Linear independence of localized magnon states

TL;DR

This paper rigorously analyzes the linear independence of localized magnon states in various frustrated lattices, establishing independence in many cases and dependence in the pyrochlore lattice, impacting understanding of ground state degeneracy.

Contribution

It provides rigorous proofs of linear independence for localized magnon states in several important frustrated lattices, advancing theoretical understanding of their ground state structure.

Findings

Localized magnon states are linearly independent in orthogonal, isolated, kagome, checkerboard, and star lattices.

Pyrochlore lattice exhibits localized magnon states with significant linear dependence.

Results imply finite entropy at the thermodynamic limit for many lattices due to independent localized states.

Abstract

At the magnetic saturation field, certain frustrated lattices have a class of states known as "localized multi-magnon states" as exact ground states. The number of these states scales exponentially with the number $N$ of spins and hence they have a finite entropy also in the thermodynamic limit $N\to \infty$ provided they are sufficiently linearly independent. In this article we present rigorous results concerning the linear dependence or independence of localized magnon states and investigate special examples. For large classes of spin lattices including what we called the orthogonal type and the isolated type as well as the kagom\'{e}, the checkerboard and the star lattice we have proven linear independence of all localized multi-magnon states. On the other hand the pyrochlore lattice provides an example of a spin lattice having localized multi-magnon states with considerable linear dependence.