Landau Expansion for the Kugel-Khomskii $t_{2g}$ Hamiltonian

TL;DR

This paper develops a variational mean-field approach to analyze the Kugel-Khomskii $t_{2g}$ Hamiltonian, revealing symmetry-driven degeneracies and the effects of perturbations like spin-orbit coupling and next-nearest-neighbor hopping on magnetic and orbital order.

Contribution

It introduces a site-specific density matrix variational method for the KK model and explores how perturbations break symmetries and influence magnetic and orbital states.

Findings

Mean-field susceptibility is dispersionless in certain directions due to symmetries.

Spin-orbit coupling induces dispersion but preserves gapless spin-wave spectrum.

Next-nearest-neighbor hopping leads to exotic states with zero net spin moments.

Abstract

The Kugel-Khomskii (KK) Hamiltonian for the titanates describes spin and orbital superexchange interactions between $d^1$ ions in an ideal perovskite structure in which the three $t_{2g}$ orbitals are degenerate in energy and electron hopping is constrained by cubic site symmetry. In this paper we implement a variational approach to mean-field theory in which each site, $i$, has its own $n \times n$ single-site density matrix $\rhov(i)$, where $n$, the number of allowed single-particle states, is 6 (3 orbital times 2 spin states). The variational free energy from this 35 parameter density matrix is shown to exhibit the unusual symmetries noted previously which lead to a wavevector-dependent susceptibility for spins in $\alpha$ orbitals which is dispersionless in the $q_\alpha$-direction. Thus, for the cubic KK model itself, mean-field theory does not provide wavevector `selection', in agreement with rigorous symmetry arguments. We consider the effect of including various perturbations. When spin-orbit interactions are introduced, the susceptibility has dispersion in all directions in ${\bf q}$-space, but the resulting antiferromagnetic mean-field state is degenerate with respect to global rotation of the staggered spin, implying that the spin-wave spectrum is gapless. This possibly surprising conclusion is also consistent with rigorous symmetry arguments. When next-nearest-neighbor hopping is included, staggered moments of all orbitals appear, but the sum of these moments is zero, yielding an exotic state with long-range order without long-range spin order. The effect of a Hund's rule coupling of sufficient strength is to produce a state with orbital order.