Ambiguity in renormalization of X-junction between Luttinger liquid wires

TL;DR

This paper investigates the renormalization of conductances in a four-lead Luttinger liquid junction, revealing an ambiguity in RG equations due to a hidden Z_2 symmetry linked to particle-hole symmetry.

Contribution

It introduces the first-order RG equations for four-lead junctions and uncovers an ambiguity in conductance-based descriptions, resolved by a sign choice related to S-matrix properties.

Findings

RG equations cannot be solely expressed in conductances

Ambiguity resolved by a sign choice at the S-matrix level

Two distinct RG flows exist from any conductance point

Abstract

We study four-lead junction of semi-infinite wires by using fermionic representation in the scattering state formalism. The model of spinless fermions with short-range Luttinger liquid type interaction is used. We find the renormalization group (RG) equations for the conductances of the system in the first order of fermionic interaction. In contrast to the well-known cases of two-lead and three-lead junctions, we show that the RG equations cannot be expressed solely in terms of conductances. The appearing ambiguity is resolved by choosing a certain sign defined at the level of S-matrix, but hidden in conductances. The origin of this Z_2 symmetry is traced back to the particle-hole symmetry of our Hamiltonian. We demonstrate two distinct RG flows from any point in the space of conductances. The scaling exponents at the fixed points are not affected by these observations.