Transconducting transition for a dynamic boundary coupled to several Luttinger liquids

We study a dynamic boundary, e.g. a mobile impurity, coupled to N independent Tomonaga-Luttinger liquids (TLLs) each with interaction parameter K. We demonstrate that for N>2 there is a quantum phase transition at K>1/2, where the TLL phases lock together at the particle position, resulting in a non-zero transconductance equal to e^2/Nh. The transition line terminates for strong coupling at K=1- 1/N, consistent with results at large N. Another type of a dynamic boundary is a superconducting (or Bose-Einstein condensate) grain coupled to N>2 TLLs, here the transition signals also the onset of a relevant Josephson coupling.