Out-of-equilibrium dynamics and thermalization of string order

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Out-of-equilibrium dynamics and thermalization of string order

Leonardo Mazza, Davide Rossini, Manuel Endres, and Rosario Fazio

Phys. Rev. B 90, 020301(R) – Published 11 July 2014

Abstract

We investigate the equilibration dynamics of string order in one-dimensional quantum systems. After initializing a spin-1 chain in the Haldane phase, the time evolution of nonlocal correlations following a sudden quench is studied by means of matrix-product-state-based algorithms. Thermalization occurs only for scales up to a horizon growing at a well defined speed, due to the finite maximal velocity at which string correlations can propagate, related to a Lieb-Robinson bound. The persistence of string ordering at finite times is nontrivially related to symmetries of the quenched Hamiltonian. A qualitatively similar behavior is found for the string order of the Mott insulating phase in the Bose-Hubbard chain. This paves the way towards an experimental testing of our results in present cold-atom setups.

Received 11 November 2013
Revised 3 May 2014

DOI:https://doi.org/10.1103/PhysRevB.90.020301

Authors & Affiliations

Leonardo Mazza¹, Davide Rossini¹, Manuel Endres², and Rosario Fazio¹

¹NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56126 Pisa, Italy
²Max-Planck-Institut für Quantenoptik, D-85748 Garching, Germany

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Issue

Vol. 90, Iss. 2 — 1 July 2014

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Images

Figure 1
(a) Sketch of a product state featuring the dilute antiferromagnetic order detected by ${\hat{O}}_{i, j}^{z}$ . (b)–(e) String correlations for a quench of $| Ψ_{0} 〉$ with Hamiltonian (2), $Δ = 0.2$ . (b) $〈 {\hat{O}}_{i, i + ℓ}^{x} 〉 (t)$ as a function of the length $ℓ$ . Each line refers to a different time, ranging from $t = 0$ $ℏ / J$ (dark red) to $t = 5.0$ $ℏ / J$ (bright red), with spacing $0.2$ $ℏ / J$ . The blue line represents string correlations of the thermal state ${\hat{ρ}}_{th} (β)$ , with $β = 1.6$ $J^{- 1}$ . (c) $〈 {\hat{O}}_{i, i + ℓ}^{x} 〉 (t)$ as a function of $t$ . Each line refers to a different $ℓ$ , ranging from $ℓ = 1$ (dark green) to $ℓ = 30$ (bright green), with spacing 1. (d), (e) $〈 {\hat{O}}_{i, i + ℓ}^{z} 〉 (t)$ with color coding and axes as in (b) and (c), respectively.
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Figure 2
String correlations for a quench with Hamiltonian (2), $Δ = 0.2$ . (a) Subtracted string correlations $| 〈 {\hat{O}}_{i, i + ℓ}^{x} 〉 (t) - O^{x} (t) |$ as a function of time. Color code as in Fig. 1. (b) Spreading of string correlations. Space-time dependence of $t_{F_{0}} (ℓ)$ for $F_{0} = 10^{- 4}$ (blue diamonds) and $10^{- 3}$ (red dots). Time dependence of $L_{th} (t)$ (green empty circles). (c) Subtracted string correlations $| 〈 {\hat{O}}_{i, i + ℓ}^{x} 〉 (t) - O^{x} (t) |$ as a function of length. Color code as in Fig. 1. The blue line represents string correlations of the thermal state ${\hat{ρ}}_{th} (β)$ , with $β = 1.6$ $J^{- 1}$ . (d) Illustration of the construction of $L_{th} (t)$ .
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Figure 3
String correlations $〈 O_{i, i + ℓ}^{x} 〉 (t)$ for a quench with Hamiltonian (2) for $Δ = 1.2, 1.4$ , and $1.6$ , as a function of length $ℓ$ (left panels) and time $t$ (right panels). Left panels: Each line refers to a different time, ranging from $t = 0$ $ℏ / J$ (dark red) to $t = 3.4$ $ℏ / J$ (bright red), with spacing $0.2$ $ℏ / J$ . Blue lines represent string correlations of the thermal states ${\hat{ρ}}_{th} (β)$ , with $β = 1.92$ , $1.48$ , and $1.2$ $J^{- 1}$ , respectively. Right panels: Each line refers to a different length, ranging from $ℓ = 1$ (dark green) to $ℓ = 30$ (bright green), with spacing 1.
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Figure 4
String correlations $〈 {\hat{O}}_{i, i + ℓ}^{(BH)} 〉 (t)$ as a function of $ℓ$ for a quench with Hamiltonian (5), from $U_{0} = 40 J$ to $U = 15 J$ . (a) Each line refers to a different time, ranging from $t = 0$ $ℏ / J$ (dark red) to $t = 4.2$ $ℏ / J$ (bright red) with spacing $0.1$ $ℏ / J$ . The blue line represents string correlations of the grand canonical state ${\hat{ρ}}_{gc} (β, μ)$ , with $β = 0.788 J^{- 1}$ , and $μ = 0.475 U_{Q}$ . (b) Each line refers to a different length, ranging from $ℓ = 1$ (dark green) to $ℓ = 26$ (bright green), with spacing 1. (c), (d) Mean and variance of the particle number for ${\hat{ρ}}_{gc} (β, μ)$ as a function of $μ$ , for the $β$ matching the initial state energy: $0.578$ , $0.622$ , $0.67$ , $0.724$ , and $0.786 J^{- 1}$ , respectively.
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