Spin-orbit effects in Na{}_{4}Ir{}_{3}O{}_{8}, a hyper-kagomé  lattice antiferromagnet

# Spin-orbit effects in Na$_4$Ir$_3$O$_8$, a hyper-kagomé lattice antiferromagnet

## Abstract

We consider spin-orbit coupling effects in NaIrO, a material in which Ir spins form an hyper-kagomé  lattice, a three-dimensional network of corner-sharing triangles. We argue that both low temperature thermodynamic measurements and the impurity susceptibility induced by dilute substitution of Ti for Ir are suggestive of significant spin-orbit effects. Because of uncertainties in the crystal-field parameters, we consider two limits in which the spin-orbit coupling is either weak or strong compared to the non-cubic atomic splittings. A semi-microscopic calculation of the exchange Hamiltonian confirms that indeed large antisymmetric Dzyaloshinskii-Moriya (DM) and/or symmetric exchange anisotropy may be present. In the strong spin-orbit limit, the Ir-O-Ir superexchange contribution consists of unfrustrated strong symmetric exchange anisotropy, and we suggest that spin-liquid behavior is unlikely. In the weak spin-orbit limit, and for strong spin-orbit and direct Ir-Ir exchange, the Hamiltonian consists of Heisenberg and DM interactions. The DM coupling is parametrized by a three component DM vector (which must be determined empirically). For a range of orientation of this vector, frustration is relieved and an ordered state occurs. For other orientations, even the classical ground states are very complex. We perform spin-wave and exact diagonalization calculations which suggest the persistence of a quantum spin liquid in the latter regime. Applications to NaIrO and broader implications are discussed.

###### pacs:
75.10.-b,75.10.Jm,75.25.+z

## I Introduction

Geometrically frustrated antiferromagnetism is a rich subject enjoying considerable theoretical and experimental attention over several decades of research.Moessner (2001); Ramirez (1994) Such systems are realized by materials containing magnetic ions in which the strongest antiferromagnetic exchanges occur on a network of bonds containing many triangular units. The most celebrated examples are the two-dimensional kagomé (corner-sharing triangles) lattice and three-dimensional pyrochlore (corner-sharing tetrahedron) lattice. In ideal classical models, these lattices support highly degenerate ground states which prevent order down to very low temperature. Instead, the spins continue to fluctuate strongly despite significant correlations induced by the frustrated interactions. Systems in this regime are dubbed (classical) spin liquids, or cooperative paramagnets. A major goal in the field is to ascertain whether such spin liquids might also occur even in the zero temperature limit, in which both quantum effects and many non-ideal features of the materials must be taken into account. The answer to this question is quite subtle, due to many competing effects that can come into play. Quantum and thermal fluctuations may break the ground state degeneracy and actually induce magnetic order, an effect known as order-by-disorder.Gvozdikova and Zhitomirsky (2005); henley (1987); Villain et al. (1980); Bergman et al. (2007) This effect, however, is understood theoretically only in the large spin, limit, in which spins behave semi-classically. Nevertheless, some models even with the smallest possible spins, , seem at least qualitatively to follow the order-by-disorder scenario. Conversely, in other models with small spin, quantum spin liquids have been shown to occur. No general theory to predict which of these two tendencies obtains exists at present.

Despite this lack of theoretical discrimination, experimentalists have forged onward in recent years, uncovering a number of promising candidate quantum spin liquid materials with small spin on geometrically frustrated lattices. These include an organic magnet, -(ET)Cu(CN), containing spin- moments on a slightly spatially anisotropic triangular lattice, ZnCu(OH)Cl, an inorganic realization of a spatially isotropic spin- kagome antiferromagnet, and very recently the cubic material NaIrO  which realizes an hyper-kagomé antiferromagnet, in which spin- moments reside on a three-dimensional network of corner-sharing trianglesOkamoto et al. (2007) – see Fig. 1. None of these compounds exhibit indications of magnetic ordering. The interpretation of the first two materials, however, is complicated by the appearance of inhomogeneous magnetic moments at low temperature in -(ET)Cu(CN) Shimizu et al. (2006), and by fairly high levels of substitutional disorder (Zn for Cu) in ZnCu(OH)Cl. By contrast, the Ir moments are expected to be well ordered in NaIrO , due to the much larger ionic radius of Ir compared to Na and O.

