# Classical Spin Spirals in Frustrated Magnets from Free-Fermion Band Topology

## TL;DR

This paper establishes a precise mathematical link between classical frustrated spin systems exhibiting spiral magnetic orders and free-fermion band structures, revealing topological features in both contexts.

## Contribution

It introduces an explicit lattice construction connecting classical spin spirals with free-fermion band topology, highlighting the role of topological band structures in frustrated magnets.

## Key findings

- Classical spin spiral manifolds correspond to Fermi surfaces of free-fermion models.
- Examples include triangular lattice 120° order linked to Dirac points in honeycomb models.
- fcc spiral lines relate to Dirac nodal lines in diamond lattice models.

## Abstract

The formation of coplanar spin spirals is a common motif in the magnetic ordering of many frustrated magnets. For classical antiferromagnets, geometric frustration can lead to a massively degenerate ground state manifold of spirals whose propagation vectors can be described, depending on the lattice geometry, by points (triangular), lines (fcc), surfaces (frustrated diamond) or completely flat bands (pyrochlore). Here we demonstrate an exact mathematical correspondence of these spiral manifolds of classical antiferromagnets with the Fermi surfaces of free-fermion band structures. We provide an explicit lattice construction relating the frustrated spin model to a corresponding free-fermion tight-binding model. Examples of this correspondence relate the 120$^\circ$ order of the triangular lattice antiferromagnet to the Dirac nodal structure of the honeycomb tight-binding model or the spiral line manifold of the fcc antiferromagnet to the Dirac nodal line of the diamond tight-binding model. We discuss implications of topological band structures in the fermionic system to the corresponding classical spin system.