These are lecture notes for a series of lectures given at the Les Houches Summer School on Integrability in Atomic and Condensed Matter Physics, 30 July to 24 August 2018. The same series of lectures ...has also been given at the Tokyo Institute of Technology, October 2019. I overview in a pedagogical fashion the main aspects of the theory of generalised hydrodynamics, a hydrodynamic theory for quantum and classical many-body integrable systems. Only very basic knowledge of hydrodynamics and integrable systems is assumed.
Using the theory of generalized hydrodynamics (GHD), we derive exact Euler-scale dynamical two-point correlation functions of conserved densities and currents in inhomogeneous, non-stationary states ...of many-body integrable systems with weak space-time variations. This extends previous works to inhomogeneous and non-stationary situations. Using GHD projection operators, we further derive formulae for Euler-scale two-point functions of arbitrary local fields, purely from the data of their homogeneous one-point functions. These are new also in homogeneous generalized Gibbs ensembles. The technique is based on combining a fluctuation-dissipation principle along with the exact solution by characteristics of GHD, and gives a recursive procedure able to generate $n$-point correlation functions. Owing to the universality of GHD, the results are expected to apply to quantum and classical integrable field theory such as the sinh-Gordon model and the Lieb-Liniger model, spin chains such as the XXZ and Hubbard models, and solvable classical gases such as the hard rod gas and soliton gases. In particular, we find Leclair-Mussardo-type infinite form-factor series in integrable quantum field theory, and exact Euler-scale two-point functions of exponential fields in the sinh-Gordon model and of powers of the density field in the Lieb-Liniger model. We also analyze correlations in the partitioning protocol, extract large-time asymptotics, and, in free models, derive all Euler-scale $n$-point functions.
Let an infinite, homogeneous, many-body quantum system be unitarily evolved for a long time from a state where two halves are independently thermalized. One says that a non-equilibrium steady state ...emerges if there are nonzero steady currents in the central region. In particular, their presence is a signature of ballistic transport. We analyze the consequences of the current observable being a conserved density; near equilibrium this is known to give rise to linear wave propagation and a nonzero Drude peak. Using the Lieb–Robinson bound, we derive, under a certain regularity condition, a lower bound for the non-equilibrium steady-state current determined by equilibrium averages. This shows and quantifies the presence of ballistic transport far from equilibrium. The inequality suggests the definition of “nonlinear sound velocities”, which specialize to the sound velocity near equilibrium in non-integrable models, and “generalized sound velocities”, which encode generalized Gibbs thermalization in integrable models. These are bounded by the Lieb–Robinson velocity. The inequality also gives rise to a bound on the energy current noise in the case of pure energy transport. We show that the inequality is satisfied in many models where exact results are available, and that it is saturated at one-dimensional criticality.
Based on the method of hydrodynamic projections we derive a concise
formula for the Drude weight of the repulsive Lieb-Liniger
\delta
δ
-Bose
gas. Our formula contains only quantities which are ...obtainable from the
thermodynamic Bethe ansatz. The Drude weight is an infinite-dimensional
matrix, or bilinear functional: it is bilinear in the currents, and each
current may refer to a general linear combination of the conserved
charges of the model. As a by-product we obtain the dynamical two-point
correlation functions involving charge and current densities at small
wavelengths and long times, and in addition the scaled covariance matrix
of charge transfer. We expect that our formulas extend to other
integrable quantum models.
Generalized hydrodynamics (GHD) was proposed recently as a
formulation of hydrodynamics for integrable systems, taking into account
infinitely-many conservation laws. In this note we further develop ...the
theory in various directions. By extending GHD to all commuting flows of
the integrable model, we provide a full description of how to take into
account weakly varying force fields, temperature fields and other
inhomogeneous external fields within GHD. We expect this can be used,
for instance, to characterize the non-equilibrium dynamics of
one-dimensional Bose gases in trap potentials. We further show how the
equations of state at the core of GHD follow from the continuity
relation for entropy, and we show how to recover Euler-like equations
and discuss possible viscosity terms.
A geometric viewpoint on generalized hydrodynamics Doyon, Benjamin; Spohn, Herbert; Yoshimura, Takato
Nuclear physics. B,
January 2018, 2018-01-00, 2018-01-01, Letnik:
926, Številka:
C
Journal Article
Recenzirano
Odprti dostop
Generalized hydrodynamics (GHD) is a large-scale theory for the dynamics of many-body integrable systems. It consists of an infinite set of conservation laws for quasi-particles traveling with ...effective (“dressed”) velocities that depend on the local state. We show that these equations can be recast into a geometric dynamical problem. They are conservation equations with state-independent quasi-particle velocities, in a space equipped with a family of metrics, parametrized by the quasi-particles' type and speed, that depend on the local state. In the classical hard rod or soliton gas picture, these metrics measure the free length of space as perceived by quasi-particles; in the quantum picture, they weigh space with the density of states available to them. Using this geometric construction, we find a general solution to the initial value problem of GHD, in terms of a set of integral equations where time appears explicitly. These integral equations are solvable by iteration and provide an extremely efficient solution algorithm for GHD.
We study the dynamics of the entanglement in one-dimensional critical quantum systems after a local quench in which two independently thermalized semi-infinite halves are joined to form a homogeneous ...infinite system and left to evolve unitarily. We show that under certain conditions a nonequilibrium steady state (NESS) is reached instantaneously as soon as the entanglement interval is within the light cone emanating from the contact point. In this steady state, the exact expressions for the entanglement entropy and the logarithmic negativity are in agreement with the steady state density matrix being a boosted thermal state, as expected. We derive various general identities: relating the negativity after the quench with unequal left and right initial temperatures with that where the left and right temperatures are equal; and relating these with the negativity in equilibrium thermal states. In certain regimes the resulting expressions can be analytically evaluated. Immediately after the interval intersects the light cone, we find logarithmic growth. For a very long interval, we find that the negativity approaches a plateau after sufficiently long times, different from its NESS value. The NESS value is reached instantly as soon as the entire interval is contained in the light cone. This provides a theoretical framework explaining recently obtained numerical results.
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 space-time 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 space-time 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.
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