Recent two worksHopkinson et al. (2007); Lawler et al. () assumed the nearest neighbor antiferromagnetic Heisenberg model for NaIrO. In Ref. Hopkinson et al., 2007, the authors treated the spin as a classical spin. By a large- mean field theory and classical Monte Carlo simulation, they found that the classical ground states are highly degenerate and a nematic order emerges at low temperatures in the Heisenberg model () via “order by disorder”, representing the dominance of coplanar spin configuration. In Ref. Lawler et al., , the authors presented a large- method and studied both the semi-classical spin and quantum spin regimes. In the semi-classical limit, they predicted that an unusual coplanar magnetically ordered ground state is stabilized with no local “weather vane” modes. While in the quantum limit, a gapped topological spin liquid emerges.

Due to the large atomic number () of Ir, however, we should carefully consider the role of spin-orbit coupling, whose leading effect in localized electron systems is the Dzyaloshinskii-Moriya (DM) interaction in the weak spin-orbit coupling limt. In fact, DM interactions have been argued to play an important role even in ZnCu(OH)Cl, with much less relativistic Cu () moments.Rigol and R.R.P.Singh (2007) The DM interaction reduces the full SU(2) spin-rotational invariance of the Heisenberg Hamiltonian to the discrete time-reversal symmetry (in addition to coupling spin transformations to the discrete point group operations of the lattice). On general grounds, this is expected to lower the degeneracy of the classical ground state manifold. However, depending upon the detailed form of the DM coupling, varying degrees of degeneracy remain, indicative of different amounts of frustration. The tendency of the system to retain the order of the classical ground state is certainly also variable, and warrants investigation. This is one of the motivations of the present study.

Another motivation comes directly from the experiments in Ref.Okamoto et al., 2007, several aspects of which are suggestive of the presence of spin-orbit coupling. First, the “Wilson ratio” , is observed to grow with cooling at low temperature, following a power-law , with . Here is the magnetic susceptibility and is the specific heat. As will be discussed in Sec. II, such a low temperature behavior is incompatible with any spin-rotationally invariant phase of matter supporting well-defined quasi-particle excitations. To our knowledge, it is at odds with all known theoretical models of quantum spin liquids, and seems highly unlikely on general grounds. Taking into account the observed field-independence (up to 12 Tesla) of the specific heat brings the behavior even further into disagreement with spin-rotationally invariant theories. Second, samples in which a fraction of Ir atoms are substituted by Ti (which are in a non-magnetic Ti state) display a Curie component in the susceptibility linearly proportional to with a strongly suppressed amplitude, of approximately of a spin- moment per Ti. As we also show in Sec. II.2, such behavior is also at odds with any simple spin-rotationally invariant low-temperature phase (assuming no clustering of the Ti atoms), though some more exotic. All these observations, however, are readily reconciled by assuming the presence of spin-rotational symmetry breaking. Given the lack of any observed magnetic ordering, explicit and substantial spin-orbit interactions would appear to be a likely candidate.

In Sec. III, we consider an explicit semi-microscopic calculation of the exchange Hamiltonian in the presence of spin-orbit coupling. We consider both super-exchange through the intermediate O atoms, and direct exchange between closest pairs of Ir spins. The results depend crucially upon the relative magnitude of the spin orbit coupling constant and the non-cubic splittings of the multiplet. This is quantified by two dimensionless ratios of to the two energy splittings and of the orbital levels in the absence of spin-orbit. When is the largest energy scale – the “strong spin orbit limit” – the “spin” has a substantial orbital angular momentum component, while in the opposite “weak spin orbit limit”, it is predominantly microscopic spin angular momentum. Indeed, the -factor has opposite sign in the two limits. Which if either limit applies is the most fundamental physical question to be understood concerning the nature of magnetism in NaIrO. We are not aware of any calculations or direct experimental measurements that indicate whether NaIrO is in the weak or strong spin-orbit limits, or intermediate between these situations. Instead we will address this question by comparing the expected phenomenology for the two cases to experimental observations.

