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arXiv:2207.02188

Counter-rotating spiral, zigzag, and 120$^\circ$ orders from coupled-chain analysis of Kitaev-Gamma-Heisenberg model, and relations to honeycomb iridates

Wang Yang, Alberto Nocera, Chao Xu, Hae‐Young Kee, Ian K. Affleck

Published Jul 5, 2022• 3 citations
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Abstract

We study the nearest neighboring spin-1/2 Kitaev-Heisenberg-Gamma ($KJΓ$) model on the honeycomb lattice in the parameter region of ferromagnetic (FM) Kitaev and antiferromagnetic (AFM) Heisenberg couplings relevant for honeycomb iridates, using a coupled-chain analysis. Starting from the gapless Luttinger liquid phase of a decoupled $KJΓ$ chain, the inter-chain interactions in the two-dimensional model is treated within a self-consistent mean field approach based on the Luttinger liquid theory. In the FM Gamma region, our analysis recovers the reported 120$^\circ$ magnetic order, previously obtained by classical analysis and exact diagonalization method. On the other hand, new physics is revealed in the AFM Gamma region, where three magnetic orders are found, including 120$^\circ$, commensurate counter-rotating spiral, and zigzag orders. Interestingly, the two first order phase transition lines separating these three magnetic orders merge at a single point at $K = -2 Γ$ and $J=0$, which is predicted to be a quantum critical point. The current theory captures the experimentally observed counter-rotating spiral order in $α$-Li$_22$IrO$_3$ and the zigzag order in Na$_2$IrO$_3$, thereby indicating that the spin-1/2 $KJΓ$ model may serve as a minimal model for honeycomb iridates. Limitations of the mean field theory presented in this work and the $J \rightarrow 0$ regime are also discussed.

DOI
10.48550/arxiv.2207.02188

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We study the nearest neighboring spin-1/2 Kitaev-Heisenberg-Gamma (KJΓKJΓKJΓ) model on the honeycomb lattice in the parameter region of ferromagnetic (FM) Kitaev and antiferromagnetic (AFM) Heisenberg couplings relevant for honeycomb iridates, using a coupled-chain analysis. Starting from the gapless Luttinger liquid phase of a decoupled KJΓKJΓKJΓ chain, the inter-chain interactions in the two-dimensional model is treated within a self-consistent mean field approach based on the Luttinger liquid theory. In the FM Gamma region, our analysis recovers the reported 120∘^\circ∘ magnetic order, previously obtained by classical analysis and exact diagonalization method. On the other hand, new physics is revealed in the AFM Gamma region, where three magnetic orders are found, including 120∘^\circ∘, commensurate counter-rotating spiral, and zigzag orders. Interestingly, the two first order phase transition lines separating these three magnetic orders merge at a single point at K=−2ΓK = -2 ΓK=−2Γ and J=0J=0J=0, which is predicted to be a quantum critical point. The current theory captures the experimentally observed counter-rotating spiral order in ααα-Li22_222​2IrO3_33​ and the zigzag order in Na2_22​IrO3_33​, thereby indicating that the spin-1/2 KJΓKJΓKJΓ model may serve as a minimal model for honeycomb iridates. Limitations of the mean field theory presented in this work and the J→0J \rightarrow 0J→0 regime are also discussed.
The unconventional magnetic orderings found in Kitaev spin liquid candidates suggest frustration of spin interactions and raise a possibility of nearby spin liquid phases. In particular, counter-rotating spiral ordering in three-dimensional (3D) iridates is striking, and understanding the microscopic mechanism of such ordering may provide routes to 3D Kitaev spin liquids. We study a minimal 3D model including Kitaev KKK and a symmetric off-diagonal bond-dependent $\mathrm{\ensuremath{\Gamma}}$ interaction on the hyperhoneycomb lattice by using exact diagonalization (ED). An ED cluster was chosen to satisfy a hidden SU(2) symmetry when $K=\mathrm{\ensuremath{\Gamma}}$ via a 12-site unitary transformation. The magnetic ordering in the transformed basis then generically maps to the counter-rotating noncoplanar spiral order of the original spin when the moment is pinned along a certain direction. When KKK and $\mathrm{\ensuremath{\Gamma}}$ are negative, a small positive Heisenberg interaction favors the (1,1,0) direction consistent with the reported spiral order in $\ensuremath{\beta}\ensuremath{-}{\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}$. Two other ordered phases close to the spiral order and spin liquids are identified. Our findings offer a relevant set of microscopic parameters, which in turn guides a way to approach possible Kitaev spin liquids.
We study a spin model on honeycomb lattice with two types of Heisenberg exchange couplings, JJJ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}$, where JJJ is for the conventional spin and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}$ is for rotated spin. When J=0J=0J=0, the system is either in a stripy antiferromagnetic order ($\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}<0$, ferromagnetic for rotated spin) or a zigzag antiferromagnetic order ($\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}>0$, antiferromagnetic for rotated spin). The competition between two ferromagnetic orders or between two antiferromagnetic orders induces Kitaev's spin liquid phase, characterized by the exactly solvable Kitaev model ($J=\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}$). Our model can be applied to layered Mott insulators AAA2{}_{2}2​IrO3{}_{3}3​ (A=LiA=\text{Li}A=Li, Na). For a monolayer of Li2{}_{2}2​IrO3{}_{3}3​, 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 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.
In this work, we perform a combination of analytical and numerical studies on the phase diagram of the spin-1/2 Kitaev-Heisenberg chain in the region of negative Kitaev and positive Heisenberg couplings. Apart from the antiferromagnetic phase, we find a magnetically ordered phase with left-left-right-right order and a gapless phase with central charge value c=1c=1c=1, resolving the existing contradictory results in literature regarding this parameter region. In particular, the origin of the left-left-right-right order is clarified based on a perturbative Luttinger liquid analysis. The left-left-right-right phase is further shown to persist in the Kitaev-Heisenberg-Gamma model when a small nonzero Gamma interaction is introduced. Using a coupled-chain method, we also demonstrate that the one-dimensional (1D) left-left-right-right order gives a quasi-1D explanation for the 2D stripy order of the same model on the honeycomb lattice.
Motivated by recent experiments on $\ensuremath{\beta}$-Li2{}_{2}2​IrO3{}_{3}3​, we study the phase diagram of the Heisenberg-Kitaev model on a three-dimensional lattice of tricoordinated Ir4+{}^{4+}4+, dubbed the hyperhoneycomb lattice. The lattice geometry of this material, along with Ir4+{}^{4+}4+ ions carrying Jeff=1/2{J}_{\mathrm{eff}}=1/2Jeff​=1/2 moments, suggests that the Heisenberg-Kitaev model may effectively capture the low-energy spin-physics of the system in the strong-coupling limit. Using a combination of semiclassical analysis, exact solution, and slave-fermion mean-field theory, we find, in addition to the spin liquid, four different magnetically ordered phases depending on the parameter regime. All four magnetic phases---the N\'eel, the polarized ferromagnet, the skew-stripy, and the skew-zig-zag---have collinear spin ordering. The three-dimensional Z2{Z}_{2}Z2​ spin liquid, which extends over an extended parameter regime around the exactly solvable Kitaev point, has a gapless Majorana mode with a deformed Fermi circle (codimensions, dc=2{d}_{c}=2dc​=2). We discuss the effect of the magnetic field and finite temperature on different phases that may be relevant for future experiments.
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