Abstract
Due to the combination of a substantial spinorbit coupling and correlation effects, iridium oxides hold a prominent place in the search for novel quantum states of matter, including, e.g., Kitaev spin liquids and topological Weyl states. We establish the promise of the very recently synthesized hyperhoneycomb iridate βLi_{2}IrO_{3} in this regard. A detailed theoretical analysis reveals the presence of large ferromagnetic firstneighbor Kitaev interactions, while a secondneighbor antiferromagnetic Heisenberg exchange drives the ground state from ferro to zigzag order via a threedimensional Kitaev spin liquid and an incommensurate phase. Experiment puts the system in the latter regime but the Kitaev spin liquid is very close and reachable by a slight modification of the ratio between the second and firstneighbor couplings, for instance via strain.
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Introduction
In magnetism, frustration refers to the existence of competing exchange interactions that cannot be simultaneously satisfied. Such effects can spawn new states of matter with quite exotic physical properties. Most famous in this regard are the different kinds of quantum spin liquids (QSL’s) that emerge from frustrated spin couplings^{1}. In these collective states of matter quantum fluctuations are so strong that they disorder the spins even at the lowest temperatures. The types of QSL states that then emerge range from chiral ones^{2,3} to Z_{2} topological spin liquids^{4,5,6} carrying fractionalized excitations. Both experimentally and theoretically such QSL’s have been observed and intensely studied in twodimensional (2D) systems^{1,2,3,4,5,6,7,8,9}. How this situation carries over to three spatial dimensions (3D), in which tendencies towards formation of longrange ordered magnetic states are in principle stronger and the disordering effect of quantum fluctuations therefore less potent, is largely unexplored. This is not only due to the limitations of theoretical and numerical approaches in 3D but also to the the sparsity of relevant candidate materials^{10}. Very recently the latter however fundamentally changed through the synthesis of insulating Li_{2}IrO_{3} polymorphs^{11,12,13} in which the magnetic moments of Ir^{4+} ions form 3D honeycomb structures with threefold coordination. Here we concentrate on the βLi_{2}IrO_{3} polymorph, which forms a socalled hyperhoneycomb lattice, see Fig. 1. Such a lattice might in principle support a 3D Kitaev spin liquid^{14,15,16,17}, a direct counterpart of its lowerdimensional, 2D equivalent^{18,19,20}.
The 2D KitaevHeisenberg model on the honeycomb lattice is characterised by the presence of large uniaxial symmetric magnetic couplings that cyclically permute on the bonds of a given hexagonal ring^{18,19,20}. A QSL phase is present in this model if the ratio between the Kitaev interaction K and Heisenberg coupling J is larger than 8^{20}. Quasi2D honeycomb compounds initially put forward for the experimental realization of the KitaevHeisenberg Hamiltonian are 5d^{5} and 4d^{5} j ≈ 1/2 systems^{19,21} such as Na_{2}IrO_{3}, αLi_{2}IrO_{3} and Li_{2}RhO_{3}. Subsequent measurements evidenced, however, either antiferromagnetically ordered^{22,23,24,25} or spinglass^{26} ground states in these materials.
The three factors that complicate a straightforward materialisation of the Kitaev QSL ground state in the quasi2D honeycomb compounds are the presence of (i) appreciable additional exchange anisotropies^{27,28,29}, (ii) two crystallographically inequivalent IrIr bonds and (iii) longerrange magnetic interactions between second and thirdneighbor iridium moments^{22,23,27,30,31}. These additional interactions push quasi2D Na_{2}IrO_{3} and αLi_{2}IrO_{3} towards the formation of longrange antiferromagnetic (AF) order at temperatures below 15 K. Also the 3D honeycomb system βLi_{2}IrO_{3} orders magnetically: at 38 K the spins form an incommensurate (IC) ordering pattern^{32} with strong ferromagnetic (FM) correlations^{11}. Apparently additional interactions beyond only the nearestneighbor (NN) Kitaev and Heisenberg ones are relevant also in the 3D system. This leaves two main challenges: first, one would like to precisely quantify the different magnetic exchange interactions between the Ir moments and second, one should like to determine how far away the magnetic ground state is from a Kitaevtype 3D QSL. Here we meet these challenges through a combination of ab initio quantum chemistry calculations by which we determine the NN magnetic couplings in βLi_{2}IrO_{3} and exact diagonalization (ED) of the resulting effective spin Hamiltonian, on large clusters, to determine how far βLi_{2}IrO_{3} is situated from the QSL ground state in the magnetic phase diagram.