In the strong spin-orbit limit, when the dominant mechanism is Ir-O-Ir superexchange, we find an highly anisotropic effective spin Hamiltonian, in which two spin components on each bond interact antiferromagnetically while the third interacts ferromagnetically. Specifically,

 H=∑⟨ij⟩JϵμijSμiSμj, (1)

where is a permutation of chosen appropriately for each bond (see Sec. III.6) to specify the two antiferromagnetic and one ferromagnetic direction. We call Eq. (1) the “strong anisotropy” Hamiltonian.

Somewhat surprisingly, the remaining three cases: strong spin orbit and direct exchange, and weak spin-orbit and superexchange or direct exchange, all lead to approximately isotropic Heisenberg interactions. For the weak spin-orbit limit, this is guaranteed, but it is certainly not in the strong spin-orbit case. The dominant spin-rotational symmetry breaking effect, which is perturbative in all three regimes, is the DM interaction. The effective Hamiltonian has the form

 H=∑⟨ij⟩[JSi⋅Sj+Dij⋅(Si×Sj)]. (2)

Here is the same for all bonds, and estimated as from the measured Curie-Weiss temperature . Symmetry strongly restricts the structure of this effective magnetic Hamiltonian for hyper-kagomé. The full set of DM vectors may be fixed by just three parameters. That is, on any one bond is arbitrary (by symmetry), but that choice determines all remaining in the system. It is convenient to choose the local coordinate system , where is the component aligned with the bond, is normal to the triangle plane in hyper-kagomé  lattice, lies in the triangle plane but perpendicular to the bond (see Fig. 3). The semi-microscopic calculations in Sec. III.7 confirm that all three components are non-vanishing, and gives a quantitative understanding of them. Due to the large and considerable uncertainties in estimating the non-cubic energy splittings, it is difficult to estimate the overall magnitude of the DM terms, but there is no reason they need be particularly small, though the perturbative estimates are presumably valid only for or so. A naïve estimation is obtained by noting that in this limit the ratios of DM to exchange are expected to be of the same order as the shift of the -factor, i.e. . From the measured moment , assuming we are in this limit would give or so.

In Sec. IV, we considered the strong anisotropy Hamiltonian, Eq. (1) in the classical approximation. Remarkably, unlike the Heisenberg model which is macroscopically degenerate (i.e. its ground states are specified by a number of continuous parameters proportional to the number of spins), the system in this limit has an almost unique ground state. We find a continuous two parameter manifold of ground states, in which any one spin can be specified arbitrarily after which all others are determined. This is still a (small) accidental degeneracy, since the system has itself only discrete (space-group and time-reversal) symmetries which do not protect any continuous degeneracies. Nevertheless, this degeneracy is presumably insufficient to prevent ordering in a classical system. The behavior in the physical quantum problem is not known, but one would expect that an ordered phase of the same symmetry as the classical one is rather likely, and there is little reason to suppose a significant suppression of the ordering temperature relative to the Curie-Weiss scale. The disagreement of these expectations with the experimental observations suggests that it is the weakly-anisotropic DM Hamiltonian rather than this one which is most appropriate. We however return to this question in more detail in Sec. VII.

In Sec. V, we turn to the weak anisotropy limit, and first explore the classical phase diagram of Eq. (2). In general, even this optimization problem is highly non-trivial, given the large unit cell of the hyper-kagomé lattice, and the possibility that the magnetic unit cell of the ground states may be yet larger. In the special case and , however, it is possible to solve this problem exactly. The degeneracy is broken completely to a single Kramer’s pair of coplanar ground states, for which the magnetic unit cell is equal to the crystallographic one. These may in this sense be considered states. One is drawn in Fig.2. We call this the “windmill” state. By several approximate methods, we establish the form of the phase diagram in the general -- parameter space. Generically the windmill state distorts to a “canted windmill” state (still with ), occupying a finite region of the phase diagram. In addition, one finds a wide range of incommensurate phase, in which the ordering wavevector is non-zero and generically irrational in reciprocal lattice coordinates. Owing to the breaking of space group symmetries, the incommensurate phase retains more of the frustration-induced degeneracy.

