Overcomplete Bases for S = 1 Spin Liquids

TL;DR

This paper explores the overcomplete basis of singlet pair states in spin-1 systems, showing their relevance to spin liquids and resonating-valence-bond states, with generalizations to higher symmetries.

Contribution

It introduces overcomplete bases for $S=1$ and higher integer spin systems, linking them to spin liquid states and symmetry generalizations.

Findings

Product states of two-body singlets form overcomplete bases for many-body singlet states.

Odd number of spins can be decomposed into superpositions of three-body and two-body singlet configurations.

Results extend to systems with $SO(2S+1)$ and $SU(n)$ symmetry.

Abstract

For a $S=1$ system with even number of spins, the product states of two-body singlets, called the singlet pair states (SPSs), are overcomplete bases for the Hilbert space of many-body singlets. If the system contains odd number of spins, a singlet state can be decomposed as a superposition of all of the following configurations, in each configuration three of the spins form a three-body singlet and the remaining form two-body singlet pairs. This indicates that a $S=1$ spin liquid is especially a resonating-valence bond state. Although this conclusion is no longer valid for $SO(3)$ symmetric $S>1$ systems, it can be generalized to an integer spin-$S$ system if it has an enlarged $SO(2S+1)$ symmetry. Similar results can also be obtained for systems with $SU(n)$ symmetry.