Order-by-disorder and spin-orbital liquids in a distorted Heisenberg-Kitaev model

TL;DR

This paper investigates how lattice distortions influence the emergence of ordered and spin-orbital liquid phases in a distorted Heisenberg-Kitaev model, revealing the stability of gapless phases and the fragility of gapped Z2 phases.

Contribution

It demonstrates the role of anisotropies and exchange interactions in selecting ordered states and destabilizing certain spin-liquid phases in a realistic model for iridates.

Findings

Ordered states arise via order-by-disorder sensitive to anisotropy.

Gapless spin-liquid phase remains stable under distortions.

Gapped Z2 phase quickly breaks down with Heisenberg exchange inclusion.

Abstract

The microscopic modeling of spin-orbit entangled $j=1/2$ Mott insulators such as the layered hexagonal Iridates Na$_2$IrO$_3$ and Li$_2$IrO$_3$ has spurred an interest in the physics of Heisenberg-Kitaev models. Here we explore the effect of lattice distortions on the formation of the collective spin-orbital states which include not only conventionally ordered phases but also gapped and gapless spin-orbital liquids. In particular, we demonstrate that in the presence of spatial anisotropies of the exchange couplings conventionally ordered states are formed through an order-by-disorder selection which is not only sensitive to the type of exchange anisotropy but also to the relative strength of the Heisenberg and Kitaev couplings. The spin-orbital liquid phases of the Kitaev limit -- a gapless phase in the vicinity of spatially isotropic couplings and a gapped Z$_2$ phase for a dominant spatial anisotropy of the exchange couplings -- show vastly different sensitivities to the inclusion of a Heisenberg exchange. While the gapless phase is remarkably stable, the gapped Z$_2$ phase quickly breaks down in what might be a rather unconventional phase transition driven by the simultaneous condensation of its elementary excitations.