Constraints on conductances for Y-junction of quantum wires

TL;DR

This paper analyzes the constraints on conductance matrices in a Y-junction of three quantum wires without interactions, revealing geometric restrictions and their implications for bosonization analysis.

Contribution

It characterizes the set of possible conductance matrices in a non-interacting Y-junction and links geometric properties to bosonization applicability.

Findings

Conductance matrix values form a four-dimensional body touching the unit sphere at six points.

Unitarity of the scattering matrix imposes strict restrictions on conductance values.

Bosonization analysis is valid only near these six specific points.

Abstract

We consider the Y-junction, connecting three quantum wires, in the scattering states formalism. In the absence of fermionic interaction we analyze the restrictions on the values of conductance matrix, imposed by the unitarity of scattering S-matrix. Using the combination of numerical and analytical results, we describe the four-dimensional body of values of reduced conductance matrix. We show that this body touches the unit sphere at six points only, in accordance with Birkhoff-von Neumann theorem. It implies that the Abelian bosonization analysis for the vanishing interaction strength can be performed only in the vicinity of these six points.