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arXiv:1306.4306 · Physical Review Letters

Ballistic Spreading of Entanglement in a Diffusive Nonintegrable System

HyungWon Kim, David A. Huse

Published Sep 20, 2013• 557 citations
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Abstract

We study the time evolution of the entanglement entropy of a one-dimensional nonintegrable spin chain, starting from random nonentangled initial pure states. We use exact diagonalization of a nonintegrable quantum Ising chain with transverse and longitudinal fields to obtain the exact quantum dynamics. We show that the entanglement entropy increases linearly with time before finite-size saturation begins, demonstrating a ballistic spreading of the entanglement, while the energy transport in the same system is diffusive. Thus, we explicitly demonstrate that the spreading of entanglement is much faster than the energy diffusion in this nonintegrable system.

DOI
10.1103/physrevlett.111.127205

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We study the time evolution of the entanglement entropy of a one-dimensional nonintegrable spin chain, starting from random nonentangled initial pure states. We use exact diagonalization of a nonintegrable quantum Ising chain with transverse and longitudinal fields to obtain the exact quantum dynamics. We show that the entanglement entropy increases linearly with time before finite-size saturation begins, demonstrating a ballistic spreading of the entanglement, while the energy transport in the same system is diffusive. Thus, we explicitly demonstrate that the spreading of entanglement is much faster than the energy diffusion in this nonintegrable system.
We study the unitary time evolution of the entropy of entanglement of a one-dimensional system between the degrees of freedom in an interval of length and its complement, starting from a pure state which is not an eigenstate of the Hamiltonian. We use path integral methods of quantum field theory as well as explicit computations for the transverse Ising spin chain. In both cases, there is a maximum speed v of propagation of signals. In general the entanglement entropy increases linearly with time t up to , after which it saturates at a value proportional to , the coefficient depending on the initial state. This behaviour may be understood as a consequence of causality.
The entangling power and operator entanglement entropy are state-independent measures of entanglement. Their growth and saturation is examined in the time-evolution operator of quantum many-body systems that can range from the integrable to the fully chaotic. An analytically solvable integrable model of the kicked transverse-field Ising chain is shown to have ballistic growth of operator von Neumann entanglement entropy and exponentially fast saturation of the linear entropy with time. Surprisingly, a fully chaotic model with longitudinal fields turned on shares the same growth phase, and is consistent with a random matrix model that is also exactly solvable for the linear entropy entanglements. However, an examination of the entangling power shows that its largest value is significantly less than the nearly maximal value attained by the nonintegrable one. The importance of long-range spectral correlations, and not just the nearest-neighbor spacing, is pointed out in determining the growth of entanglement in nonintegrable systems. Finally, an interesting case that displays some features peculiar to both integrable and nonintegrable systems is briefly discussed.
Entanglement propagation provides a key routine to understand quantum many-body dynamics in and out of equilibrium. Entanglement entropy (EE) usually approaches to a subsaturation known as the Page value S[over ˜]_{P}=S[over ˜]-dS (with S[over ˜] the maximum of EE and dS the Page correction) in, e.g., the random unitary evolutions. The ballistic spreading of EE usually appears in the early time and will be deviated far before the Page value is reached. In this work, we uncover that the magnetic field that maximizes the EE robustly induces persistent ballistic spreading of entanglement in quantum spin chains. The linear growth of EE is demonstrated to persist until the maximal S[over ˜] (along with a flat entanglement spectrum) is reached. The robustness of ballistic spreading and the enhancement of EE under such an optimal control are demonstrated, considering particularly perturbing the initial state by random pure states (RPSs). These are argued as the results from the endomorphism of the time evolution under such an entanglement-enhancing optimal control for the RPSs.
We investigate the dynamics of quantum entanglement after a global quench and uncover a qualitative difference between the behavior of the von Neumann entropy and higher Rényi entropies. We argue that the latter generically grow sub-ballistically, as ∝sqrt[t], in systems with diffusive transport. We provide strong evidence for this in both a U(1) symmetric random circuit model and in a paradigmatic nonintegrable spin chain, where energy is the sole conserved quantity. We interpret our results as a consequence of local quantum fluctuations in conserved densities, whose behavior is controlled by diffusion, and use the random circuit model to derive an effective description. We also discuss the late-time behavior of the second Rényi entropy and show that it exhibits hydrodynamic tails with three distinct power laws occurring for different classes of initial states.
We study three aspects of quantum nonequilibrium dynamics; (1) transport of conserved quantities, (2) entanglement spreading, and (3) construction of local operators which slowly relax to thermal equilibrium. Motivated by recent progresses in ultracold atom experiments, we first analyze transport phenomena of a population imbalanced two-component fermi gas with arbitrary strength of inter-species interaction in three dimension. Using the Boltzmann kinetic equation at dilute regime, we obtain the transport coefficients of linear responses to gradients of temperature and chemical potential imbalance. We identify the magneto-caloric effects, and determine how these effects depend on interaction strength and population imbalance. Then, we propose an experimental protocol to observe these effects in an experiment with ultracold atoms. Next, we study entanglement spreading in a one-dimensional quantum Ising chain with longitudinal and transverse fields, which is diffusive and nonintegrable. Fully diagonalizing the Hamiltonian matrix, we explicitly show that the entanglement spreading is ballistic, thus faster than diffusive transport of conserved quantities. We provide a local spreading picture of entanglement entropy in terms of logarithmic negativity. Then, we discuss a role of energy conservation in entanglement spreading through analyzing a Floquet system. Lastly, we construct local operators that relax slowly to equilibrium with the same one-dimensional Hamiltonian. By showing that the Hamiltonian satisfies the eigenstate thermalization hypothesis, we first conclude that this model relaxes local operators to thermal equilibrium. Then, we systematically construct local operators that relax slower than conventional diffusive modes.
We explore the dynamics of the entanglement entropy near equilibrium in highly entangled pure states of two quantum-chaotic spin chains undergoing unitary time evolution. We examine the relaxation to equilibrium from initial states with either less or more entanglement entropy than the equilibrium value, as well as the dynamics of the spontaneous fluctuations of the entanglement that occur in equilibrium. For the spin chain with a time-independent Hamiltonian and thus an extensive conserved energy, we find slow relaxation of the entanglement entropy near equilibration. Such slow relaxation is absent in a Floquet spin chain with a Hamiltonian that is periodic in time and thus has no local conservation law. Therefore, we argue that slow diffusive energy transport is responsible for the slow relaxation of the entanglement entropy in the Hamiltonian system.