Measurement-induced phase transition: A case study in the nonintegrable model by density-matrix renormalization group calculations

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Measurement-induced phase transition: A case study in the nonintegrable model by density-matrix renormalization group calculations

Qicheng Tang and W. Zhu

Phys. Rev. Research 2, 013022 – Published 7 January 2020

Abstract

We study the effect of local projective measurements on the quantum quench dynamics. As a concrete example, a one-dimensional Bose-Hubbard model is simulated by the matrix product state and time-evolving block decimation. We map out a global phase diagram in terms of the measurement rate in spatial space and time domain, which demonstrates a volume-to-area law entanglement phase transition. When the measurement rate reaches the critical value, we observe a logarithmic growth of entanglement entropy as the subsystem size or evolved time increases. Moreover, we find that the probability distribution of the single-site entanglement entropy distinguishes the volume and area law phases, similar to the case of disorder-induced many-body localization. We also investigate the scaling behavior of entanglement entropy and mutual information between two separated sites, which is indicative of a single universality class and thus suggests a possible unified description of this transition.

Received 3 September 2019

DOI:https://doi.org/10.1103/PhysRevResearch.2.013022

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)

Localization Nonequilibrium statistical mechanics

Matrix product states

Statistical Physics & ThermodynamicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Qicheng Tang ^1,2,3,* and W. Zhu^2,3,†

¹Zhejiang University, Hangzhou 310027, China
²School of Science, Westlake University, Hangzhou 310024, China
³Institute of Natural Sciences, Westlake Institute of Advanced Study, Hangzhou 310024, China

^*tangqicheng@westlake.edu.cn
^†zhuwei@westlake.edu.cn

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Vol. 2, Iss. 1 — January - March 2020

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Figure 1
(a) A diagrammatic representation of the quench dynamics with local projective measurements. The quench dynamics is from a trivial product state described by MPS (gray rectangles), and the unitary time evolution background is build with several unitary layers described by MPO (blue rectangles). The local projective measurements (green rounded squares) are applied only after several unitary layers, which are chosen randomly with measurement rate in time $N_{t}$ . For each measured layers, the local projective measurements are posited randomly with measurement rate in space $P_{x}$ . (b) Two-dimensional phase diagram of the entanglement phase transition as a function of the measurement rate in time $N_{t}$ and in space $P_{x}$ , where $A$ and $B$ represent the disentangling and entangling phases respectively. The background colors represent the value of the von Neumann entropy for long-time steady states with subsystem size $L = 8$ averaged over different (up to 900) random realizations.
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Figure 2
The dynamics of the von Neumann entropy with subsystem size $L = 8$ for different values of $P_{x}$ ; here the results for $N_{t} = 50$ are chosen. The results for different (up to 900) random realizations are presented in the form of a two-dimensional color-coded histogram. [(a)–(c)] The full dynamics of the entropy with time up to $T = 30$ , where the von Neumann entropy already reaches the platform saturated by the subsystem size. [(d)–(f)] The short-time dynamics of the entropy plotted in $ln t$ . The red line shows a $ln t$ growth of the averaged entropy. The inset in panel (e) shows a fitting in form $S = a ln t + b$ , with $a \approx 0.22 \pm 0.02$ and $b \approx 0.58 \pm 0.02$ .
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Figure 3
(a) The spacial distribution of the von Neumann entropy for long-time steady states, with fixed $N_{t} = 50$ , after averaging over hundreds of random realizations. Different curves from top to bottom correspond to different values of $P_{x} = 0.02, 0.025, 0.03, 0.035, 0.04, 0.045, 0.05, 0.055, 0.06, 0.07, 0.08$ . (b) The von Neumann entropy of long-time steady states for $P_{x} \sim P_{x, c}$ . The results for 240 different random realizations are presented in the form of a two-dimensional color-coded histogram. The red line shows the averaged data. (c) The red line shows the averaged von Neumann entropy of long-time steady states for $P_{x} \sim P_{x, c}$ plotted in $ln L$ . The black line shows the result of a fitting in the form $S = a ln L + b$ , with $a \approx 0.21 \pm 0.01$ and $b \approx 0.41 \pm 0.01$ .
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Figure 4
Probability distribution of the single-site von Neumann entropy for different values $P_{x}$ , with fixed $N_{t} = 50$ . (a) Deep in volume-law phase and (b) near the critical point and in area-law phase.
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Figure 5
(a) The von Neumann entropy of long-time steady states with different subsystem size $L$ as a function of the measurement rate in space $P_{x}$ , with fixed $N_{t} = 50$ . (b) Data collapse of the same data presented in panel (a) by scaling form in Eq. (2) with $P_{x, c} = 0.06$ and $ν = 2$ .
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Figure 6
Decay of the the mutual information $I_{A, B}$ in spatial distance $r$ . (a) The mutual information $I_{A, B}$ as a function of $ln r$ for different values of $P_{x}$ , with fixed $N_{t} = 50$ . (b) Linear fitting of the mutual information in form $ln I_{A, B} = a ln r + b$ with the critical measurement rate $P_{x} = 0.06$ . The data for different total system sizes $L_{0} = 36$ , 48 are plotted by green and red lines, respectively. The fitting results are $a (L_{0} = 36) \approx - 2.58 \pm 0.65, b (L_{0} = 36) \approx 0.02 \pm 1.99$ and $a (L_{0} = 48) \approx - 3.12 \pm 0.57, b (L_{0} = 48) \approx 0.68 \pm 1.89$ , which give critical exponents $Δ (L_{0} = 36) \approx 1.29$ and $Δ (L_{0} = 48) \approx 1.56$ .
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