The Fine Structure of the Phonon in One Dimension from Quantum Hydrodynamics

TL;DR

This paper demonstrates that quantum hydrodynamics with phonon dispersion can effectively describe the nonlinear physics of one-dimensional phonons, capturing singular behaviors near energetic thresholds without complex refermionization.

Contribution

It introduces a hydrodynamic approach with phonon dispersion to analyze nonlinear Luttinger liquids, avoiding refermionization or fictitious impurities.

Findings

Identifies singular behaviors near phonon and soliton thresholds.

Uses quadratic dispersion model linked to the Benjamin--Ono equation.

Provides benchmarks against the Calogero--Sutherland model.

Abstract

We show that the resonant interactions between phonons in one dimension may be treated consistently within Quantum Hydrodynamics by the introduction of phonon dispersion. In this way the physics of a nonlinear Luttinger liquid may be described in terms of hydrodynamic (i.e. bosonized) variables without recourse to refermionization or the introduction of fictitious impurities. We focus on the calculation of the dynamic structure factor for a model with quadratic dispersion, which has the Benjamin--Ono equation of fluid dynamics as its equation of motion. We find singular behavior in the vicinity of upper and lower energetic thresholds corresponding to phonon and soliton branches of the classical theory, which may be benchmarked against known results for the Calogero--Sutherland model.