Fine structure of the phonon in one dimension from quantum hydrodynamics

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 the introduction of impurities at the outset, and gives a complementary view on the mobile impurity model from the hydrodynamics. 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.

Figure 1

Dynamical structure factor $S(q,\omega )$ indicated by gray scale between the phonon $\epsilon \left(q\right)$ and soliton (dashed) branches $E\left(q\right)$, with the power-law behavior of Eq. (5) as $\omega$ approaches the threshold at given momentum $p$ (lower inset). The upper inset shows a snapshot of the Lorentzian profile $V\left(x\right)$ of the soliton Eq. (4) in real space.