Exact ground states for coupled spin trimers

TL;DR

This paper identifies exact ground states in certain geometrically frustrated Heisenberg spin systems composed of coupled antiferromagnetic spin trimers, providing conditions for these states to be ground states and analyzing their properties.

Contribution

It introduces a method to determine inter-trimer couplings that yield exact product ground states in frustrated spin systems, including classical and quantum cases.

Findings

Exact conditions for ground states in coupled spin trimer systems.

Numerical determination of critical inter-trimer couplings in specific models.

Proof that trimer chains with these ground states are gapped.

Abstract

We consider a class of geometrically frustrated Heisenberg spin systems which admit exact ground states. The systems consist of suitably coupled antiferromagnetic spin trimers with integer spin quantum numbers $s$ and their ground state $\Phi$ will be the product state of the local singlet ground states of the trimers. We provide linear equations for the inter-trimer coupling constants which are equivalent to $\Phi$ being an eigenstate of the corresponding Heisenberg Hamiltonian and sufficient conditions for $\Phi$ being a ground state. The classical case $s\to\infty$ can be completely analyzed. For the quantum case we consider a couple of examples, where the critical values of the inter-trimer couplings are numerically determined. These examples include chains of corner sharing tetrahedra as well as certain spin tubes. $\Phi$ is proven to be gapped in the case of trimer chains. This follows from a more general theorem on quantum chains with product ground states.