Transconducting Transition for a Dynamic Boundary Coupled to Several Tomonaga-Luttinger Liquids

The question on how a quantum bath can affect the dynamical properties of a quantum degree of freedom is a long standing and important one. Here 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 a critical value of K >= 1/2, where the TLL phases lock together at the particle position, resulting in a nonzero transconductance equal to e2/(Nh). We analyze this transition line and compare it with large N predictions. We discuss experimental applications both for cold atomic or condensed matter realizations. B. H. gratefully acknowledges funding by the German DFG through the DIP programme [FO703/2-1]. P. L.D. acknowledges support from ANR Grant No. ANR-17-CE30-0027-01 RaMaTraF. This work was supported in part by the Swiss National Science Foundation under Division II.