Measurement-induced phase transition in a chaotic classical many-body system

Josef Willsher, Shu-Wei Liu, Roderich Moessner, and Johannes Knolle

Phys. Rev. B 106, 024305 – Published 13 July 2022

Abstract

Local measurements in quantum systems are projective operations which act to counteract the spread of quantum entanglement. Recent work has shown that local, random measurements applied to a generic volume-law entanglement generating many-body system are able to force a transition into an area-law phase. This work shows that projective operations can also force a similar classical phase transition; we show that local projections in a chaotic system can freeze information dynamics. In rough analogy with measurement-induced phase transitions, this is characterized by an absence of information spreading instead of entanglement entropy. We leverage a damage-spreading model of the classical transition to predict the butterfly velocity of the system both near to and away from the transition point. We map out the full phase diagram and show that the critical point is shifted by local projections, but remains in the directed percolation universality class. We discuss the implication for other classical chaotic many-body systems and the relation to synchronization transitions.

Received 21 March 2022
Accepted 6 July 2022

DOI:https://doi.org/10.1103/PhysRevB.106.024305

Physics Subject Headings (PhySH)

Cellular automata Classical statistical mechanics Quantum chaos Random walks

Condensed Matter, Materials & Applied PhysicsStatistical Physics & Thermodynamics

Authors & Affiliations

Josef Willsher ^1,*, Shu-Wei Liu ², Roderich Moessner², and Johannes Knolle ^1,3,4

¹Department of Physics TQM, Technische Universität München, James-Franck-Straße 1, 85748 Garching, Germany
²Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
³Munich Center for Quantum Science and Technology (MCQST), 80799 Munich, Germany
⁴Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom

^*joe.willsher@tum.de

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Vol. 106, Iss. 2 — 1 July 2022

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Images

Figure 1
(a) Schematic of the Kauffman cellular automaton: a configuration of Boolean elements $σ (x, t) = \pm 1$ evolves in time under local rules. The effects of an initial perturbation are studied with the spatiotemporal decorrelator which measures the difference between two copies differing only by such a perturbation. Both copies evolve under the same local Kauffman CA rules (as demonstrated in the inset by three such rules at a given position and time), which are randomly generated anew at each time step. The decorrelator is represented by coloring sites of copy $B$ which differ from the reference copy $A$ ; in the chaotic phase of the model, the effect of the perturbation is able to lead to a widening footprint of the decorrelator in space. The single site with initial damage is additionally marked red in both copies. Measurements are represented by projecting the reference copy onto the perturbed copy and are shown with blue vertical lines. (b) Damage-spreading model: the decorrelator defined between two copies of a single realization of the Kauffman CA, evolving in time. The measurement rate $q$ is tuned to induce a phase transition into the frozen phase.
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Figure 2
Phase transition of a Kauffman CA with projective measurements diagnosed by the OTOC, with connectivity $K = 4$ . Left panel: (a) Phase diagram showing the analytic boundary between frozen and chaotic phases. Its functional form is described in the text and confirmed with the data collapse in the critical regime; numerically fixed by precise measurements of the measurement-free system and error bars not shown. The region above the dashed black line $q_{CS}$ highlights where the observable $v_{b}$ in the chaotic phase is best described by the critical scaling (CS) DP model [8]; the determination of the crossover point and its corresponding uncertainty is performed using data as plotted in Fig. 5 (see Sec. 5). Center panels plot the OTOCs demonstrating at different points of the phase diagram (control parameter $p = 0.4$ , system size $N = 2048$ ) showing (b) exponential decay of correlations $q > q_{c}$ , (c) critical power-law spreading of correlations $q = q_{c}$ , and (d) light-cone spreading of the OTOC in the chaotic phase ( $q < q_{c}$ ) at the butterfly velocity $v_{b}$ , which is dependent on the parameters $(p, q)$ . This is always less than the maximum velocity at which information could travel, $v_{\max} = K$ . The right panel (e) plots the Hamming distance $H (t)$ for a range of $q$ with $p = 0.4, N = 2048$ . It shows linear growth in the chaotic phase, power-law critical growth (with DP exponent $θ$ [37, 38]) at $q_{c}$ , and exponential decay of perturbations caused by measurements above the critical rate.
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Figure 3
Main figure: Scaled boundary distribution for different realizations of the $p = 0.4, q = 0.4$ model. Top-left inset: unscaled distribution. Bottom: each gray point is the boundary position of one realization, plotted in $x − t$ space. The red line is the averaged boundary position.
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Figure 4
Scaling collapse of the numerically calculated decorrelator $D (v, t)$ in the long-time limit (colored lines) performed for a range of different measurement rates $q$ at fixed $p = 0.4$ . For each specific $q$ , data are obtained from the value of $D$ in the long-time limit. The horizontal axis is rescaled by $\sqrt{μ}$ and centered on $v_{b}$ , as calculated from the boundary model accounting for the effect of measurements. The fit line is the analytic scaling form (see main text) [8] which has VDLE exponent $β = 2$ for $v > v_{b}$ .
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Figure 5
Numerical butterfly velocity $v_{b}$ with varied measurement rate $q$ and fixed $p = 0.4$ (gray dots). The data are well described by the boundary model (dashed line) at low $q < q_{CS}$ , and by the critical scaling DP model (solid line) in the range $q_{CS} \leq q \leq q_{c}$ , with $q_{CS}$ shown to be approximately 0.3 in this instance. Data are collected on a $K = 4$ system with system size $N = 2048$ .
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Figure 6
Scaling collapse of the butterfly velocity as a function of measurement rate performed for the range $p = 0.2 X − 0.5$ ; the dashed line highlights the power-law behavior with critical exponent $z^{- 1}$ . The logarithm of butterfly velocity is plotted and shifted by the appropriate normalization $log (A)$ , defined in terms of $q$ in the main text. The horizontal axis is the distance of the measurement rate from the critical rate $q - q_{c}$ . The figure inset shows the same data on a semilogarithmic plot and without the data collapse.
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