Directional Ordering of Fluctuations in a Two-Dimensional Compass Model

Abstract

In the Mott insulating phase of the transition metal oxides, the effective orbital-orbital interaction is directional both in the orbital space and in the real space. We discuss a classical realization of directional coupling in two dimensions. Despite extensive degeneracy of the ground state, the model exhibits partial orbital ordering in the form of directional ordering of fluctuations at low temperatures stabilized by an entropy gap. Transition to the disordered phase is shown to be in the Ising universality class through exact mapping and multicanonical Monte Carlo simulations.PACS numbers: 64.60. Cn,05.10.Ln,75.30.Et,75.40.Cx Recently, there has been growing interest in the effects of orbital degeneracy in the physics of transition metal oxide insulators [1,2]. In these systems, the dominating energy scales for d-electrons on transition metal (TM) ions are the on-site Coulomb repulsion (which freezes out the charge degrees of freedom), the Hund's rule coupling, and the crystal field due to the surrounding oxygen ions. The latter two together determine the degeneracies and degrees of freedom of spin and orbital on each transition metal ions. Spins and orbitals on neighboring TM ions can then be coupled through the superexchange mechanism. In the case of orbitals, they are also coupled through the phonon-mediated cooperative JahnTeller mechanism [1]. These couplings determine the low temperature properties of these systems.Because orbital coupling is intrinsically directional[3], orbital ordering brings up some unusual questions. Especially interesting is when the coupling along a given bond direction is Ising like, but with different Ising axes along different bond directions [4]. The Hamiltonian for orbitals is then given bywhere τ is an isospin operator representing the orbital degree of freedom, and n ij is a unit vector giving the Ising axis for the bond ij . For example, for e g orbitals in Perovskite structures, n ij for the three different bond directions are coplanar and oriented relative to each other by 120• , giving rise to the so-called 120• Model [4,5,6]. This model is also applicable to t 2g orbitals on the three bonds of a honeycomb lattice as in planes of V 2 O 3 [7,8]. On the other hand, for t 2g orbitals on Perovskite structures, the relevant model is the Compass Model [1,4,10] with n ij = x, y, z for the three bond directions. The Compass Model may also feature as part of the spinspin coupling of t 2g orbitals when spin-orbit interaction is taken into account [11]. A common feature of both the Compass Model and the 120• Model is the competition between bonds in different directions, with the resulting frustration leading to macroscopic degeneracy of the classical ground state.In this Letter, we report analytical and numerical results on a classical version of (1) in 2D. The highly anisotropic coupling gives rise to interesting interplay between one-and two-dimensional ordering, between continuous and discrete spin physics, and between slow and fast modes. Our main result is that at low ...

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