Entanglement dynamics in confining spin chains

Stefano Scopa, Pasquale Calabrese, and Alvise Bastianello

Phys. Rev. B 105, 125413 – Published 16 March 2022

Abstract

The confinement of elementary excitations induces distinctive features in the non-equilibrium quench dynamics. One of the most remarkable is the suppression of entanglement entropy, which in several instances turns out to oscillate rather than grow indefinitely. While the qualitative physical origin of this behavior is clear, till now no quantitative understanding away from the field theory limit was available. Here we investigate this problem in the weak quench limit, when mesons are excited at rest, hindering entropy growth and exhibiting persistent oscillations. We provide analytical predictions of the entire entanglement dynamics based on a Gaussian approximation of the many-body state, which captures numerical data with great accuracy and is further simplified to a semiclassical quasiparticle picture in the regime of weak confinement. Our methods are valid in general and we apply explicitly to two prototypical models: the Ising chain in a tilted field and the experimentally relevant long-range Ising model.

Received 2 December 2021
Revised 4 March 2022
Accepted 8 March 2022

DOI:https://doi.org/10.1103/PhysRevB.105.125413

Physics Subject Headings (PhySH)

Confinement Entanglement entropy

Nonequilibrium systems

Ising model

Statistical Physics & Thermodynamics

Authors & Affiliations

Stefano Scopa ¹, Pasquale Calabrese^1,2, and Alvise Bastianello ^3,4,5

¹SISSA and INFN, via Bonomea 265, 34136 Trieste, Italy
²International Centre for Theoretical Physics (ICTP), I-34151, Trieste, Italy
³Department of Physics, Technical University of Munich, 85748 Garching, Germany
⁴Institute for Advanced Study, 85748 Garching, Germany
⁵Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 München, Germany

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Issue

Vol. 105, Iss. 12 — 15 March 2022

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Images

Figure 1
Illustration of the quasiparticle picture in the confined model: Small quenches mostly excite pairs of domain walls with zero total momentum, which are then confined into mesons with zero velocity. In the dilute regime, mesons behave as stationary extended particles with semiclassical trajectory $d_{t} (k)$ in Eq. (5). The entanglement entropy measures the shared information among the two subsystems $A \cup B$ mediated by the mesons sat at the boundaries between the two regions.
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Figure 2
von Neumann entropy dynamics of the confined Ising model (1). We show different quenches of the transverse field on different columns and different values of $h_{∥}$ on different rows. In each panel, we compare the semiclassical prediction (6) (dashed line), the quantum prediction in Gaussian approximation (full line), and numerical data obtained with time-dependent DMRG simulations (symbols). Notice the validity of the approach also for quenches starting from the false vacuum (g)–(h).
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Figure 3
(Left) Rényi-2 and (right) von Neumann entropy dynamics for the long-range Ising model (9) with different exponent $α$ on different rows. The system is initially prepared in a ferromagnetic state with all spins up. The quantum prediction in Gaussian approximation (full line) is compared with numerical data (dashed line) obtained with global subspace expansion time-dependant variational principle [55].
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Figure 4
Comparison of the tensor network simulations (symbols) with the single-meson prediction obtained for two different initial wave function: (i) the wave function $K (k) = - i tan (θ_{k}^{{\bar{h}}_{⊥}} - θ_{k}^{h_{⊥}})$ for the transverse field quench with no additional corrections (dash-dot line); (ii) the wave function $K (k) = - i tan (θ_{k}^{{\bar{h}}_{⊥}} - θ_{k}^{h_{⊥}}) - i h_{∥} \bar{σ} v (k) / ε^{2} (k)$ in Eq. (4), obtained with a renormalized initial state (full line). The figure clearly shows that only a properly renormalized initial state $| {\tilde{ψ}}_{0} 〉$ gives a good description of the entanglement dynamics.
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Figure 5
(a) Large quench in the transverse field: ${\bar{h}}_{⊥} = 0 \to h_{⊥} = 0.5$ , at $h_{∥} = 0.1$ . Inset: exact density of quasiparticles $n_{k}$ (full line) significantly deviates from its approximation (dashed-dot line) in the dilute regime. In this regime, both analytical predictions for the Von Neumann entropy (quasiparticle picture–dashed line; quantum Gaussian treatment–full line) fail to predict the observed numerical behavior (symbols) at finite times. (b) Example of Schwinger mechanism: ${\bar{h}}_{⊥} = 0.5 \to h_{⊥} = 0.25$ and strong confinement $h_{∥} = 0.4$ . The analytical predictions (quasiparticle picture–dashed line; quantum Gaussian treatment–thick full line) significantly deviates from the numerical data (thin full line). The difference $δ S_{A}$ between the numerical data and the quantum prediction clearly shows a linear growth of entanglement due to the spontaneous creation/annihilation of new particles.
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Figure 6
(a) Single spin flip energy profile of a finite-size chain with $N = 150$ sites for different values of $α$ . For $α ≲ 2$ , the profile significantly deviates from a flat one and leads to a non-negligible boundary-meson interaction term. In each of the curves, we subtract the max value of the energy for a better comparison. (b) Rényi-2 entropy dynamics for the long-range Ising model [see Eq. (9)] with exponent $α \leq 2$ obtained with variational methods (dashed line). The emergence of meson-meson and meson-boundary interactions produces a linear drift, which is not captured by the quantum prediction in single-meson approximation (full line). In each of the right panels, we set the transverse field $h_{⊥}$ such that $h_{⊥} / U (1) ≪ 1$ .
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