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arXiv:1101.1145 · Physical Review B

Possible proximity of the Mott insulating iridate Na 2 IrO 3 to a topological phase: Phase diagram of the Heisenberg-Kitaev model in a magnetic field

Hong‐Chen Jiang, Zheng‐Cheng Gu, Xiao-Liang Qi, Simon Trebst

Published Jun 6, 2011• 216 citations
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Abstract

Motivated by the recent experimental observation of a Mott insulating state for the layered iridate Na${}_{2}$IrO${}_{3}$, 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.

DOI
10.1103/physrevb.83.245104

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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⟨111\ensuremath⟩\ensuremath{\langle}111\ensuremath{\rangle}\ensuremath⟨111\ensuremath⟩ 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.
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.
We show that the topological Kitaev spin liquid on the honeycomb lattice is extremely fragile against the second-neighbor Kitaev coupling K 2 , which has recently been shown to be the dominant perturbation away from the nearest-neighbor model in iridate Na 2 IrO 3 , and may also play a role in α-RuCl 3 and Li 2 IrO 3 .This coupling naturally explains the zigzag ordering (without introducing unrealistically large longer-range Heisenberg exchange terms) and the special entanglement between real and spin space observed recently in Na 2 IrO 3 .Moreover, the minimal K 1 -K 2 model that we present here holds the unique property that the classical and quantum phase diagrams and their respective order-by-disorder mechanisms are qualitatively different due to the fundamentally different symmetries of the classical and quantum counterparts.
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The exact solution of Kitaev's spin-1/21/21/2 honeycomb spin-liquid model has sparked an intense search for Mott insulators hosting bond-dependent Kitaev interactions, of which Na2IrO3{\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}Na2​IrO3​ and $\ensuremath{\alpha}\text{\ensuremath{-}}\mathrm{Ru}{\mathrm{Cl}}_{3}$ are prime examples. Subsequently, it has been proposed that also spin-1 and spin-3/23/23/2 analogs of Kitaev interactions may occur in materials with strong spin-orbit coupling. As a minimal model to describe these Kitaev materials, we study the Heisenberg-Kitaev Hamiltonian in a consistent 1/S1/S1/S expansion, with SSS being the spin size. We present a comprehensive study of this model in the presence of an external magnetic field applied along two different directions, [001] and [111], for which an intricate classical phase diagram has been reported. In both settings, we employ spin-wave theory in a number of ordered phases to compute phase boundaries at the next-to-leading order in 1/S1/S1/S and show that quantum corrections substantially modify the classical phase diagram. More broadly, our work presents a consistent route to investigate the leading quantum corrections in spin models that break spin-rotational symmetry.
Tuning into the Kitaev spin liquid phase from a new perspective in Li2_22​IrO3_33​
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 the quantum many-body instabilities of the t−JK−JHt -J_{\mathrm{K}} - J_{\mathrm{H}}t−JK​−JH​ Kitaev-Heisenberg Hamiltonian on the honeycomb lattice as a minimal model for a doped spin-orbit Mott insulator. This spin-1/21/21/2 model is believed to describe the magnetic properties of the layered transition-metal oxide Na2_22​IrO3_33​. We determine the ground-state of the system with finite charge-carrier density from the functional renormalization group (fRG) for correlated fermionic systems. To this end, we derive fRG flow-equations adapted to the lack of full spin-rotational invariance in the fermionic interactions, here represented by the highly frustrated and anisotropic Kitaev exchange term. Additionally employing a set of Ward identities for the Kitaev-Heisenberg model, the numerical solution of the flow equations suggests a rich phase diagram emerging upon doping charge carriers into the ground-state manifold (Z2\mathbb{Z}_2Z2​ quantum spin liquids and magnetically ordered phases). We corroborate superconducting triplet ppp-wave instabilities driven by ferromagnetic exchange and various singlet pairing phases. For filling δ>1/4δ> 1/4δ>1/4, the ppp-wave pairing gives rise to a topological state with protected Majorana edge-modes. For antiferromagnetic Kitaev and ferromagnetic Heisenberg exchange we obtain bond-order instabilities at van Hove filling supported by nesting and density-of-states enhancement, yielding dimerization patterns of the electronic degrees of freedom on the honeycomb lattice. Further, our flow equations are applicable to a wider class of model Hamiltonians.