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

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