Long time correlations of nonlinear Luttinger liquids

TL;DR

This paper reviews the limitations of Luttinger liquid theory in capturing the real-time dynamics of nonlinear one-dimensional systems, highlighting the effects of band curvature, high-frequency oscillations, and diffusion phenomena.

Contribution

It provides a detailed analysis of the breakdown of Luttinger liquid theory due to nonlinear dispersion and introduces the impact of diffusion and umklapp scattering on correlation decay.

Findings

Perturbation theory singularities from band curvature effects.

High frequency oscillations influence long-time correlations.

Diffusion effects challenge exponential decay assumptions.

Abstract

An overview is given of the limitations of Luttinger liquid theory in describing the real time equilibrium dynamics of critical one-dimensional systems with nonlinear dispersion relation. After exposing the singularities of perturbation theory in band curvature effects that break the Lorentz invariance of the Tomonaga-Luttinger model, the origin of high frequency oscillations in the long time behaviour of correlation functions is discussed. The notion that correlations decay exponentially at finite temperature is challenged by the effects of diffusion in the density-density correlation due to umklapp scattering in lattice models.