Hyperelliptic curves for multi-channel quantum wires and the multi-channel Kondo problem

TL;DR

This paper uses hyperelliptic curves to analyze the current and magnetization in multi-channel quantum wires and Kondo problems, revealing dualities between weak and strong coupling regimes.

Contribution

It introduces a novel approach using hyperelliptic curves to connect weak and strong coupling regimes in multi-channel quantum systems.

Findings

Contour integrals over hyperelliptic curves describe physical quantities.

Dualities between weak and strong coupling are demonstrated.

The same hyperelliptic curve applies to different screening cases in Kondo problems.

Abstract

We study the current in a multi-channel quantum wire and the magnetization in the multi-channel Kondo problem. We show that at zero temperature they can be written simply in terms of contour integrals over a (two-dimensional) hyperelliptic curve. This allows one to easily demonstrate the existence of weak-coupling to strong-coupling dualities. In the Kondo problem, the curve is the same for under- and over-screened cases; the only change is in the contour.