The ab initio results show that the NN exchange in βLi_{2}IrO_{3} is mostly FM, with relatively weak FM Heisenberg couplings of a few meV, large FM Kitaev interactions in the range of 10–15 meV and additional anisotropies not included in the plain KitaevHeisenberg model. The sign and magnitude of secondneighbor Heisenberg couplings we determine from fits of the ED calculations to the experimental magnetization data. This secondneighbor effective coupling comes out as J_{2} ≈ 0.2–0.3 meV and is thus small and AF. Remarkably, this AF J_{2} stabilizes an IC magnetic structure that puts the system to be only a jot apart from the transition to a QSL ground state. Our findings provide strong theoretical motivation for further investigations on the material preparation side. The Kitaev QSL phase might be achieved by for instance epitaxial strain and relaxation in βLi_{2}IrO_{3} thin films, slightly modifying the J_{2}/K ratio.
Results
Quantum Chemistry Calculations
Quantum chemistry calculations were first performed for the onsite dd excitations, on embedded clusters consisting of one central octahedron and the three adjacent octahedra (for technical details, see Supplementary Information (SI) and ref. 33). Reference completeactivespace (CAS) multiconfigurational wave functions^{34} were in this case generated with an active orbital space defined by the five 5d functions at the central Ir site. While all possible occupations are allowed within the set of Ir 5d orbitals, double occupancy is imposed in the CAS calculations on the O 2p levels and other lowerenergy orbitals. The selfconsistent optimization was here carried out for an average of four states, i.e., and the states of maximum spin multiplicity associated with each of the and configurations. We then subsequently performed multireference configurationinteraction (MRCI) calculations^{34} with single and double excitations out of the Ir 5d and O 2p shells at the central octahedron. MRCI relative energies, without and with spinorbit coupling (SOC), are listed in Table 1.
Due to slight distortion of the O cage^{11} and possibly anisotropic fields associated with the extended surroundings, the degeneracy of the Ir t_{2g} levels is lifted. Without SOC, the Ir states are spread over an energy window of ≈0.1 eV (see Table 1). Similar results were earlier reported for the quasi2D honeycomb iridates^{33}. The lowsymmetry fields additionaly remove the degeneracy of the j = 3/2 spinorbit quartet. With orbitals optimized for an average of 5d^{5} states, i.e., , , and , the j = 3/2like components lie at 0.82 and 0.86 eV above the j ≈ 1/2 doublet, by MRCI + SOC computations (see Table 1). If the reference active space in the prior CAS selfconsistentfield (CASSCF) calculation^{34} is restricted to only three (t_{2g}) orbitals and five electrons, the relative energies of the j ≈ 3/2 components in the subsequent MRCI + SOC treatment are somewhat lower, 0.69 and 0.73 eV. The Ir t_{2g} to e_{g} transitions require excitation energies of at least 3 eV according to the MRCI data in Table 1, similar to values computed for αLi_{2}IrO_{3}^{33}.
