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arXiv:1806.04688 · Physical review. B./Physical review. B

Counter-rotating spiral order in three-dimensional iridates: Signature of hidden symmetry in the Kitaev- Γ model

P. Peter Stavropoulos, Andrei Catuneanu, Hae‐Young Kee

Published Sep 4, 2018• 21 citations
arXivHTMLPDF

Abstract

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 $K$ 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 $K$ 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.

DOI
10.1103/physrevb.98.104401

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

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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.

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Predominance of the Kitaev interaction in a three-dimensional honeycomb iridate: From ab initio to spin model

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Motivated by the recent synthesis of two insulating ${\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}$ polymorphs, where ${\mathrm{Ir}}^{4+}$ ${S}_{\mathrm{eff}}=1/2$ moments form 3D (``harmonic'') honeycomb structures with threefold coordination, we study magnetic Hamiltonians on the resulting $\ensuremath{\beta}\text{\ensuremath{-}}{\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}$ hyperhoneycomb lattice and $\ensuremath{\gamma}\text{\ensuremath{-}}{\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}$ stripyhoneycomb lattice. Experimentally measured magnetic susceptibilities suggest that Kitaev interactions, predicted for the ideal ${90}^{\ensuremath{\circ}}$ Ir-O-Ir bonds, are sizable in these materials. We first consider pure Kitaev interactions, which lead to an exactly soluble 3D quantum spin liquid (QSL) with emergent Majorana fermions and ${\mathrm{Z}}_{2}$ flux loops. Unlike 2D QSLs, the 3D QSL is stable to finite temperature, with ${T}_{c}\ensuremath{\approx}|K|/100$. On including Heisenberg couplings, exact solubility is lost. However, by noting that the shortest closed loop $\ensuremath{\ell}$ is relatively large in these structures, we construct an $\ensuremath{\ell}\ensuremath{\rightarrow}\ensuremath{\infty}$ approximation by defining the model on the Bethe lattice. The phase diagram of the Kitaev-Heisenberg model on this lattice is obtained directly in the thermodynamic limit, using tensor network states and the infinite-system time-evolving-block-decimation (iTEBD) algorithm. Both magnetically ordered and gapped QSL phases are found, the latter being identified by an entanglement fingerprint.

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Unveiling magnetic interactions of ruthenium trichloride via constraining direction of orbital moments: Potential routes to realize a quantum spin liquid

Yusheng Hou, Hongjun Xiang, X. G. Gong

Recent experiments reveal that the honeycomb ruthenium trichloride $\ensuremath{\alpha}\ensuremath{-}\mathrm{RuC}{\mathrm{l}}_{3}$ is a prime candidate of the Kitaev quantum spin liquid (QSL). However, there is no theoretical model which can properly describe its experimental dynamical response due to the lack of a full understanding of its magnetic interactions. Here, we propose a general scheme to calculate the magnetic interactions in systems (e.g., $\ensuremath{\alpha}\ensuremath{-}\mathrm{RuC}{\mathrm{l}}_{3}$) with nonnegligible orbital moments by constraining the directions of orbital moments. With this scheme, we put forward a minimal ${J}_{1}\ensuremath{-}{K}_{1}\ensuremath{-}{\mathrm{\ensuremath{\Gamma}}}_{1}\ensuremath{-}{J}_{3}\ensuremath{-}{K}_{3}$ model for $\ensuremath{\alpha}\ensuremath{-}\mathrm{RuC}{\mathrm{l}}_{3}$ and find that: (I) The third nearest neighbor (NN) antiferromagnetic Heisenberg interaction ${J}_{3}$ stabilizes the zigzag antiferromagnetic order; (II) The NN symmetric off-diagonal exchange ${\mathrm{\ensuremath{\Gamma}}}_{1}$ plays a pivotal role in determining the preferred direction of magnetic moments and generating the spin wave gap. An exact diagonalization study on this model shows that the Kitaev QSL can be realized by suppressing the NN symmetric off-diagonal exchange ${\mathrm{\ensuremath{\Gamma}}}_{1}$ and the third NN Heisenberg interaction ${J}_{3}$. Thus, we not only propose a powerful general scheme for investigating the intriguing magnetism of ${J}_{\mathrm{eff}}=1/2$ magnets, but also point out future directions for realizing the Kitaev QSL in the honeycomb ruthenium trichloride $\ensuremath{\alpha}\ensuremath{-}\mathrm{RuC}{\mathrm{l}}_{3}$.

