Quantum spins on star graphs and the Kondo model

TL;DR

This paper models quantum spins on a star graph, revealing a connection to the 4-channel Kondo model and non-Fermi liquid behavior, and introduces a solvable XY model variant.

Contribution

It establishes a novel mapping of the star graph XX model to a Kondo Hamiltonian with spin-1 representation and constructs a quadratic XY model for direct diagonalization.

Findings

Equivalent to 4-channel Kondo model in continuum limit

Demonstrates non-Fermi liquid behavior

Provides a solvable XY model variant

Abstract

We study the XX model for quantum spins on the star graph with three legs (i.e., on a Y-junction). By performing a Jordan-Wigner transformation supplemented by the introduction of an auxiliary space we find a Kondo Hamiltonian of fermions, in the spin 1 representation of su(2), locally coupled with a magnetic impurity. In the continuum limit our model is shown to be equivalent to the 4-channel Kondo model coupling spin-1/2 fermions with a spin-1/2 impurity and exhibiting a non-Fermi liquid behavior. We also show that it is possible to find a XY model such that - after the Jordan-Wigner transformation - one obtains a quadratic fermionic Hamiltonian directly diagonalizable.