While the quantum chemistry results for the onsite excitations in βLi_{2}IrO_{3} resemble very much the data for the quasi2D honeycomb iridates, the computed intersite effective interactions show significant differences. The latter were estimated by MRCI + SOC calculations for embedded fragments having two edgesharing IrO_{6} octahedra in the active region. As detailed in earlier work^{27,35,36}, the ab initio quantum chemistry data for the lowest four spinorbit states describing the magnetic spectrum of two NN octahedra is mapped in our scheme onto an effective spin Hamiltonian including both isotropic Heisenberg exchange and symmetric anisotropies. Yet the spinorbit calculations, CASSCF or MRCI, incorporate all nine triplet and nine singlet states that arise from the twoIrsite – configuration (see SI). The MRCI treatment includes the Ir 5d electrons and the O 2p electrons at the two bridging ligand sites.
MRCI + SOC results for the NN effective couplings are listed in Table 2. The two, structurally different sets of IrIr links are labeled B1 and B2, see Fig. 1. For each of those, the O ions are distributed around the Ir sites such that the IrOIr bond angles deviate significantly from 90°. While the B1 links display effective D_{2} pointgroup symmetry (the effective symmetry of a block of two NN octahedra is dictated not only by the precise arrangement of the O ions coordinating the two magnetically active Ir sites but also by the symmetry of the extended surroundings), the B2 bonds possess C_{i} symmetry, slightly away from C_{2h} due to small differences between the IrO bond lengths on the Ir_{2}O_{2} plaquette of two Ir ions and two bridging ligands (2.025 vs 2.023 Å^{11}). The absence of an inversion center allows a nonzero antisymmetric exchange on the B1 links. However, our analysis shows this antisymmetric DzyaloshinskiiMoriya coupling is the smallest effective parameter in the problem — two orders of magnitude smaller than the dominant NN interactions, i.e., the Kitaev exchange. On this basis and further symmetry considerations (see the discussion in refs 27, 35, 36, 37), the effective spin Hamiltonian for the B1 links is assumed D_{2h}like and in the local Kitaev reference frame (with the z axis perpendicular to the Ir_{2}O_{2} plaquette and x, y within the plane of the plaquette^{19,27}) it reads
where and are pseudospin 1/2 operators, K defines the Kitaev component and Γ_{xy} is the only nonzero offdiagonal coupling of the symmetric anisotropic tensor.
For the B2 units of edgesharing IrO_{6} octahedra, the effective spin Hamiltonian reads in the local Kitaev coordinate frame as
We find for the B2 links that slight distortions lowering the bond symmetry from C_{2h} to C_{i} have minor effects on the computed wave functions and the quantum chemistry data can be safely mapped onto a C_{2h} model. For C_{2h} symmetry, the elements of the symmetric anisotropic tensor are such that Γ_{zx} = −Γ_{yz}.
The wave functions for the lowlying four states in the twoIrsite problem can be conveniently expressed in terms of 1/2 pseudospins as in Table 2. In D_{2} symmetry (B1 links) these pseudospin wave functions, singlet Φ_{S} and triplet Φ_{1}, Φ_{2}, Φ_{3}, transform according to the A_{u}, B_{2}, B_{1} and A_{u} irreducible representations, respectively. For (nearly) C_{2h} symmetry (B2 links), Φ_{S}, Φ_{1}, Φ_{2} and Φ_{3} transform according to A_{g}, B_{u}, B_{u} and A_{u}, respectively. The amount of Φ_{S}–Φ_{3} (B1) and Φ_{1}–Φ_{2} (B2) mixing (see Table 2) is determined by analysis of the “full” spinorbit wave functions obtained in the quantum chemistry calculations.
As seen in Table 2, for each set of IrIr links in βLi_{2}IrO_{3}, B1 and B2, both J and K are FM. In contrast, J is AF for all pairs of Ir NN’s in honeycomb Na_{2}IrO_{3}^{27} and features different signs for the two types of IrIr links in αLi_{2}IrO_{3}^{35}. The Kitaev exchange, on the other hand, is found to be large and FM in all 213 compounds, see Table 2 and refs 27 and 35. In addition to the Kitaev coupling, sizable offdiagonal symmetric anisotropic interactions are predicted. In βLi_{2}IrO_{3}, these are FM for the B1 bonds and show up with both + and − signs for the B2 links (the sign of these terms is with respect to the local Kitaev reference frame), see Table 2.