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A Functional Renormalization Group Perspective on Quantum Spin Liquids in Three-Dimensional Frustrated Magnets

Finn Lasse Buessen

When sizable quantum fluctuations and strong frustration mechanisms act in concert to repel the formation of conventional long-range order in quantum magnets, they can make way for massively entangled spin liquid phases which may imbue the material with extraordinary properties. \nThe search for such curious phases of matter has proceeded for several decades, gaining extra momentum some fifteen years ago when Kitaev proposed an analytically solvable model for a quantum spin liquid with anyonic excitations on the basis of realistic microscopic spin exchange terms [Kitaev, Annals of Physics 321, 2 (2006)]. Attempts to identify different models or materials which harbor quantum spin liquid ground states have kept researchers -- experimentalists and theorists alike -- in suspense ever since. However, the simulation of quantum many-body systems poses a serious challenge even to modern numerical techniques, particularly in the case of frustrated quantum magnetism in three spatial dimensions. Such models evade tractability by many established approaches, leaving a methodological void. \n \nIn this thesis, we report on recent progress in cutting-edge implementations of the pseudo-fermion functional renormalization group (pf-FRG), which has originally been proposed by Reuther and Wölfle in the context of two-dimensional frustrated quantum magnetism with highly symmetric spin interactions [Reuther and Wölfle, Phys. Rev. B 81, 144410 (2010)]. Reflecting the growing interest in models with SU(2) symmetry-breaking spin exchange terms that has followed the unearthing of Kitaev's honeycomb model, we present a generalized implementation of the pf-FRG which is suited to numerically simulate arbitrary microscopic models with diagonal or off-diagonal two-spin interactions, even in three-dimensional frustrated quantum magnets. We provide insight into the inner workings of the method which has emerged over the course of the last couple of years, arguing that the pf-FRG formalism simultaneously combines aspects of a large-S expansion as well as a large-N expansion on equal footing, thus being able to resolve the subtle interplay between magnetic ordering tendencies and disruptive quantum fluctuations. \n \nMoreover, on a case by case basis we explore the stability of quantum spin liquids in paradigmatic models of frustrated quantum magnetism and elucidate the joint action of geometric frustration, exchange frustration, and quantum fluctuations to inhibit the formation of magnetic long-range order. Examples include: (i) a Heisenberg spin model on the three-dimensional diamond lattice where geometric frustration arises from antiferromagnetic next-nearest neighbor interactions. The additional competition with nearest neighbor interactions leads to an unusually large ground state degeneracy already on classical level. The quantum-to-classical transition is studied by systematically varying the spin length and the different roles of quantum fluctuations and thermal fluctuations are discussed. The theory is applied to interpret experiments on the spin-liquid candidate NiRh2O4; (ii) a Heisenberg antiferromagnet on the face centered cubic lattice which is augmented by competing bond-directional interactions, evolving around the concurrent manifestation of geometric frustration and exchange frustration. A situation is identified where both mechanisms collude to give rise to an unusually large degree of frustration over a wide parameter regime. We discuss related experimental findings on the highly frustrated iridate compound Ba2CeIrO6; (iii) a Heisenberg antiferromagnet on the kagome lattice with additional Dzyaloshinskii-Moriya interactions. The robustness of the unperturbed kagome antiferromagnet's spin liquid ground state against low-symmetry Dzyaloshinskii-Moriya interactions is investigated. Implications for the spin-liquid candidate herbertsmithite are discussed.

