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.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.

arXiv:1704.04409Nuclear Physics BDec 6, 2017172 citations

Integrable Many-Particle Systems

V. I. Inozemtsev

WORLD SCIENTIFIC (EUROPE) eBooksSep 27, 20221 citation

Quantum integrable systems

Miki Wadati, Taro Nagao, Kazuhiro Hikami

Physica D Nonlinear PhenomenaSep 1, 199310 citations

Functional Integral approach to quasi-particles in Fermi Liquid theory

Tai Kai Ng

In this paper we propose a new way of organizing many-body perturbation theory in the Path-integral formulation where a set of quasi-particle wave-functionals $ψ$ 's are introduced and are identified with quasi-particles in Landau Fermi liquid theory. We show how Fermi liquid theory can be obtained through $ψ$ 's in this new framework of perturbation theory, where the only assumption is adiabaticity between non-interacting and interacting states and the quasi-particle renormalization factor z does not appear explicitly. Consequences of our new formulation are discussed.In this paper we propose a new way of organizing many-body perturbation theory in the Path-integral formulation where a set of quasi-particle wave-functionals $ψ$'s are introduced and are identified with quasi-particles in Landau Fermi liquid theory. We show how Fermi liquid theory can be obtained through $ψ$'s in this new framework of perturbation theory, where the only assumption is adiabaticity between non-interacting and interacting states and the quasi-particle renormalization factor z does not appear explicitly. Consequences of our new formulation are discussed.

arXiv:cond-mat/9706033Jun 4, 19971 citation

Integrable systems for particles with internal degrees of freedom

Joseph A. Minahan, Alexios P. Polychronakos

arXiv:hep-th/9206046Physics Letters BMar 1, 1993160 citations

Thermalization of dipole oscillations in confined systems by rare collisions

Maxim Khodas, Alex Levchenko

We study the relaxation of the center of mass or dipole oscillations in the system of interacting fermions confined spatially. With the confinement frequency ${\ensuremath{\omega}}_{\ensuremath{\perp}}$ fixed, the particles were considered to freely move along one (quasi-1D) or two (quasi-2D) spatial dimensions. We have focused on the regime of rare collisions, such that the inelastic collision rate $1/{\ensuremath{\tau}}_{\mathrm{in}}\ensuremath{\ll}{\ensuremath{\omega}}_{\ensuremath{\perp}}$. The dipole oscillation relaxation rate $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}$ is obtained at three different levels: by direct perturbation theory, solving the integral Bethe-Salpeter equation, and applying the memory function formalism. As long as anharmonicity is weak, $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}\ensuremath{\ll}1/{\ensuremath{\tau}}_{\mathrm{in}}$, the three methods are shown to give identical results. In the quasi-2D case, $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}\ensuremath{\ne}0$ at zero temperature. In quasi-1D system, $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}\ensuremath{\propto}{T}^{3}$ if the Fermi energy ${E}_{F}$ lies below the critical value ${E}_{F}<3{\ensuremath{\omega}}_{\ensuremath{\perp}}/4$. Otherwise, unless the system is close to integrability, the rate $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}$ has a temperature dependence similar to that in quasi-2D. In all cases, the relaxation results from the excitation of particle-hole pairs propagating along unconfined directions resulting in the relationship $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}\ensuremath{\propto}1/{\ensuremath{\tau}}_{\mathrm{in}}$, with the inelastic rate $1/{\ensuremath{\tau}}_{\mathrm{in}}\ensuremath{\ne}0$ as the phase-space opens up at finite energy of excitation ${\ensuremath{\omega}}_{\ensuremath{\perp}}$. While $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}\ensuremath{\propto}{\ensuremath{\tau}}_{\mathrm{in}}$ in the hydrodynamic regime, ${\ensuremath{\omega}}_{\ensuremath{\perp}}\ensuremath{\ll}1/{\ensuremath{\tau}}_{\mathrm{in}}$, in the regime of rare collisions, ${\ensuremath{\omega}}_{\ensuremath{\perp}}\ensuremath{\gg}1/{\ensuremath{\tau}}_{\mathrm{in}}$, we obtain the opposite trend $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}\ensuremath{\propto}1/{\ensuremath{\tau}}_{\mathrm{in}}$.We study the relaxation of the center of mass or dipole oscillations in the system of interacting fermions confined spatially. With the confinement frequency ${\ensuremath{\omega}}_{\ensuremath{\perp}}$ fixed, the particles were considered to freely move along one (quasi-1D) or two (quasi-2D) spatial dimensions. We have focused on the regime of rare collisions, such that the inelastic collision rate $1/{\ensuremath{\tau}}_{\mathrm{in}}\ensuremath{\ll}{\ensuremath{\omega}}_{\ensuremath{\perp}}$. The dipole oscillation relaxation rate $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}$ is obtained at three different levels: by direct perturbation theory, solving the integral Bethe-Salpeter equation, and applying the memory function formalism. As long as anharmonicity is weak, $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}\ensuremath{\ll}1/{\ensuremath{\tau}}_{\mathrm{in}}$, the three methods are shown to give identical results. In the quasi-2D case, $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}\ensuremath{\ne}0$ at zero temperature. In quasi-1D system, $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}\ensuremath{\propto}{T}^{3}$ if the Fermi energy ${E}_{F}$ lies below the critical value ${E}_{F}<3{\ensuremath{\omega}}_{\ensuremath{\perp}}/4$. Otherwise, unless the system is close to integrability, the rate $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}$ has a temperature dependence similar to that in quasi-2D. In all cases, the relaxation results from the excitation of particle-hole pairs propagating along unconfined directions resulting in the relationship $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}\ensuremath{\propto}1/{\ensuremath{\tau}}_{\mathrm{in}}$, with the inelastic rate $1/{\ensuremath{\tau}}_{\mathrm{in}}\ensuremath{\ne}0$ as the phase-space opens up at finite energy of excitation ${\ensuremath{\omega}}_{\ensuremath{\perp}}$. While $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}\ensuremath{\propto}{\ensuremath{\tau}}_{\mathrm{in}}$ in the hydrodynamic regime, ${\ensuremath{\omega}}_{\ensuremath{\perp}}\ensuremath{\ll}1/{\ensuremath{\tau}}_{\mathrm{in}}$, in the regime of rare collisions, ${\ensuremath{\omega}}_{\ensuremath{\perp}}\ensuremath{\gg}1/{\ensuremath{\tau}}_{\mathrm{in}}$, we obtain the opposite trend $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}\ensuremath{\propto}1/{\ensuremath{\tau}}_{\mathrm{in}}$.

