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arXiv:1706.00020 · Physical review. B./Physical review. B

Entanglement and quantum transport in integrable systems

Vincenzo Alba

Published Jun 22, 2018• 106 citations
arXivHTMLPDF

Abstract

Understanding the entanglement structure of out-of-equilibrium many-body systems is a challenging, yet revealing task. Here, the author investigates the entanglement dynamics after a quench from a piecewise homogeneous state. This is the prototypical setup for studying quantum transport. By using a generalized hydrodynamic approach for integrable models and the quasiparticle picture for the entanglement dynamics, he conjectures a formula for the entanglement production rate after joining two semi-infinite homogeneous states, as well as for the steady-state entanglement entropy of a finite region. Interestingly, both quantities are determined by the physics, describing the interface between the two reservoirs.

DOI
10.1103/physrevb.97.245135

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Understanding the entanglement structure of out-of-equilibrium many-body systems is a challenging, yet revealing task. Here, the author investigates the entanglement dynamics after a quench from a piecewise homogeneous state. This is the prototypical setup for studying quantum transport. By using a generalized hydrodynamic approach for integrable models and the quasiparticle picture for the entanglement dynamics, he conjectures a formula for the entanglement production rate after joining two semi-infinite homogeneous states, as well as for the steady-state entanglement entropy of a finite region. Interestingly, both quantities are determined by the physics, describing the interface between the two reservoirs.
We investigate the dynamics of bipartite entanglement after the sudden junction of two leads in interacting integrable models. By combining the quasiparticle picture for the entanglement spreading with Generalised Hydrodynamics we derive an analytical prediction for the dynamics of the entanglement entropy between a finite subsystem and the rest. We find that the entanglement rate between the two leads depends only on the physics at the interface and differs from the rate of exchange of thermodynamic entropy. This contrasts with the behaviour in free or homogeneous interacting integrable systems, where the two rates coincide.
Abstract The large-scale properties of homogeneous states after quantum quenches in integrable systems have been successfully described by a semiclassical picture of moving quasiparticles. Here we consider the generalisation for the entanglement evolution after an inhomogeneous quench in noninteracting systems in the framework of generalised hydrodynamics. We focus on the protocol where two semi-infinite halves are initially prepared in different states and then joined together, showing that a proper generalisation of the quasiparticle picture leads to exact quantitative predictions. If the system is initially prepared in a quasistationary state, we find that the entanglement entropy is additive and it can be computed by means of generalised hydrodynamics. Conversely, additivity is lost when the initial state is not quasistationary; yet the entanglement entropy in the large-scale limit can be exactly predicted in the quasiparticle picture, provided that the initial state is low entangled.
Understanding the general principles underlying strongly interacting quantum states out of equilibrium is one of the most important tasks of current theoretical physics.With experiments accessing the intricate dynamics of many-body quantum systems, it is paramount to develop powerful methods that encode the emergent physics.Up to now, the strong dichotomy observed between integrable and nonintegrable evolutions made an overarching theory difficult to build, especially for transport phenomena where spacetime profiles are drastically different.We present a novel framework for studying transport in integrable systems: hydrodynamics with infinitely many conservation laws.This bridges the conceptual gap between integrable and nonintegrable quantum dynamics, and gives powerful tools for accurate studies of spacetime profiles.We apply it to the description of energy transport between heat baths, and provide a full description of the current-carrying nonequilibrium steady state and the transition regions in a family of models including the Lieb-Liniger model of interacting Bose gases, realized in experiments.
Entanglement and entropy are key concepts standing at the foundations of quantum and statistical mechanics. Recently, the study of quantum quenches revealed that these concepts are intricately intertwined. Although the unitary time evolution ensuing from a pure state maintains the system at zero entropy, local properties at long times are captured by a statistical ensemble with nonzero thermodynamic entropy, which is the entanglement accumulated during the dynamics. Therefore, understanding the entanglement evolution unveils how thermodynamics emerges in isolated systems. Alas, an exact computation of the entanglement dynamics was available so far only for noninteracting systems, whereas it was deemed unfeasible for interacting ones. Here, we show that the standard quasiparticle picture of the entanglement evolution, complemented with integrability-based knowledge of the steady state and its excitations, leads to a complete understanding of the entanglement dynamics in the space-time scaling limit. We thoroughly check our result for the paradigmatic Heisenberg chain.
These are lecture notes for a short course given at the Les Houches Summer School on ``Integrability in Atomic and Condensed Matter Physics'', in summer 2018. Here, I pedagogically discuss recent advances in the study of the entanglement spreading during the non-equilibrium dynamics of isolated integrable quantum systems. I first introduce the idea that the stationary thermodynamic entropy is the entanglement accumulated during the non-equilibrium dynamics and then join such an idea with the quasiparticle picture for the entanglement spreading to provide quantitive predictions for the time evolution of the entanglement entropy in arbitrary integrable models, regardless of the interaction strength.
Quantum impurity models (QIMs) are ubiquitous throughout physics. As simplified toy models they provide crucial insights for understanding more complicated strongly correlated systems, while in their own right are accurate descriptions of many experimental platforms. In equilibrium, their physics is well understood and have proven a testing ground for many powerful theoretical tools, both numerical and analytical, in use today. Their nonequilibrium physics is much less studied and understood. However, the recent advancements in nonequilibrium integrable quantum systems through the development of generalized hydrodynamics (GHD) coupled with the fact that many archetypal QIMs are in fact integrable presents an enticing opportunity to enhance our understanding of these systems. We take a step towards this by expanding the framework of GHD to incorporate integrable interacting QIMs. We present a set of Bethe-Boltzmann type equations which incorporate the effects of impurity scattering and discuss the new aspects which include entropy production. These impurity GHD equations are then used to study a bipartioning quench wherein a relevant backscattering impurity is included at the location of the bipartition. The density and current profiles are studied as a function of the impurity strength and expressions for the entanglement entropy and full counting statistics are derived.