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 nonzero 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 a Bose-Einstein condensate) grain coupled to N≥2 TLLs; here the transition signals also the onset of a relevant Josephson coupling.