Origin of Biquadratic Exchange Interactions in a Mott Insulator as a Driving Force of Spin Nematic Order

TL;DR

This paper investigates the origin of biquadratic exchange interactions in Mott insulators with two orbitals per site, revealing the role of twisted ring exchange processes in driving spin nematic order.

Contribution

It identifies the specific cyclic exchange processes responsible for biquadratic interactions and analyzes how Coulomb interactions influence their strength, clarifying conditions for nematic order emergence.

Findings

Biquadratic exchange arises from twisted ring exchange processes.

Inter-orbital Coulomb interactions suppress biquadratic terms.

Degenerate two-orbital Mott insulators are not ideal for nematic order studies.

Abstract

We consider a series of Mott insulators in unit of two orbitals each hosting spin-1/2 electron, and by pairing two spin-1/2 into spin-1 triplet, derive the effective exchange interaction between the adjacent units via fourth order perturbation theory. It turns out that the biquadratic exchange interaction between spin-1, which is one of the origins of the nematic order, arises only in processes where the four different electrons exchange cyclically along the twisted loop, which we call "twisted ring exchange" processes. We show that the term becomes the same order with the Heisenberg exchange interactions when the on-orbital Coulomb interaction is not too large. Whereas, the inter-orbital Coulomb interactions give rise to additional processes that cancel the twisted ring exchange, and strongly suppresses the biquadratic term. The Mott insulator with two electrons on degenerate two orbitals is thus not an ideal platform to study such nematic orders.