arXiv:1802.05161Physical review. B./Physical review. BAug 13, 20180 citations

In this paper we propose a new way of organizing many-body perturbation theory in the Path-integral formulation where a set of quasi-particle wave-functionals

ψ

's are introduced and are identified with quasi-particles in Landau Fermi liquid theory. We show how Fermi liquid theory can be obtained through

ψ

's in this new framework of perturbation theory, where the only assumption is adiabaticity between non-interacting and interacting states and the quasi-particle renormalization factor z does not appear explicitly. Consequences of our new formulation are discussed.

{E}_{F}

lies below the critical value ${E}_{F}<3{\ensuremath{\omega}}_{\ensuremath{\perp}}/4$. Otherwise, unless the system is close to integrability, the rate $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}$ has a temperature dependence similar to that in quasi-2D. In all cases, the relaxation results from the excitation of particle-hole pairs propagating along unconfined directions resulting in the relationship $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}\ensuremath{\propto}1/{\ensuremath{\tau}}_{\mathrm{in}}$, with the inelastic rate $1/{\ensuremath{\tau}}_{\mathrm{in}}\ensuremath{\ne}0$ as the phase-space opens up at finite energy of excitation ${\ensuremath{\omega}}_{\ensuremath{\perp}}$. While $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}\ensuremath{\propto}{\ensuremath{\tau}}_{\mathrm{in}}$ in the hydrodynamic regime, ${\ensuremath{\omega}}_{\ensuremath{\perp}}\ensuremath{\ll}1/{\ensuremath{\tau}}_{\mathrm{in}}$, in the regime of rare collisions, ${\ensuremath{\omega}}_{\ensuremath{\perp}}\ensuremath{\gg}1/{\ensuremath{\tau}}_{\mathrm{in}}$, we obtain the opposite trend $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}\ensuremath{\propto}1/{\ensuremath{\tau}}_{\mathrm{in}}$.