A key question is whether the DM interactions, expected on physical grounds and invoked phenomenologically to explain the experimental properties discussed above, are consistent with the observed spin liquid behavior of NaIrO, i.e. the lack of any ordering down to the very low temperatures of . The breaking of degeneracy by DM might be expected to reduce quantum fluctuations and thereby lead to ordering, in conflict with experiment. To study this possibility, we carried out spin wave calculations of the excitation gap and the quantum correction to the classical ordered moment. Indeed, we find that deep inside the phases, the quantum correction is not too large, which leads us to expect that the spin- system exhibits the classical order. However, we find very large quantum corrections elsewhere in the phase diagram, even for fairly substantial . In our results, small excitation gap will lead to a large quantum correction to classical ordered moment. Decreasing the excitation gap by changing the DM vector will eventually destroy the classical ordered moment completely. In this regime, the large quantum effects invalidate the spin-wave treatment and indeed leave open the possibility of a quantum spin liquid, consistent with experiment. To further confirm the results and treatment of spin wave theory, we implemented exact diagonalization on a small cluster (six triangles with spins). The excitation gap obtained from numerical data of specific heat qualitatively agrees with the prediction of spin wave theory.

The remainder of this paper is organized as follows. In Sec. III we discuss the symmetry allowed DM vector components and calculate the exchange spin Hamiltonian with a microscopic theory for both strong and weak spin-orbit coupling. In Sec. IV we discuss the classical ground states of the strong anisotropic exchange Hamiltonian obtained from Ir-O-Ir superexchange in the strong spin-orbit coupling limit. In Sec. V we will turn to look at the weak anisotropy Hamiltonian, namely, the nearest-neighbor Heisenberg model with small DM interactions. We first present the magnetic ordered state when then discuss the more general case when nonvanishing and components are present in the system. In Sec. VI, we present a linear spin wave theory to find the zero temperature quantum correction to the magnetically ordered phase and compare with exact diagonalization. Finally, a discussion of our main results and their relevance to NaIrO  is given in Sec. VII.

## Ii Thermodynamics of spin-rotationally invariant magnetic phases

In this section, we discuss some apparent constraints on the low temperature susceptibility and specific heat in spin-rotationally invariant phases of matter. As described in the introduction, these constraints appear to be violated in NaIrO, which we take as an indication of the presence of substantial spin-orbit interactions.

### ii.1 Clean system

We take spin-rotational invariance to mean the existence of global SU(2) spin symmetry. According to standard quantum mechanics, this implies that all states may be chosen as eigenstates of and , where is the operator for total spin. The choice of axis being arbitrary, we take it along the axis of any applied field. The effect of the field on the system is then entirely described by the term

 HH=−H∑iSzi=−HSzTOT, (3)

where we have absorbed the (presumed known) -factor, Bohr magneton, etc. into the definition of . One observes from Eq. (3) that is diagonal in the basis, and thus the Hamiltonian eigenstates themselves are independent of field, and only the eigenvalues change. Focusing on the states rather than their energies, we may say that the only effect of the field upon the system in equilibrium is to modify the occupation probabilities of states. In this sense, the magnetic field is a thermodynamic perturbation, and the susceptibility is a thermodynamic quantity, determined only by the density of states. The specific heat is of course also such a thermodynamic quantity, determined from the same density of states. Thus they are connected.

Specifically, the specific heat is a measure of the full density of states for all excitations above the ground state, irrespective of their spin quantum numbers. The susceptibility, however, only counts those excitations which carry non-zero spin along the field. The possibility of spin-less excitations allows some independence of the two: by introducing more states, one can increase arbitrarily while leaving unchanged. However, the converse is not true. It would seem difficult to increase without also contributing to . The only way in which this can be done is to introduce states with very large (which then contribute a large amount to ) but very low energy (and hence do not contribute much to ). This case corresponds to a system on the verge of a ferromagnetic instability.

Without fine-tuning to such a point, we are led to expect that, in the presence of SU(2) symmetry, the Wilson ratio,

 R=Tχcv (4)

should have an upper bound, corresponding to all excitations contributing both to and . This can indeed be shown provided we assume the system can be described by a non-magnetic ground state and non-interacting quasiparticles characterized by a spin quantum number. We define the density of state and for boson or fermion excitations carrying spin , respectively. The specific heat is

 cv=∂T∑m∫∞0dϵϵ[gbm(ϵ)nb(ϵ)+gfm(ϵ)nf(ϵ)], (5)

where

 nb/f(ϵ)=1eβϵ∓1. (6)

One obtains

 cv=k2BT4∑m∫∞0dxx2[gbm(kBTx)sinh2(x/2)+gfm(kBTx)cosh2(x/2)]. (7)

Now consider the susceptibility

 χ = ∂H∑m∫∞0dϵm[gbm(ϵ)nb(ϵ−Hm) (8) +gfm(ϵ)nf(ϵ−Hm)]∣∣H=0.

