The vicinity of hyper-honeycomb β-Li₂IrO₃ to a three-dimensional Kitaev spin liquid state

Abstract

Due to the combination of a substantial spin-orbit 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 hyper-honeycomb iridate β-Li₂IrO₃ in this regard. A detailed theoretical analysis reveals the presence of large ferromagnetic first-neighbor Kitaev interactions, while a second-neighbor antiferromagnetic Heisenberg exchange drives the ground state from ferro to zigzag order via a three-dimensional 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 first-neighbor couplings, for instance via strain.

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 couplings¹. 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 Z₂ topological spin liquids^4,5,6 carrying fractionalized excitations. Both experimentally and theoretically such QSL’s have been observed and intensely studied in two-dimensional (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 long-range 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 materials¹⁰. Very recently the latter however fundamentally changed through the synthesis of insulating Li₂IrO₃ polymorphs^11,12,13 in which the magnetic moments of Ir⁴⁺ ions form 3D honeycomb structures with threefold coordination. Here we concentrate on the β-Li₂IrO₃ polymorph, which forms a so-called hyper-honeycomb 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 lower-dimensional, 2D equivalent^18,19,20.

Figure 1

(a) Ir hyper-honeycomb lattice of β-Li₂IrO₃. The Ir-Ir links along the c axis, associated with equilateral Ir₂O₂ plaquettes¹¹ and labeled B1, are shown in four different colors. B1 links located at 0.125, 0.375, 0.625 and 0.875 on the a axis are shown in green, yellow, orange and red color, respectively. B2 bonds connecting the B1 links are shown in dual colors. O ions around four of the Ir sites are also shown. (b) Projection of the unit cell on the bc plane.

The 2D Kitaev-Heisenberg 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 8²⁰. Quasi-2D honeycomb compounds initially put forward for the experimental realization of the Kitaev-Heisenberg Hamiltonian are 5d⁵ and 4d⁵ j ≈ 1/2 systems^19,21 such as Na₂IrO₃, α-Li₂IrO₃ and Li₂RhO₃. Subsequent measurements evidenced, however, either antiferromagnetically ordered^22,23,24,25 or spin-glass²⁶ ground states in these materials.

The three factors that complicate a straightforward materialisation of the Kitaev QSL ground state in the quasi-2D honeycomb compounds are the presence of (i) appreciable additional exchange anisotropies^27,28,29, (ii) two crystallographically inequivalent Ir-Ir bonds and (iii) longer-range magnetic interactions between second- and third-neighbor iridium moments^{22,23,27,30,31}. These additional interactions push quasi-2D Na₂IrO₃ and α-Li₂IrO₃ towards the formation of long-range antiferromagnetic (AF) order at temperatures below 15 K. Also the 3D honeycomb system β-Li₂IrO₃ orders magnetically: at 38 K the spins form an incommensurate (IC) ordering pattern³² with strong ferromagnetic (FM) correlations¹¹. Apparently additional interactions beyond only the nearest-neighbor (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 Kitaev-type 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 β-Li₂IrO₃ and exact diagonalization (ED) of the resulting effective spin Hamiltonian, on large clusters, to determine how far β-Li₂IrO₃ is situated from the QSL ground state in the magnetic phase diagram.

The ab initio results show that the NN exchange in β-Li₂IrO₃ 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 Kitaev-Heisenberg model. The sign and magnitude of second-neighbor Heisenberg couplings we determine from fits of the ED calculations to the experimental magnetization data. This second-neighbor effective coupling comes out as J₂ ≈ 0.2–0.3 meV and is thus small and AF. Remarkably, this AF J₂ 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 β-Li₂IrO₃ thin films, slightly modifying the J₂/K ratio.

Results

Quantum Chemistry Calculations

Quantum chemistry calculations were first performed for the on-site d-d 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 complete-active-space (CAS) multiconfigurational wave functions³⁴ 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 lower-energy orbitals. The self-consistent 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 configuration-interaction (MRCI) calculations³⁴ with single and double excitations out of the Ir 5d and O 2p shells at the central octahedron. MRCI relative energies, without and with spin-orbit coupling (SOC), are listed in Table 1.

