Kinetic Theory for Interacting Luttinger Liquids

TL;DR

This paper develops a kinetic theory framework for interacting Luttinger Liquids with cubic resonant interactions, enabling analysis of their non-equilibrium dynamics and thermalization processes.

Contribution

It derives a closed set of kinetic equations for non-equilibrium Luttinger Liquids, incorporating phonon dressing and lifetime, and applies field theory methods to analyze relaxation and thermalization.

Findings

Phonons remain well-defined quasi-particles with long lifetimes.

The theory recovers known results in low-density limits, including Andreev's scaling.

Thermalization exhibits exponential initial relaxation followed by algebraic decay.

Abstract

We derive a closed set of equations for the kinetics and non-equilibrium dynamics of interacting Luttinger Liquids with cubic resonant interactions. In the presence of these interactions, the Luttinger phonons become dressed but still well defined quasi-particles, characterized by a life-time much larger then the inverse energy. This enables the separation of forward time dynamics and relative time dynamics into slow and fast dynamics and justifies the so-called Wigner approximation, which can be seen as a "local-time approximation" for the relative dynamics. Applying field theoretical methods in the Keldysh framework, i.e. kinetic and Dyson-Schwinger equations, we derive a closed set of dynamic equations, describing the kinetics of normal and anomalous phonon densities, the phonon self-energy and vertex corrections for an arbitrary non-equilibrium initial state. In the limit of low phonon densities, the results from self-consistent Born approximation are recaptured, including Andreev's scaling solution for the quasi-particle life-time in a thermal state. As an application, we compute the relaxation of an excited state to its thermal equilibrium. \red{ While the intermediate time dynamics displays exponentially fast relaxation, the last stages of thermalization are governed by algebraic laws. This can be traced back to the importance of energy and momentum conservation at the longest times, which gives rise to dynamical slow modes.