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Phase transitions for a collective coordinate coupled to Luttinger liquids

We study various realizations of collective coordinates, e.g. the position of a particle, the charge of a Coulomb box or the phase of a Bose or a superconducting condensate, coupled to Luttinger liquids (LL) with N flavors. We find that for Luttinger parameter 1/2<K<1 there is a phase transition from a delocalized phase into a phase with a periodic potential at strong coupling. In the delocalized phase the dynamics is dominated by an effective mass, i.e. diffusive in imaginary time, while on the transition line it becomes dissipative. At K=1/2 there is an additional transition into a localized phase with no diffusion at zero temperature.