Ambiguity in renormalization of the conductance of an X-junction between quantum wires with a Luttinger-type interaction

We study the conductance renormalization for the four-lead junction of semi-infinite wires in the scattering state formalism. We use the model of spinless fermions with a short-range Luttinger-type interaction and find the renormalization group (RG) equations for the conductances of the system in the first order of the fermionic interaction. In contrast to the well-known cases of two-lead and three-lead junctions, the four-lead case does not admit a formulation of the RG equations solely in terms of conductances. The arising ambiguity with arbitrariness in the choice of the sign of $S$-matrix elements related to identical conductance matrices might be connected with a symmetry of the particle–hole type in the Hamiltonian. We show that there are two distinct RG flows from any initial point in the space of conductances. The discovered ambiguity does not affect the scaling exponents at the fixed points.