Universal scaling laws for correlation spreading in quantum systems with short- and long-range interactions

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Universal scaling laws for correlation spreading in quantum systems with short- and long-range interactions

Lorenzo Cevolani, Julien Despres, Giuseppe Carleo, Luca Tagliacozzo, and Laurent Sanchez-Palencia

Phys. Rev. B 98, 024302 – Published 5 July 2018

Abstract

The spreading of correlations after a quantum quench is studied in a wide class of lattice systems, with short- and long-range interactions. Using a unifying quasiparticle framework, we unveil a rich structure of the correlation cone, which encodes the footprints of several microscopic properties of the system. When the quasiparticle excitations propagate with a bounded group velocity, we show that the correlation edge and correlation maxima move with different velocities that we derive. For systems with a divergent group velocity, especially relevant for long-range interacting systems, the correlation edge propagates slower than ballistic. In contrast, the correlation maxima propagate faster than ballistic in gapless systems but ballistic in gapped systems. Our results shed light on existing experimental and numerical observations and pave the way to the next generation of experiments. For instance, we argue that the dynamics of correlation maxima can be used as a witness of the elementary excitations of the system.

Received 8 June 2017
Revised 20 June 2018

DOI:https://doi.org/10.1103/PhysRevB.98.024302

Physics Subject Headings (PhySH)

Long-range interactions Quantum correlations in quantum information Quantum entanglement

Many-body techniques

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

Authors & Affiliations

Lorenzo Cevolani¹, Julien Despres², Giuseppe Carleo^3,4, Luca Tagliacozzo⁵, and Laurent Sanchez-Palencia²

¹Institut für Theoretische Physik, Georg-August-Universität Göttingen, 37077 Göttingen, Germany
²Centre de Physique Théorique, Ecole Polytechnique, CNRS, Université Paris-Saclay, F-91128 Palaiseau, France
³Institute for Theoretical Physics, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093 Zurich, Switzerland
⁴Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York, New York 10010, USA
⁵Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, United Kingdom

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Issue

Vol. 98, Iss. 2 — 1 July 2018

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Images

Figure 1
Top: Spreading of the connected one-body correlation function $G (R, t) = 〈 {\hat{a}}_{R}^{†} (t) {\hat{a}}_{0} (t) 〉 - 〈 {\hat{a}}_{R}^{†} (0) {\hat{a}}_{0} (0) 〉$ for the 1D Bose-Hubbard model. (a) Superfluid phase for a quench from the initial value $U_{i} n = J$ to the final value $U_{f} n = 0.5 J$ . (b) Mott-insulator phase with $n = 1$ for a quench from $U_{i} = \infty$ to $U_{f} = 18 J$ . The solid green and dashed blue lines indicate ballistic spreading at twice the maximum group velocity $2 V_{g}^{★}$ and twice the corresponding phase velocity $2 V_{φ}^{★}$ , respectively. Bottom: Comparison between the maximum group velocity $V_{g}^{★}$ (solid green line), the corresponding phase velocity $V_{φ}^{★}$ (dashed blue line), the sound velocity $c$ (solid purple line), and fits to the LR cone velocity $V_{CE}$ (green diamonds) and to the velocity of the maxima $V_{m}$ (blue disks) for the (c) superfluid and (d) Mott insulator phases with the same initial values as for (a) and (b).
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Figure 2
Spreading of the connected spin correlation function $G (R, t) = G_{0} (R, t) - G_{0} (R, 0)$ for the following 1D models: (a) LRXY model with $α = 2.3$ , $G_{0} (R, t) = 〈 S_{R}^{z} (t) S_{0}^{z} (t) 〉 - 〈 S_{R}^{z} (t) 〉 〈 S_{0}^{z} (t) 〉$ for a quench from the ground state of the XXZ model ( $ε = 0.2$ ; see Appendix pp2) and (b) LRTI model with $α = 1.7$ , $G_{0} (R, t) = 〈 S_{R}^{x} (t) S_{0}^{x} (t) 〉 - 〈 S_{R}^{x} (t) 〉 〈 S_{0}^{x} (t) 〉$ for the quench in the polarized phase from $J_{i} / h = 0.02$ to $J_{f} / h = 1$ (see Ref. [52]). They feature a double algebraic structure (straight lines in log-log scale): a subballistic correlation edge (solid green line) and superballistic or ballistic spreading of local maxima (dashed blue lines). The white dotted line indicates ballistic spreading for reference. The light blue dashed lines are guides to the eye.
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Figure 3
Spreading of the connected spin-spin correlation function for the LRXY model with $α = 2.3$ [same data as in Fig. 2 of the main text; log-log scale]. The solid blue points correspond for each distance $R$ to the first time where the correlation reaches $2 %$ of its maximum value. The solid green line shows the power law predicted theoretically with a fitted multiplicative factor.
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Figure 4
Trajectory of the first extremum of the connected spin-spin correlation function for the LRXY model (lin-lin scale). The numerical results found from Fig. 3 (solid red line) are shown together with a fitted power law (dashed blue line).
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Figure 5
Spreading of the connected spin-spin correlation function for the LRTI model with $α = 1.7$ [same data as in Fig. 2 of the main text; log-log scale]. The solid blue points correspond for each distance $R$ to the first time where the correlation reaches $2.8 %$ of its maximum value. The solid green line shows the power law predicted theoretically with a fitted multiplicative factor.
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Figure 6
Trajectory of an extremum of the connected spin-spin correlation function for the LRTI model (lin-lin scale). It corresponds to the dashed dark blue line in Fig. 2 of the main text. The numerical results found from Fig. 5 (solid red line) are shown together with a fitted power law (dashed blue line).
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