Quantum Hydrodynamics in One Dimension beyond the Luttinger Liquid

TL;DR

This paper develops a tractable nonlinear extension of the Luttinger model for one-dimensional quantum fluids, analyzing the dynamical structure factor and singularities beyond traditional approaches, with implications for understanding phonon and soliton excitations.

Contribution

It introduces a phonon basis approach to nonlinear Luttinger models, enabling analytical calculation of singularities in the dynamical structure factor beyond standard methods.

Findings

Calculated singularities in the zero temperature dynamical structure factor.

Identified thresholds corresponding to phonon and soliton excitations.

Derived analytical expressions for power law singularities at thresholds.

Abstract

Recent years have seen the development of a rich phenomenology beyond the Luttinger Liquid model of one dimensional quantum fluids, arising from interactions between the elementary phonon excitations. It has been known for some time, however, that the straightforward inclusion of these interactions presents technical difficulties that have necessitated approaches based on refermionization or effective impurity models. In this work we show that the nonlinear extensions of the Luttinger model are tractable in the phonon basis. We present a calculation of the singularities present in the zero temperature dynamical structure factor in the semiclassical limit where the phonon dispersion is strong. A unitary transformation decouples interactions between left-- and right--moving phonons, leaving a nonlinear chiral Hamiltonian. At low momenta, this Hamiltonian has a spectrum bounded above and below by thresholds identified with phonon and soliton excitations in the semiclassical limit. The chiral dynamical structure factor therefore has support only in this region, with power law singularities at the thresholds originating in the Anderson orthogonality catastrophe, which we calculate analytically. The dynamical structure factor for the original nonchiral Hamiltonian is a convolution of this chiral correlator with a power law arising from the left--right decoupling.