One finds

 χ=14∑mm2∫∞0dx[gbm(kBTx)sinh2(x/2)+gfm(kBTx)cosh2(x/2)]. (9)

In the low temperature limit, we may approximate by its small argument behavior, which is usually a power-law form:

 gb/fm(ϵ)∼Ab/fmϵγb/fm. (10)

One needs obviously for the density of states to be integrable (and hence the cumulative distribution well defined). We will encounter problems with Eq. (9) if for any . This could be fixed by the inclusion of a chemical potential, whose temperature dependence we have ignored, and as usual is necessary to avoid Bose condensation of free bosons at low when their energy is close to zero. This effect, however, does not change any of the results, so we have excluded it for simplicity here.

Given Eq. (10), the specific heat will be controlled at low by the minimum exponent over all :

 γ0=min{γb/fm}. (11)

One has

 cv∼A0k2+γ0BT1+γ0, (12)

with some constant . The susceptibility is controlled by the minimum exponent for :

 γ1=min{γb/fm;m≠0}. (13)

Note that by definition, . Then

 χ∼A1Tγ1. (14)

Then the Wilson ratio becomes

 R∼R0TΥ, (15)

where and

 Υ=γ1−γ0≥0. (16)

Because , the Wilson ratio cannot diverge on lowering , and unless , actually vanishes as . In defining the Wilson ratio, we have considered only the zero field specific heat. In a field, contributions from all excitations with will be field dependent. So unless the mode is dominant in , the specific heat should be expected to be field dependent. Conversely, field independence of the specific heat requires that the excitations dominate . In this case, we have , and the equality is not satisfied. Thus a field-independent low-temperature specific heat would be expected to correspond to a vanishing Wilson ratio as . This makes the observed divergence of on lowering in NaIrO even more at odds with the theoretical expectation for an SU(2) invariant system.

A few comments are in order. First, while we have assumed power-law forms for the low energy density of states, this is not essential. We believe the lack of low temperature divergence in is very robust within the quasiparticle picture. Beyond the quasiparticle approximation, the situation is less clear, and we do not have a definitive proof of this behavior of . However, we do not know of any single theoretical counter-examples in the literature for SU(2) invariant low temperature phases.

If SU(2) symmetry (or more specifically, invariance under spin rotations about the measurement axis) is broken, however, one readily and indeed almost generically observes this behavior. This is quite familiar from the case of ordered antiferromagnets in two or three dimensions. These are well-known to display a non-vanishing constant zero temperature uniform susceptibility and a power-law specific heat due to spin wave excitations, hence a Wilson ratio obeying Eq. (15) with however . This arises because the ground state itself is modified continuously by the introduction of a magnetic field. Semi-classically, the magnetic field leads to a smooth canting of the antiferromagnetic moments in the field direction, linearly proportional to the applied field.

This phenomena is, however, not limited to systems with spontaneous symmetry breaking. It occurs whenever the effective Hamiltonian for the low temperature phase does not conserve the spin component along the magnetic field. As an extreme example, one may consider the case of two spin- spins coupled together by antiferromagnetic exchange and DM interaction:

 H2=JS1⋅S2−D^z⋅S1×S2−H(Sx1+Sx2), (17)

where we have chosen the DM vector along the axis, and therefore oriented the field along so that it couples to a non-conserved magnetization. One can readily diagonalize the Hamiltonian, and find that in zero field has a unique ground state with a gap . Nevertheless, the susceptibility is non-zero when :

 χ=∂Sxi∂H∣∣∣H=0=√J2+D2−JJ(√J2+D2+J). (18)

Because of the gap, the specific heat of the dimer is activated at low temperature, and hence the dimer’s Wilson ratio diverges exponentially at low temperature. In general, a non-zero limit for the low temperature susceptibility is always to be expected once SU(2) symmetry breaking perturbations are taken into account. The specific heat, however, is insensitive to symmetry, and remains a true probe of low energy modes.

