2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/108525The dissipation induced by a metallic gate on the low-energy properties of interacting 1D electron liquids is studied. As function of the distance to the gate, or the electron density in the wire, the system undergoes a quantum phase transition from the Tomonaga-Luttinger liquid state to two kinds of dissipative phases, one of them with a finite spatial correlation length. We also define a dual model, which describes an attractive one dimensional metal with a Josephson coupling to a dirty metallic lead.5 pages, 2 EPS figures; v2: improved figure for phase diagram, added discussion, corrected typosMesoscale and Nanoscale PhysicsStrongly Correlated Electronstext