Unusual Symmetries in the Kugel-Khomskii Hamiltonian

TL;DR

This paper investigates the symmetries of the Kugel-Khomskii Hamiltonian in cubic titanates, revealing how orbital conservation leads to unique symmetry properties and the necessity of symmetry-breaking for observed order.

Contribution

It uncovers unusual symmetries in the Hamiltonian and explains the conditions under which long-range order and gaps can emerge in these systems.

Findings

Orbital conservation leads to additional symmetries.

No long-range spin order at nonzero temperatures without symmetry breaking.

Spin-orbit coupling introduces order but maintains gapless excitations.

Abstract

The Kugel-Khomskii Hamiltonian for cubic titanates describes spin and orbital superexchange interactions between $d^1$ ions having three-fold degenerate $t_{2g}$ orbitals. Since orbitals do not couple along "inactive" axes, perpendicular to the orbital planes, the total number of electrons in $|\alpha >$ orbitals in any such plane and the corresponding total spin are both conserved. A Mermin-Wagner construction shows that there is no long-range spin ordering at nonzero temperatures. Inclusion of spin-orbit coupling allows such ordering, but even then the excitation spectrum is gapless due to a continuous symmetry. Thus, the observed order and gap require more symmetry breaking terms.