Magnetic Phase Diagram
Having established the nature and the magnitude of the NN effective spin couplings, we now turn to the magnetic phase diagram of βLi_{2}IrO_{3}. In addition to the NN MRCI + SOC data of Table 2, we have to take into account explicitly the secondneighbor Heisenberg interactions. Due to the 3D nature of the iridium lattice, with alternate rotation of two adjacent B2 bonds around the B1 link with which both share an Ir ion, one can safely assume that the thirdneighbor exchange is vanishingly small. Results of ED calculations for an extended (pseudo)spin Hamiltonian including the MRCI NN interactions and a variable secondneighbor Heisenberg coupling parameter J_{2} are shown in Fig. 2. Different types of clusters were considered, with either 16, 20 or 24 Ir sites. The 24site cluster used in ED calculations with periodic boundary conditions is displayed in Fig. 2(a) while the structure of the smaller clusters is detailed in SI.
In order to investigate the magnetic properties of βLi_{2}IrO_{3}, we calculated the static spinstructure factor along two paths denoted as θ (bcdiagonal) and ϕ (abdiagonal) in Fig. 2(a), where the distance between neighboring B1 bonds is taken as 1. The results for several J_{2} values with the 24site cluster are plotted in Fig. 2(b). The propagation vector for each path , determined as the wave number q providing a maximum of S(q), is plotted in Fig. 2(c). For J_{2} = 0 the ground state is characterized by longrange FM order, i.e., , consistent with a previous classical Monte Carlo study^{38,39}. Given the strong FM character of the NN exchange, ground states different from FM order are only obtained for finite AF J_{2}. With increasing strength of the AF J_{2}, q_{θ} develops finite values starting at J_{2} = J_{2,c1} and reaches π at J_{2} = J_{2,c2} whereas q_{ϕ} is finite but small in the range J_{2,c1} < J_{2} < J_{2,c2} and zero otherwise. This evidences two magnetic phase transitions, from FM to IC order and further to a commensurate ground state. The latter commensurate structure corresponds to zigzag AF order, a schematic picture of which is shown in Fig. 2(d). The ED results for the four different types of periodic clusters are here in good overall agreement, as shown in Fig. 2(e). Some differences arise only with respect to the precise position of the critical points.
An intriguing feature is the appearance of a SL state in between the FM and IC phases. Since the total spin 2 S/N falls off rapidly and continuously near J_{2} = J_{2,c1} [see Fig. 2(c)], the FM ground state is expected to change into SL before reaching the IC regime. It can be confirmed by a structureless static spinstructure factor, like nearly flat qdependence of S(q) at J_{2} = 0.65 in Fig. 2(b). In Fig. 2(e) we also provide the critical values marking the transition between the FM and SL states. This was estimated as the point where any of the expectation values turn negative, which implies a collapse of longrange FM order. Importantly, we find that the SL phase shows up in each of the four different types of periodic clusters. A more detailed analysis of the spinspin correlations is provided in SI.
Discussion
Typically, a commensuratetoIC transition critical point tends to be overestimated by using periodicity. For estimating more precisely the critical J_{2} values we therefore additionally studied clusters with open boundary conditions along the c direction. Also, for a direct comparison between our ED results and the experimentally observed magnetic structure, we introduce an additional path δ (acdiagonal), sketched in Fig. 3(a). The size of the cluster along a and b has insignificant effect on the computed critical J_{2} values because (abdiagonal) is either zero, around the critical points (periodic 24site cluster), or very small, in the IC phase (periodic 16 and 20site clusters), as seen in Fig. 2(c).
