## Department of Physics Papers

#### Document Type

Journal Article

#### Date of this Version

3-16-2004

#### Publication Source

Physical Review B

#### Volume

69

#### Start Page

094409-1

#### Last Page

094409-19

#### DOI

10.1103/PhysRevB.69.094409

#### Abstract

The Kugel-Khomskii (KK) Hamiltonian describes spin and orbital superexchange interactions between *d*^{1} ions in an ideal cubic 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×n single-site density matrix ρ(*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 wave-vector-dependent susceptibility for spins in α orbitals which is dispersionless in the *q*_{α} 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 **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.

#### Recommended Citation

Harris, A.,
Aharony, A.,
Entin-Wohlman, O.,
Korenblit, I.,
&
Yildirim, T.
(2004).
Landau Expansion for the Kugel-Khomskii *t*_{2g} Hamiltonian.
*Physical Review B,*
*69*
094409-1-094409-19.
http://dx.doi.org/10.1103/PhysRevB.69.094409

**Date Posted:** 12 August 2015

This document has been peer reviewed.

## Comments

At the time of publication, author Taner Yildirim was affiliated with the National Institute of Standards and Technology, Gaithersburg, Maryland. Currently, he is a faculty member in the Materials Science and Engineering Department at the University of Pennsylvania.