflect profound changes in many-body wave functions due to observation, where the entanglement signatures just represent the tip of the iceberg.

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Vol. 11, Iss. 4 — October - December 2021

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Images

Figure 1
Illustration of competing Hamilton and measurement dynamics in the two-site toy model. (a),(b) Single trajectory dynamics. (a) For strong monitoring $g = 10$ , the system is pinned to dark states for most of the time, with weak, short-time fluctuations. (b) A dominant Hamilton dynamics ( $g = 0.5$ ) leads to depinning, with almost no time spent in the dark states. (c) Trajectory averaged correlation functions. Quantities linear in the state are insensitive to the competition ratio, while the correlation function nonlinear in the state witnesses a monitoring dependence, which develops a nonanalytic behavior in the thermodynamic limit.
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Figure 2
(a) Phase diagram of the non-Hermitian sine-Gordon field theory, obtained from numerical integration of the flow equations (52) and (53). In the gray region the theory flows toward the Gaussian fixed point located at the origin. The phase boundary is signaled by the black dotted line. For comparison we also show the transition line for the Hermitian (thermal equilibrium) theory. Panels (b) and (c) display the renormalization group flow toward the strong coupling fixed point (b) and the Gaussian fixed point (c) for the non-Hermitian sine-Gordon theory and the corresponding Hermitian sine-Gordon theory (with comparably larger starting value of $K$ ). The initial parameters of the curves (i) and (iii)-(v) are displayed in (a); the remaining curves correspond to $| K | = 5.2 π$ for (ii) and $| K | = 3 π$ in (vi). All curves start at $λ / A = 3$ .
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Figure 3
(a) Keldysh representation of a single replica. Time evolution of the noise averaged density matrix is captured by two Keldysh contours running horizontally; the spatial dimension is represented by the vertical lines. The trace operation is represented by a black line connecting final times $t_{f}$ , for a selected position $x$ . The noise average (lower blue line) yields the usual path integral representation of a Lindblad equation, with characteristic contour coupling terms. (b) Keldysh representation of $n$ replicas. In addition to the coupling of contours within each replica, the average over the stochastic term couples all replicas (upper red lines). The trace operation acts on each replica individually, and is indicated by black lines.
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Figure 4
Illustration of the coupling of the fields $ϕ_{X}$ to the measurement noise at a fixed position $x$ and for different times $t$ . (a) In the replica basis ${ϕ_{σ, X}^{(l)}}$ , each field couples to the same (spatiotemporal) measurement noise $d W_{t}$ (the individual fields are given an offset with respect to each other for better visibility). (b) In the Fourier basis ${ϕ_{σ, X}^{(k)}}$ , defined in Eq. (67), the $k = 0$ or center-of-mass mode couples to an enhanced noise $\sqrt{n} d W_{t}$ while all the relative modes ( $k > 0$ ) decouple from $d W_{t}$ and are noise-free.
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Figure 5
Prefactor of the logarithmic scaling function (effective central charge) $c (γ)$ obtained from integrating the RG equation (53) for the parameter $K$ . For $λ = 0$ , we observe the scaling $c (γ) \sim γ^{- 0.5}$ (red line and inset). For $λ \neq 0$ , a transition into the area law phase happens at a critical measurement strength $γ_{c} (λ)$ , where $c (γ)$ drops to zero. In the weak measurement regime, $c (γ)$ increases with $λ$ . The lines correspond to $λ = 0.2 γ$ (light gray, solid), $λ = 0.25 γ$ (dark gray, dashed), and $λ = 0.3 γ$ (black, dotted).
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Figure 6
Graphical representation of the reduced partition sum. Over the interval $A$ , consecutive replicas are coupled. For the rest of the system $R \ A$ , replicas are decoupled from each other. The singular gauge $A_{μ}^{(k)}$ appears as charges of opposite sign at the boundaries of $A$ .
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