We study [1] 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>1 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). Another type of a dynamic boundary is a superconducting (or a Bose-Einstein condensate) grain coupled to N>1 TLLs, here the transition signals also the onset of a relevant Josephson coupling.

[1] B. Horovitz, T. Giamarchi and P. Le Doussal, Phys. Rev. Lett. 121, 166803 (2018) https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.121.166803