### ii.2 Impurity susceptibility

In NaIrO, the introduction of non-magnetic impurities (substitution of Ti for Ir) was observed to give rise to a Curie component with a reduced effective moment of per Ti. We would like to argue that a spin liquid state with such a large reduction from the moment of a free spin, , is unlikely in the absence of spin-orbit interactions, but quite likely when they are invoked.

Suppose the Hamiltonian has global SU(2) spin-rotational symmetry in the absence of an applied magnetic field. Then a spin-liquid ground state, which, by definition, does not break SU(2) symmetry, must be a spin singlet, i.e. a state of total spin . Its excitations can therefore by characterized by spin quantum numbers. Representations of SU(2) always have integer or half-integer spin, and in particular for all these the projection of the total spin along any field axis is a multiple of .

Now consider a single impurity. It may be a strong perturbation locally, but does not perturb the Hamiltonian far from itself. Again presuming spin-orbit can be neglected, the ground state of this system should be a spin eigenstate, though not necessarily non-zero. Nevertheless, it can be classified by a total spin which is a multiple of an half integer. It is natural to expect that the ground state multiplet of a single impurity controls the impurity susceptibility (but see below). Allowing now for an external field, this is simply described as in the previous subsection by Eq. (3). Since the low energy states are still good representations of SU(2), and only the total spin projection enters Eq. (3), we will obtain an effective moment which is at a minimum (if it is non-zero) per impurity.

The caveat in this argument is the possibility of a Kondo-like effect. If the spin liquid state is gapless, then there is a possibility for an impurity moment to be “screened” by the bulk degrees of freedom. Still, the possibility of a fractional impurity moment is delicate. Most Kondo effects either completely screen the moment (as in the single channel case, leading to ) or to weaker temperature dependence of the impurity susceptibility (e.g. in the two-channel model, which has a non-trivial Kondo fixed point). Thus most types of Kondo effect do not allow for such behavior. Recently, it has been suggested that some spin liquids might sustain a critical fixed line of Kondo fixed points, connected to the free impurity fixed point. This situation can in fact lead to a renormalized Curie constant.Kolezhuk et al. (2006) It would indeed be appealing should such an exotic possibility be realized in NaIrO, but we should allow for simpler explanations.

As is well-known, the effective moment of ions in solids varies widely from the quantized values expected from SU(2) symmetric considerations. This is of course due to spin-orbit coupling. In general, with spin-orbit interactions present, the ground state of an impurity can be expected to be a Kramer’s singlet or a Kramer’s doublet. In the latter case, it will behave energetically (i.e. in specific heat) as a spin- spin, but will have in general a non-trivial g-tensor describing its coupling to a field. This reflects a change in the effective moment. Thus there is no “quantization” of the effective moment once spin-orbit coupling is substantial. The observed fractional effective moment in NaIrO is perhaps another indication in this direction.

## Iii Spin-Orbit coupling in the hyper-kagomé lattice

In this section, we discuss the form of the spin-orbit modifications to the isotropic Heisenberg Hamiltonian. This is not directly calculable from semi-microscopic considerations without some assumptions about the local energetics due to crystal field splittings. Therefore we consider below a number of cases.

### iii.1 Symmetry allowed DM vector components

In several cases, we will find that the dominant effect of spin-orbit coupling is to induce Dzyaloshinskii-Moriya (DM) interactions between the nearest-neighbor spins. Therefore before attempting any calculations, it is instructive to first consider the symmetry constraints upon them. Generally, DM interactions are rather highly constrained. For instance, they are absent if there is an inversion center between the two spins in question (this is not the case in NaIrO). The compound NaIrO has cubic symmetry, described by the space group P432, and consequently has a number of point group symmetries. For our purposes, it is useful to consider a unconventional set of generators of these symmetries. Specifically, the full point group can be generated from the set of rotations around a local axis at each site. Due to this symmetry, all the hyper-kagomé sites and bonds are equivalent. In Table. 5, we list the directions of the axes () for every site in the unit cell (see Fig. 3 for the labeling). The rotational symmetries relate the DM vectors of any two bonds. That is, given the DM vector on any one hyper-kagomé bond, all others are determined. This one DM vector, however, is itself entirely unconstrained by the P432 symmetry.