Table 1 Ir⁴⁺ 5d⁵ multiplet structure in β-Li₂IrO₃, all numbers in eV.

Due to slight distortion of the O cage¹¹ 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 quasi-2D honeycomb iridates³³. The low-symmetry fields additionaly remove the degeneracy of the j = 3/2 spin-orbit quartet. With orbitals optimized for an average of 5d⁵ states, i.e., , , and , the j = 3/2-like 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 self-consistent-field (CASSCF) calculation³⁴ 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 α-Li₂IrO₃³³.

While the quantum chemistry results for the on-site excitations in β-Li₂IrO₃ resemble very much the data for the quasi-2D honeycomb iridates, the computed intersite effective interactions show significant differences. The latter were estimated by MRCI + SOC calculations for embedded fragments having two edge-sharing IrO₆ octahedra in the active region. As detailed in earlier work^27,35,36, the ab initio quantum chemistry data for the lowest four spin-orbit 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 spin-orbit calculations, CASSCF or MRCI, incorporate all nine triplet and nine singlet states that arise from the two-Ir-site – 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 Ir-Ir 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 Ir-O-Ir bond angles deviate significantly from 90°. While the B1 links display effective D₂ point-group 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 Ir-O bond lengths on the Ir₂O₂ plaquette of two Ir ions and two bridging ligands (2.025 vs 2.023 Ź¹). The absence of an inversion center allows a nonzero antisymmetric exchange on the B1 links. However, our analysis shows this antisymmetric Dzyaloshinskii-Moriya 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 Ir₂O₂ plaquette and x, y within the plane of the plaquette^19,27) it reads

Table 2 MRCI splittings among the four low-lying magnetic states and effective exchange couplings (meV) for two NN IrO₆ octahedra in β-Li₂IrO₃.

where and are pseudospin 1/2 operators, K defines the Kitaev component and Γ_xy is the only non-zero off-diagonal coupling of the symmetric anisotropic tensor.

For the B2 units of edge-sharing IrO₆ 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 low-lying four states in the two-Ir-site problem can be conveniently expressed in terms of 1/2 pseudospins as in Table 2. In D₂ symmetry (B1 links) these pseudospin wave functions, singlet Φ_S and triplet Φ₁, Φ₂, Φ₃, transform according to the A_u, B₂, B₁ and A_u irreducible representations, respectively. For (nearly) C_2h symmetry (B2 links), Φ_S, Φ₁, Φ₂ and Φ₃ transform according to A_g, B_u, B_u and A_u, respectively. The amount of Φ_S–Φ₃ (B1) and Φ₁–Φ₂ (B2) mixing (see Table 2) is determined by analysis of the “full” spin-orbit wave functions obtained in the quantum chemistry calculations.

As seen in Table 2, for each set of Ir-Ir links in β-Li₂IrO₃, B1 and B2, both J and K are FM. In contrast, J is AF for all pairs of Ir NN’s in honeycomb Na₂IrO₃²⁷ and features different signs for the two types of Ir-Ir links in α-Li₂IrO₃³⁵. 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 off-diagonal symmetric anisotropic interactions are predicted. In β-Li₂IrO₃, 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 β-Li₂IrO₃. In addition to the NN MRCI + SOC data of Table 2, we have to take into account explicitly the second-neighbor 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 third-neighbor exchange is vanishingly small. Results of ED calculations for an extended (pseudo)spin Hamiltonian including the MRCI NN interactions and a variable second-neighbor Heisenberg coupling parameter J₂ are shown in Fig. 2. Different types of clusters were considered, with either 16, 20 or 24 Ir sites. The 24-site 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.