The value of the propagation vector along the δpath is shown in Fig. 3(b) as function of J_{2} for various cluster “lengths” in the c direction. The inset displays a finitesize scaling analysis for the critical values. In the infinitelength limit, we find J_{2,c1} = 0.02 and J_{2,c2} = 1.43 meV. The corresponding phase diagram is provided in Fig. 3(c). Similar critical points, i.e., J_{2,c1} = 0.02 and J_{2,c2} = 1.48 meV, are obtained for (see SI).
As shown in Fig. 3(b), the dropdown of 2 S/N near J_{2} = J_{2,c1} is more clearly seen than in the case of periodic clusters because the formation of IC order is not hindereded for open clusters. Defining the FMSL J_{2,c1} critical value as the point where turns negative for any (i, j) pair, the SL phase in the vicinity of J_{2} ≈ J_{2,c1} = 0.02 meV would have a width of about 0.01 J_{2}. In other words, a very tiny FM J_{2} coupling may drive the system from FM order to a SL state. With further increasing J_{2}, the system goes through an IC phase to AF zigzag order at J_{2} = 1.43 meV.
To finally determine the value of J_{2} in βLi_{2}IrO_{3}, we fitted the magnetization curve obtained by ED calculations at T = 0 K [see Fig. 3(d)] to the experimental data at T = 5 K^{11}. Such an exercise yields J_{2} = 0.2–0.3 meV, i.e., J_{2} ≈ 0.1 J_{2,c2}, so that the system is relatively far from the instability to zigzag order but very close to the transition to the SL ground state. Since with increasing J_{2} the propagation vector of the IC phase increases smoothly from that of the SL to that of the zigzag state , longwavelength IC order with a small propagation vector is expected for βLi_{2}IrO_{3}. By performing a finitesize scaling analysis of at J_{2} = J_{2,c1}(N) + 0.28 meV, we obtain for J_{2} = 0.3 meV in the infinitelength limit. An experimentbased estimate for can be extracted from recent magnetic resonant xray diffraction data^{32} [see Fig. 3(f)]; the spins on sites A and B (their distance is three lattice spacings) have almost opposite directions, which leads to . That fits reasonably well our theoretical estimate. The stabilization of an IC state by J_{2} couplings has been previously discussed for 1D zigzag chains like the path we label here as δ in ref. 40.
The value extracted for J_{2} from our fit of the magnetization data is thus within our theoretical framework fully consistent with the experimentally observed IC magnetic order in βLi_{2}IrO_{3}. Nevertheless we find that the system is remarkably close to a threedimensional spinliquid ground state, which can be reached by a minute change of ~0.25 meV, an energy scale that corresponds to about 3 K, in the secondneighbour exchange parameter J_{2}. Changes of this order of magnitude can easily be induced by pressure or strain.
Additional Information
How to cite this article: Katukuri, V. M. et al. The vicinity of hyperhoneycomb βLi_{2}IrO_{3} to a threedimensional Kitaev spin liquid state. Sci. Rep. 6, 29585; doi: 10.1038/srep29585 (2016).
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Acknowledgements
We thank N. Bogdanov for helpful discussions. L.H. and S.N. acknowledges financial support from the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG — HO4427 and SFB1143). J.v.d.B. acknowledges support from the HarvardMIT CUA. Part of the calculations have been performed using the facilities of the Center for Information Services and High Performance Computing (ZIH) of the Technical University of Dresden.
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V.M.K. carried out the ab initio calculations and subsequent mapping of the ab initio results onto the effective spin Hamiltonian, with assistance from R.Y. and L.H. S.N. performed the exactdiagonalization calculations to obtain the magnetic phase diagram. V.M.K., S.N., J.v.d.B. and L.H. analyzed the data and wrote the paper, with contributions from all other coauthors.
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Katukuri, V., Yadav, R., Hozoi, L. et al. The vicinity of hyperhoneycomb βLi_{2}IrO_{3} to a threedimensional Kitaev spin liquid state. Sci Rep 6, 29585 (2016). https://doi.org/10.1038/srep29585
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DOI: https://doi.org/10.1038/srep29585
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