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Mesoscopic Impurities in Generalized Hydrodynamics

Huebner, Friedrich

Published Jul 2, 2023• 0 citations
OpenAlex

Abstract

We study impurities in integrable models from the viewpoint of generalized hydrodynamics (GHD). An impurity can be thought of as a boundary condition for the GHD equation, relating the state on the left and right side. We find that in interacting models it is not possible to disentangle incoming and outgoing states, which means that it is not possible to think of scattering as a mapping which maps the incoming state to the outgoing state. We then introduce a novel class of impurities, dubbed mesoscopic impurities, whose spatial size is mesoscopic (i.e.\ their size $L_{\mathrm{micro}} \ll L_{\mathrm{imp}} \ll L$ is much larger than the microscopic length scale $L_{\mathrm{micro}}$, but much smaller than the system size $L$). Due to their large size it is possible to describe mesoscopic impurities via GHD. This simplification allows one to study these impurities both analytically and numerically. These impurities show interesting non-perturbative scattering behavior, for instance non-uniqueness of solutions and a non-analytic dependence on the impurity strength. In models with one quasi-particle species and a scattering phase shift that depends on the difference of momenta only, we find that one can describe the scattering using an effective Hamiltonian. This Hamiltonian is dressed due to the interaction between particles and satisfies a self consistency fixed point equation. On the example of the hard rods model we demonstrate how this fixed point equation can be used to find explicit solutions to the scattering problem.

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Mesoscopic impurities in generalized hydrodynamics

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We study impurities in integrable models from the viewpoint of generalized hydrodynamics (GHD). An impurity can be thought of as a boundary condition for the GHD equation, relating the state on the left and right side. We find that in interacting models it is not possible to disentangle incoming and outgoing states, which means that it is not possible to think of scattering as a mapping which maps the incoming state to the outgoing state. We then introduce a novel class of impurities, dubbed mesoscopic impurities, whose spatial size is mesoscopic (i.e.\ their size Lmicro≪Limp≪LL_{\mathrm{micro}} \ll L_{\mathrm{imp}} \ll LLmicro​≪Limp​≪L is much larger than the microscopic length scale LmicroL_{\mathrm{micro}}Lmicro​, but much smaller than the system size LLL). Due to their large size it is possible to describe mesoscopic impurities via GHD. This simplification allows one to study these impurities both analytically and numerically. These impurities show interesting non-perturbative scattering behavior, for instance non-uniqueness of solutions and a non-analytic dependence on the impurity strength. In models with one quasi-particle species and a scattering phase shift that depends on the difference of momenta only, we find that one can describe the scattering using an effective Hamiltonian. This Hamiltonian is dressed due to the interaction between particles and satisfies a self consistency fixed point equation. On the example of the hard rods model we demonstrate how this fixed point equation can be used to find explicit solutions to the scattering problem.
Abstract This study focuses on the impurities in integrable models from the viewpoint of generalized hydrodynamics (GHD). An impurity can be thought of as a boundary condition for the GHD equation, relating the states on the left and right sides. It was found that, in interacting models, it is not possible to disentangle the incoming and outgoing states, which means that it is not possible to think of scattering as a mapping that maps the incoming state to the outgoing state. A novel class of impurities, dubbed mesoscopic impurities, was then introduced, whose spatial size is mesoscopic (i.e. their size L m i c r o ≪ L i m p ≪ L is much larger than the microscopic length scale L m i c r o but much smaller than the macroscopic scale L). Because of their large size, it is possible to describe mesoscopic impurities via GHD. This simplification allows one to study these impurities both analytically and numerically. These impurities show interesting non-perturbative scattering behavior, such as the nonuniqueness of the solutions and a nonanalytic dependence on the impurity strength. In models with one quasi-particle species and a scattering phase shift that depends only on the difference in momenta, the scattering can be described using an effective Hamiltonian. This Hamiltonian is dressed due to the interaction between the particles and satisfies a self-consistency fixed-point equation. In the example of the hard-rod model, it was demonstrated how this fixed-point equation can be used to find almost explicit solutions to the scattering problem by reducing it to a two-dimensional system of equations that can be solved numerically.
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.
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.
The generalized hydrodynamics (GHD) equation is the equivalent of the Euler equations of hydrodynamics for integrable models. Systems of hyperbolic equations such as the Euler equations usually develop shocks and are plagued by problems of uniqueness. We establish for the first time the existence and uniqueness of solutions to the full GHD equation and the absence of shocks, from a large class of initial conditions with bounded occupation function. We assume only absolute integrability of the two-body scattering shift. In applications to quantum models of fermionic type, this includes all commonly used physical initial states, such as locally thermal states and zero-entropy states. We show in particular that differentiable initial conditions give differentiable solutions at all times and that weak initial conditions such as the Riemann problem have unique weak solutions which preserve entropy. For this purpose, we write the GHD equation as a new fixed-point problem (announced in a companion paper). We show that the fixed point exists, is unique, and is approached, under an iterative solution procedure, in the Banach topology on functions of momenta.
Physical systems made of many interacting quantum particles can often be described by Euler hydrodynamic equations in the limit of long wavelengths and low frequencies. Recently such a classical hydrodynamic framework, now dubbed generalized hydrodynamics (GHD), was found for quantum integrable models in one spatial dimension. Despite its great predictive power, GHD, like any Euler hydrodynamic equation, misses important quantum effects, such as quantum fluctuations leading to nonzero equal-time correlations between fluid cells at different positions. Focusing on the one-dimensional gas of bosons with delta repulsion, and on states of zero entropy, for which quantum fluctuations are larger, we reconstruct such quantum effects by quantizing GHD. The resulting theory of quantum GHD can be viewed as a multicomponent Luttinger liquid theory, with a small set of effective parameters that are fixed by the thermodynamic Bethe ansatz. It describes quantum fluctuations of truly nonequilibrium systems where conventional Luttinger liquid theory fails.
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.