•Jan 1, 2019•9 citations
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 \ensuremathΓ\mathrm{\ensuremath{\Gamma}}\ensuremathΓ interaction on the hyperhoneycomb lattice by using exact diagonalization (ED). An ED cluster was chosen to satisfy a hidden SU(2) symmetry when K=\ensuremathΓK=\mathrm{\ensuremath{\Gamma}}K=\ensuremathΓ 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 \ensuremathΓ\mathrm{\ensuremath{\Gamma}}\ensuremathΓ are negative, a small positive Heisenberg interaction favors the (1,1,0) direction consistent with the reported spiral order in \ensuremathβ\ensuremath−Li2IrO3\ensuremath{\beta}\ensuremath{-}{\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}\ensuremathβ\ensuremath−Li2​IrO3​. 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.
Counter-rotating spiral, zigzag, and 120∘^\circ∘ orders from coupled-chain analysis of Kitaev-Gamma-Heisenberg model, and relations to honeycomb iridates
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 recently discovered three-dimensional hyperhoneycomb iridate, β -Li 2 IrO 3 , has raised hopes for the realization of the dominant Kitaev interaction between spin-orbit entangled local moments due to its near-ideal lattice structure. If true, this material may lie close to the sought-after quantum spin-liquid phase in three dimensions. Utilizing ab initio electronic structure calculations, we first show that the spin-orbit entangled basis, , correctly captures the low-energy electronic structure. The effective spin model derived in the strong-coupling limit supplemented by the ab initio results is shown to be dominated by the Kitaev interaction. We demonstrated that the possible range of parameters is consistent with a non-coplanar spiral magnetic order found in a recent experiment. All of these analyses suggest that β -Li 2 IrO 3 may be the closest among known materials to the Kitaev spin-liquid regime.
The Kitaev model, which hosts a quantum spin liquid (QSL) in the ground state, was originally defined on a two-dimensional honeycomb lattice, but can be straightforwardly extended to any tricoordinate lattices in any spatial dimensions. In particular, the three-dimensional (3D) extensions are of interest as a realization of 3D QSLs, and some materials like βββ-Li2_{2}2​IrO3_{3}3​, γγγ-Li2_22​IrO3_33​, and βββ-ZnIrO3_{3}3​ were proposed for the candidates. However, the phase diagrams of the models for those candidates have not been fully elucidated, mainly due to the limitation of numerical methods for 3D frustrated quantum spin systems. Here we study the Kitaev-Heisenberg model defined on a 3D hyperhoneycomb lattice, by using the pseudofermion functional renormalization group method. We show that the ground-state phase diagram contains the QSL phases in the vicinities of the pristine ferromagnetic and antiferromagnetic Kitaev models, in addition to four magnetically ordered phases, similar to the two-dimensional honeycomb case. Our results respect the four-sublattice symmetry inherent in the model, which was violated in the previous study. Moreover, we also show how the phase diagram changes with the anisotropy in the interactions. The results provide a reference for the search of the hyperhoneycomb Kitaev materials.
Motivated by the recent synthesis of two insulating Li2IrO3{\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}Li2​IrO3​ polymorphs, where Ir4+{\mathrm{Ir}}^{4+}Ir4+ Seff=1/2{S}_{\mathrm{eff}}=1/2Seff​=1/2 moments form 3D (``harmonic'') honeycomb structures with threefold coordination, we study magnetic Hamiltonians on the resulting $\ensuremath{\beta}\text{\ensuremath{-}}{\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}$ hyperhoneycomb lattice and $\ensuremath{\gamma}\text{\ensuremath{-}}{\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}$ stripyhoneycomb lattice. Experimentally measured magnetic susceptibilities suggest that Kitaev interactions, predicted for the ideal ${90}^{\ensuremath{\circ}}$ Ir-O-Ir bonds, are sizable in these materials. We first consider pure Kitaev interactions, which lead to an exactly soluble 3D quantum spin liquid (QSL) with emergent Majorana fermions and Z2{\mathrm{Z}}_{2}Z2​ flux loops. Unlike 2D QSLs, the 3D QSL is stable to finite temperature, with ${T}_{c}\ensuremath{\approx}|K|/100$. On including Heisenberg couplings, exact solubility is lost. However, by noting that the shortest closed loop $\ensuremath{\ell}$ is relatively large in these structures, we construct an $\ensuremath{\ell}\ensuremath{\rightarrow}\ensuremath{\infty}$ approximation by defining the model on the Bethe lattice. The phase diagram of the Kitaev-Heisenberg model on this lattice is obtained directly in the thermodynamic limit, using tensor network states and the infinite-system time-evolving-block-decimation (iTEBD) algorithm. Both magnetically ordered and gapped QSL phases are found, the latter being identified by an entanglement fingerprint.
