Exact Normal Modes for a System with Three Luttinger Liquids Interacting in an Elongated Trap

We consider a system of spinless fermions in a nearly one-dimensional cylindrical trap, with the Fermi level such that only the lowest-lying states with angular momenta l = +hbar, 0, or -hbar about the axis of the trap are occupied. We treat the particles in these states as comprising three separate Luttinger liquids, with the possibility for forward and backward scattering between them. We determine the normal modes in the system in the presence of this scattering, finding two sets of linear harmonic fluid modes and one set of modes that are self-coupled through a sine-Gordon interaction.