Exact Tkachenko modes and their damping in the vortex lattice regime of rapidly rotating bosons

TL;DR

This paper provides an exact analytical solution for Tkachenko modes in a vortex lattice of rapidly rotating bosons, analyzing their damping and the resulting effects on density correlations at finite temperatures.

Contribution

It presents the first exact solution of the Bogoliubov-de Gennes equations for Tkachenko modes in the lowest Landau level, including damping rates and their implications.

Findings

Tkachenko modes become strongly damped at low energies due to Beliaev and Landau damping.

Density fluctuations grow logarithmically, limiting lattice order to finite scales.

One-body density matrix decays exponentially at large distances at finite temperatures.

Abstract

We have found an exact analytical solution of the Bogoliubov-de Gennes equations for the Tkachenko modes of the vortex lattice in the lowest Landau level (LLL) in the thermodynamic limit at any momenta and calculated their damping rates. At finite temperatures both Beliaev and Landau damping leads to momentum independent damping rates in the low-energy limit, which shows that at sufficiently low energies Tkachenko modes become strongly damped. We then found that the mean square fluctuations of the density grow logarithmically at large distances, which indicates that the state is ordered in the vortex lattice only on a finite (although exponentially large) distance scale and introduces a low-momentum cut-off. Using this circumstance we showed that at finite temperatures the one-body density matrix undergoes an exponential decay at large distances.