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arXiv:1404.5490 · Physical Review B

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

Eran Sela, Hong‐Chen Jiang, Max H. Gerlach, Simon Trebst

Published Jul 14, 2014• 64 citations
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Abstract

The microscopic modeling of spin-orbit entangled $j=1/2$ Mott insulators such as the layered hexagonal iridates ${\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}$ and ${\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}$ 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 that include not only conventionally ordered phases but also gapped and gapless spin-orbital liquids. In particular, we demonstrate that in the presence of distortions, i.e., 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.

DOI
10.1103/physrevb.90.035113

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The microscopic modeling of spin-orbit entangled j=1/2j=1/2j=1/2 Mott insulators such as the layered hexagonal iridates Na2IrO3{\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}Na2​IrO3​ and Li2IrO3{\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}Li2​IrO3​ 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 that include not only conventionally ordered phases but also gapped and gapless spin-orbital liquids. In particular, we demonstrate that in the presence of distortions, i.e., 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 Z2{Z}_{2}Z2​ 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 Z2{Z}_{2}Z2​ phase quickly breaks down in what might be a rather unconventional phase transition driven by the simultaneous condensation of its elementary excitations.
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Tuning into the Kitaev spin liquid phase from a new perspective in Li2_22​IrO3_33​
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We study a spin model on honeycomb lattice with two types of Heisenberg exchange couplings, JJJ and J~\tilde JJ~, where JJJ is for the conventional spin and and J~\tilde JJ~ for rotated spin. When J~=0\tilde J=0J~=0, this is the conventional Heisenberg model. When J=0, the system is either in a stripy antiferromagnetic order(J~0\tilde J 0J~0, antiferromagnetic for rotated spin). The competition between two ferromagnetic orders or two antiferromagnetic orders induces Kitaev's spin liquid phase characterized by the exactly solvable Kitaev model (J=J~J=\tilde JJ=J~). Our model can be applied to layered Mott insulators A2_22​IrO3_33​ (A=Li, Na). For a monolayer of Li2_22​IrO3_33​, we show that it is possible to tune the controlling parameter into the Kitaev spin liquid regime by a link-dependent Rashba spin-orbital coupling.
We study quantum disordered ground states of the two-dimensional Heisenberg-Kitaev model on the triangular lattice using a Schwinger boson approach. Our aim is to identify and characterize potential gapped quantum spin liquid phases that are stabilized by anisotropic Kitaev interactions. For antiferromagnetic Heisenberg and Kitaev couplings and sufficiently small spin SSS, we find three different symmetric Z2{Z}_{2}Z2​ spin liquid phases, separated by two continuous quantum phase transitions. Interestingly, the gap of elementary excitations remains finite throughout the transitions. The first spin liquid phase corresponds to the well-known zero-flux state in the Heisenberg limit, which is stable with respect to small Kitaev couplings and develops ${120}^{\ensuremath{\circ}}$ order in the semiclassical limit at large SSS. In the opposite Kitaev limit, we find a different spin liquid ground state, which is a quantum disordered version of a magnetically ordered state with antiferromagnetic chains, in accordance with results in the classical limit. Finally, at intermediate couplings, we find a spin liquid state with unusual spin correlations. Upon spinon condensation, this state develops Bragg peaks at incommensurate momenta in close analogy to the magnetically ordered Z2 vortex crystal phase, which has been analyzed in recent theoretical works.
Motivated by the recent experimental observation of a Mott insulating state for the layered iridate Na2{}_{2}2​IrO3{}_{3}3​, we discuss possible ordering states of the effective iridium moments in the presence of strong spin-orbit coupling and a magnetic field. For a field pointing in the $\ensuremath{\langle}111\ensuremath{\rangle}$ direction---perpendicular to the hexagonal lattice formed by the iridium moments---we find that a combination of Heisenberg and Kitaev exchange interactions gives rise to a rich phase diagram with both symmetry breaking magnetically ordered phases as well as a topologically ordered phase that is stable over a small range of coupling parameters. Our numerical simulations further indicate two exotic critical points at the boundaries between these ordered phases.