Orbital order in classical models of transition-metal compounds

Abstract

Abstract. -We study the classical 120-degree and related orbital models. These are the classical limits of quantum models which describe the interactions among orbitals of transitionmetal compounds. We demonstrate that at low temperatures these models exhibit a long-range order which arises via an "order by disorder" mechanism. This strongly indicates that there is orbital ordering in the quantum version of these models, notwithstanding recent rigorous results on the absence of spin order in these systems.Introduction. -The properties of transition-metal (TM) compounds are a topic of longstanding interest. In these materials, the fractional filling of the 3d-shells in the TM ion provides a novel facet: The splitting of the t 2g and e g orbitals by the crystal field can produce situations with a single dynamical electron (or a hole) on each site along with multiple orbital degrees of freedom [1][2][3] [7]. The presence of the extra degrees of freedom raises the theoretical possibility of global, cooperative effects; i.e., orbital ordering. Such ordering may be observed via associated orbital-related magnetism and lattice distortions or, e.g., by resonant X-ray scattering techniques in which the 3d orbital order is detected by its effect on excited 4p states [8].The case for orbital ordering has been bolstered by detailed calculations and various other considerations [9]. However, alternate perspectives and various conceptual doubts have been raised concerning the entire picture of long-range orbital ordering [10,11]. In particular, at the theoretical level, a satisfactory justification of orbital ordering has not yet been provided [12]. The goal of this Letter is to present arguments which irrefutably demonstrate that orbital ordering indeed occurs. We will discuss primarily the so-called 120• -model which describes the situations when the e g orbitals are occupied by a single electron.

Discrete sliding symmetries, dualities, and self-dualities of quantum orbital compass models andp+ipsuperconducting arrays

“…The existence of sliding symmetries has profound effects on their quantum phase transitions, whose behavior only begun to be understood quite recently [12] and still remains largely unexplored. Amongst others, discrete sliding symmetries are present in spin [13,14,15], orbital [16,17,18,19,20], and superconducting array systems [21,22]. We further demonstrate that the planar orbital compass model [17,20] and the Xu-Moore model [21,22] of two dimensional p + ip superconducting arrays are, in fact, one and the same system, related by a simple duality transformation.…”

The emergence of resonating valence bond physics in spin–orbital models

“…So the Hamiltonian is only invariant if one simultaneously performs the same rotation in real space and in pseudo-spin space. For purely orbital models, this is known to have remarkable consequences [34,35,36,37]. For spin-orbital models, this implies that dimers with different orientations involve different orbital wave-functions, as can be clearly seen in phases C and C'.…”

Quantum compass model on the square lattice

“…In that respect, it is useful to emphasize that, as noticed in Refs. [8,9], the degeneracy is partly accidental and partly due to symmetry. Indeed, in addition to the lattice translational symmetries, this model has two types of discrete symmetries: (i) The Q i transformation which flips the z component of all the spins of the column r x = i, and the P j transformations which flip the x component of all spins of the line r z = j.…”

“…To see this, let us follow Ref. [8] and consider the classical version of the model, in which spins are considered as classical vectors. In that case, as shown by Nussinov et al, the ground state is highly degenerate, as in very frustrated magnets.…”