Out-of-time-order correlation in marginal many-body localized systems

Kevin Slagle, Zhen Bi, Yi-Zhuang You, and Cenke Xu

Phys. Rev. B 95, 165136 – Published 24 April 2017

Abstract

We show that the out-of-time-order correlation (OTOC) $〈 W {(t)}^{†} V {(0)}^{†} W (t) V (0) 〉$ in many-body localized (MBL) and marginal MBL systems can be efficiently calculated by the spectrum bifurcation renormalization group (SBRG). We find that in marginal MBL systems, the scrambling time $t_{scr}$ follows a stretched exponential scaling with the distance $d_{W V}$ between the operators $W$ and $V$ : $t_{scr} \sim exp (\sqrt{d_{W V} / l_{0}})$ , which demonstrates Sinai diffusion of quantum information and the enhanced scrambling by the quantum criticality in nonchaotic systems.

Received 15 December 2016
Revised 4 April 2017

DOI:https://doi.org/10.1103/PhysRevB.95.165136

Physics Subject Headings (PhySH)

Many-body localization Quantum criticality

Condensed Matter, Materials & Applied PhysicsQuantum Information, Science & Technology

Authors & Affiliations

Kevin Slagle¹, Zhen Bi¹, Yi-Zhuang You^1,2, and Cenke Xu¹

¹Department of Physics, University of California, Santa Barbara, California 93106, USA
²Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

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Vol. 95, Iss. 16 — 15 April 2017

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Images

Figure 1
Disorder average of the log of the stabilizer energy $ln ε$ vs stabilizer length $ℓ$ on a periodic lattice of 256 spins in the marginal MBL phase of the XYZ spin chain. This figure verifies the scaling $- ln ε \sim \sqrt{ℓ}$ . The two rows (of plots) correspond to different points (B and C) in the phase diagram Fig. 2 while the two columns correspond to different horizontal axes: $| i - j$ vs $\sqrt{| i - j |}$ . [The fully MBL point (A) is not shown since it is deep in the MBL phase where nearly all stabilizers are of very short length.] The stabilizer length is calculated by writing a stabilizer $τ_{a}^{z}$ in the physical basis and dropping all perturbative Schrieffer-Wolff corrections. The result is a product of Pauli operators at different sites. The stabilizer length is then the length of the shortest continuous sequence of sites (on the periodic lattice) that contains all of the Pauli operators. (Due to a slight even-odd effect, only odd stabilizer lengths are shown. $2^{16}$ disorder samples are used. Error bars denote one standard deviation statistical errors.)
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Figure 2
A ternary plot (copied from Ref. [43]) of the disorder and energy averaged Edwards-Anderson correlator vs coupling constants ( $0 < {\tilde{J}}_{x, y, z} < 1$ ) for the XYZ spin chain of length $L = 256$ . We use this plot to sketch the phase diagram. When ${\tilde{J}}_{z} > max ({\tilde{J}}_{x}, {\tilde{J}}_{y})$ , the system is in an MBL $Z_{2}$ spin glass state. When ${\tilde{J}}_{z} < {\tilde{J}}_{x} = {\tilde{J}}_{y}$ , the system is in a marginal MBL phase. (The other phases are given by permutations of $x, y, z$ .) The white dots correspond to the points in the phase diagram that are shown in Fig. 3.
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Figure 3
Disorder average (using a geometric mean [Eq. (14)]) of the out-of-time-order correlation (OTOC) [Eq. (6)] $exp avg ln | F (t) |$ of $W = σ_{i}^{x}$ and $V = σ_{j}^{y}$ showing how the light cone of the (geometric mean) OTOC depends on the time $t$ and distance $d_{W V} = | i - j |$ separation of $W$ and $V$ (on a lattice with 256 spins). Specifically, the light cone grows like $t_{scr} \sim exp (\sqrt{d_{W V} / l_{0}})$ . As in Fig. 1, the rows (of plots) correspond to different points in the phase diagram Fig. 2 while the two columns correspond to different horizontal axes: $| i - j$ vs $\sqrt{| i - j |}$ . Fits are shown for the cases when the scrambling time scaling agrees with the choice of horizontal axes. As can be seen from Fig. 4, the statistical errors and finite system size do not significantly affect the linear fit. ( $2^{9}$ disorder samples are used.)
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Figure 4
A $d_{W V} = | i - j | = 32$ slice of Fig. 3 for different system sizes (and the same three points of the phase diagram Fig. 2). That is, we plot the disorder average (using a geometric mean [Eq. (14)]) of the out-of-time-order correlation (OTOC) [Eq. (6)] $exp avg ln | F (t) |$ of $W = σ_{i}^{x}$ and $V = σ_{i + 32}^{y}$ for system sizes $L = 64 128 256$ . We see that the OTOC for $| i - j | = 32$ converges very quickly with system size and has essentially completely converged by $L = 128$ , which is only four times $| i - j |$ . Thus, we expect Fig. 3 (for which $L = 256$ ) to have converged for all $| i - j | \leq 256 / 4 = 64$ (or $\sqrt{| i - j |} \leq 8$ ). Even when $L = 64$ , which is only twice $| i - j |$ , the OTOC has already mostly converged. Error bars are statistical errors which are calculated using the bootstrap method [63, 64] and are small enough to not have a significant effect on our light cone measurements.
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