Recent experiments reveal that the honeycomb ruthenium trichloride $\ensuremath{\alpha}\ensuremath{-}\mathrm{RuC}{\mathrm{l}}_{3}$ is a prime candidate of the Kitaev quantum spin liquid (QSL). However, there is no theoretical model which can properly describe its experimental dynamical response due to the lack of a full understanding of its magnetic interactions. Here, we propose a general scheme to calculate the magnetic interactions in systems (e.g., $\ensuremath{\alpha}\ensuremath{-}\mathrm{RuC}{\mathrm{l}}_{3}$) with nonnegligible orbital moments by constraining the directions of orbital moments. With this scheme, we put forward a minimal ${J}_{1}\ensuremath{-}{K}_{1}\ensuremath{-}{\mathrm{\ensuremath{\Gamma}}}_{1}\ensuremath{-}{J}_{3}\ensuremath{-}{K}_{3}$ model for $\ensuremath{\alpha}\ensuremath{-}\mathrm{RuC}{\mathrm{l}}_{3}$ and find that: (I) The third nearest neighbor (NN) antiferromagnetic Heisenberg interaction J3{J}_{3}J3​ stabilizes the zigzag antiferromagnetic order; (II) The NN symmetric off-diagonal exchange ${\mathrm{\ensuremath{\Gamma}}}_{1}$ plays a pivotal role in determining the preferred direction of magnetic moments and generating the spin wave gap. An exact diagonalization study on this model shows that the Kitaev QSL can be realized by suppressing the NN symmetric off-diagonal exchange ${\mathrm{\ensuremath{\Gamma}}}_{1}$ and the third NN Heisenberg interaction J3{J}_{3}J3​. Thus, we not only propose a powerful general scheme for investigating the intriguing magnetism of Jeff=1/2{J}_{\mathrm{eff}}=1/2Jeff​=1/2 magnets, but also point out future directions for realizing the Kitaev QSL in the honeycomb ruthenium trichloride $\ensuremath{\alpha}\ensuremath{-}\mathrm{RuC}{\mathrm{l}}_{3}$.
When sizable quantum fluctuations and strong frustration mechanisms act in concert to repel the formation of conventional long-range order in quantum magnets, they can make way for massively entangled spin liquid phases which may imbue the material with extraordinary properties. \nThe search for such curious phases of matter has proceeded for several decades, gaining extra momentum some fifteen years ago when Kitaev proposed an analytically solvable model for a quantum spin liquid with anyonic excitations on the basis of realistic microscopic spin exchange terms [Kitaev, Annals of Physics 321, 2 (2006)]. Attempts to identify different models or materials which harbor quantum spin liquid ground states have kept researchers -- experimentalists and theorists alike -- in suspense ever since. However, the simulation of quantum many-body systems poses a serious challenge even to modern numerical techniques, particularly in the case of frustrated quantum magnetism in three spatial dimensions. Such models evade tractability by many established approaches, leaving a methodological void. \n \nIn this thesis, we report on recent progress in cutting-edge implementations of the pseudo-fermion functional renormalization group (pf-FRG), which has originally been proposed by Reuther and Wölfle in the context of two-dimensional frustrated quantum magnetism with highly symmetric spin interactions [Reuther and Wölfle, Phys. Rev. B 81, 144410 (2010)]. Reflecting the growing interest in models with SU(2) symmetry-breaking spin exchange terms that has followed the unearthing of Kitaev's honeycomb model, we present a generalized implementation of the pf-FRG which is suited to numerically simulate arbitrary microscopic models with diagonal or off-diagonal two-spin interactions, even in three-dimensional frustrated quantum magnets. We provide insight into the inner workings of the method which has emerged over the course of the last couple of years, arguing that the pf-FRG formalism simultaneously combines aspects of a large-S expansion as well as a large-N expansion on equal footing, thus being able to resolve the subtle interplay between magnetic ordering tendencies and disruptive quantum fluctuations. \n \nMoreover, on a case by case basis we explore the stability of quantum spin liquids in paradigmatic models of frustrated quantum magnetism and elucidate the joint action of geometric frustration, exchange frustration, and quantum fluctuations to inhibit the formation of magnetic long-range order. Examples include: (i) a Heisenberg spin model on the three-dimensional diamond lattice where geometric frustration arises from antiferromagnetic next-nearest neighbor interactions. The additional competition with nearest neighbor interactions leads to an unusually large ground state degeneracy already on classical level. The quantum-to-classical transition is studied by systematically varying the spin length and the different roles of quantum fluctuations and thermal fluctuations are discussed. The theory is applied to interpret experiments on the spin-liquid candidate NiRh2O4; (ii) a Heisenberg antiferromagnet on the face centered cubic lattice which is augmented by competing bond-directional interactions, evolving around the concurrent manifestation of geometric frustration and exchange frustration. A situation is identified where both mechanisms collude to give rise to an unusually large degree of frustration over a wide parameter regime. We discuss related experimental findings on the highly frustrated iridate compound Ba2CeIrO6; (iii) a Heisenberg antiferromagnet on the kagome lattice with additional Dzyaloshinskii-Moriya interactions. The robustness of the unperturbed kagome antiferromagnet's spin liquid ground state against low-symmetry Dzyaloshinskii-Moriya interactions is investigated. Implications for the spin-liquid candidate herbertsmithite are discussed.