Figure 2

(a) Sketch of the 24-site “periodic” cluster. (b) Static spin-structure factor along paths θ and ϕ, see text. (c) Propagation vectors , and total spin 2 S/N for the periodic clusters, as functions of J₂. (d) AF zigzag order on the hyper-honeycomb lattice. (e) Magnetic phase diagrams obtained for the periodic clusters.

In order to investigate the magnetic properties of β-Li₂IrO₃, we calculated the static spin-structure factor along two paths denoted as θ (bc-diagonal) and ϕ (ab-diagonal) in Fig. 2(a), where the distance between neighboring B1 bonds is taken as 1. The results for several J₂ values with the 24-site 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 J₂ = 0 the ground state is characterized by long-range 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 J₂. With increasing strength of the AF J₂, q_θ develops finite values starting at J₂ = J_2,c1 and reaches π at J₂ = J_2,c2 whereas q_ϕ is finite but small in the range J_2,c1 < J₂ < 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 J₂ = 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 spin-structure factor, like nearly flat q-dependence of S(q) at J₂ = 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 long-range 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 spin-spin correlations is provided in SI.

Discussion

Typically, a commensurate-to-IC transition critical point tends to be overestimated by using periodicity. For estimating more precisely the critical J₂ 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 (ac-diagonal), sketched in Fig. 3(a). The size of the cluster along a and b has insignificant effect on the computed critical J₂ values because (ab-diagonal) is either zero, around the critical points (periodic 24-site cluster), or very small, in the IC phase (periodic 16- and 20-site clusters), as seen in Fig. 2(c).

Figure 3

(a) Sketch of the cluster with open boundaries along the c direction. (b) Propagation vector and total spin 2 S/N for our “open” clusters, as function of J₂. Inset: finite-scaling analysis of the critical points. (c) Magnetic phase diagrams obtained by ED. (d) Experimental (see ref. 11) and theoretical magnetization curves for β-Li₂IrO₃. The latter are obtained with either J₂ = 0.2 (periodic 16- and 20-site clusters) or J₂ = 0.3 meV (periodic 24-site cluster) and the NN MRCI couplings from Table 2. (e) Finite-scaling analysis of at J₂ = 0.3 meV using the open clusters. (f) Experimental results of the magnetic structure for β-Li₂IrO₃ (see ref. 32).

The value of the propagation vector along the δ-path is shown in Fig. 3(b) as function of J₂ for various cluster “lengths” in the c direction. The inset displays a finite-size scaling analysis for the critical values. In the infinite-length 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 J₂ = 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 FM-SL J_2,c1 critical value as the point where turns negative for any (i, j) pair, the SL phase in the vicinity of J₂ ≈ J_2,c1 = 0.02 meV would have a width of about 0.01 J₂. In other words, a very tiny FM J₂ coupling may drive the system from FM order to a SL state. With further increasing J₂, the system goes through an IC phase to AF zigzag order at J₂ = 1.43 meV.

To finally determine the value of J₂ in β-Li₂IrO₃, we fitted the magnetization curve obtained by ED calculations at T = 0 K [see Fig. 3(d)] to the experimental data at T = 5 K¹¹. Such an exercise yields J₂ = 0.2–0.3 meV, i.e., J₂ ≈ 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 J₂ the propagation vector of the IC phase increases smoothly from that of the SL to that of the zigzag state , long-wavelength IC order with a small propagation vector is expected for β-Li₂IrO₃. By performing a finite-size scaling analysis of at J₂ = J_2,c1(N) + 0.28 meV, we obtain for J₂ = 0.3 meV in the infinite-length limit. An experiment-based estimate for can be extracted from recent magnetic resonant x-ray diffraction data³² [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 J₂ couplings has been previously discussed for 1D zigzag chains like the path we label here as δ in ref. 40.

The value extracted for J₂ from our fit of the magnetization data is thus within our theoretical framework fully consistent with the experimentally observed IC magnetic order in β-Li₂IrO₃. Nevertheless we find that the system is remarkably close to a three-dimensional spin-liquid 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 second-neighbour exchange parameter J₂. Changes of this order of magnitude can easily be induced by pressure or strain.