Since any single bond of the hyper-kagomé is uniquely associated with one triangle, it is natural to adopt a local coordinate system based on this triangle to describe the DM vector’s components. We denote the component aligned with the bond , the component normal to the triangle plane and the component normal to the bond but localized in the triangle plane . Three components have been illustrated in Fig. 3. If we select the direction of component axis by assigning a direction to one bond (arrows in Fig. 3), the rotation symmetry can generate the equivalent axis for other bonds (see Fig. 3). In every triangle, there is a chirality of the axis of three edges, which can be considered as the direction of axis. The cross product of and directional vector generates the direction of axis.

Such a parametrization may be applied not only for the hyper-kagomé  lattice, but for any lattice consisting of corner-sharing triangles, such as the slightly distorted kagomé lattice of Fe/Cr-jarosites.Elhajal et al. (2002a); Ballou et al. (2003); Elhajal et al. (2002b) In that example, the component is forbidden by a mirror plane symmetry. In NaIrO, there are as we said no constraints on the , and we might naively expect all three components to be non-vanishing and comparable. We will investigate this by microscopic calculations below.

### iii.2 Local electron energetics of Ir ion

Before moving to the microscopic theory of spin-orbit interactions, we need to understand the electron energy levels of the Ir ions. With coordinates taken from Table. I in Ref. Okamoto et al., 2007, two Ir and their surrounding O are drawn in Fig. 4. For A ion, the C axis orients along . Under this symmetry operation, , and . Accordingly, we can group the orbitals into even and odd parity sectors, as shown in Table. 1.

A large cubic crystal field splits the and states. The surrounding O octahedron is slightly distorted to further split all the three states. Ultimately no degeneracy is protected because the C symmetry has only one dimensional irreducible representations. The energetic ordering of orbitals shown in Fig. 5 was determined by looking at Coulomb interaction from surrounding O and ignoring the spin-orbit interaction.

### iii.3 Microscopic theory of exchange spin Hamiltonian

Though symmetry determines the allowed non-zero components of the DM interaction, it does not give any guidance as to their relative and absolute magnitudes.Moriya (1960); Elhajal et al. (2002a); Koshibae et al. (1993) In this part, we will derive the exchange spin Hamiltonian from a microscopic point of view and obtain expressions from which crude estimates of the magnitude of various terms can be obtainedMoriya (1960); Elhajal et al. (2002a); Koshibae et al. (1993) We consider both the hopping between Ir and O orbitals, and direct hopping between Ir orbitals. We also assume that the e-t splitting is much greater than the splittings among the three t states so that we can completely project out the two e states. The model is then of five electrons on on the orbitals of every Ir. Following some notations in Ref. Koshibae et al., 1993, we can write the Hamiltonian of the Ir and O sublattice as

 H=H0+Ht+HLS, (19)

where,

 H0 = ∑jmσϵmd†jmσdjmσ+∑knσϵpnp†knσpknσ (20) + Ud2∑jmm′σσ′d†jmσd†jm′σ′djm′σ′djmσ + Up2∑knn′σσ′p†knσp†kn′σ′pkn′σ′pknσ, Ht = ∑jmσ∑k(j)n(tjm,knd†jmσpknσ+H.c.) (21) +∑⟨jj′⟩∑mm′tdjm,j′m′d†jmσdj′m′σ, HLS = λ∑jℓj⋅sj. (22)

denotes the O of the neighboring Ir site , is the creation operator of an electron with spin of the th orbital of th Ir ion, is the energy of this orbital. will take . is the creation operator of an electron on the orbital with spin . The energies are measured from the lowest energy level of the Ir orbitals, and and are the Coulomb interaction constants between holes on the Ir site and O site, respectively. We assume that and are orbital-independent and ingore other “Kanamori parameters”:Kanamori (1963) the inter-orbital exchange coupling and the pair-hopping amplitude, which should be small compared with Coulomb interaction. We also ignore the Coulomb interaction between two eletrons on different intermediate O ions. Here denotes the transfer of an electron between the th orbital of Ir ion and one of the orbitals of the neighboring O ions . Similarly, is the matrix element for electron transfer between and orbitals on two nearest-neighbor (in the hyper-kagomé sense) Ir atoms. and denote the orbital and spin angular momenta at the th Ir ion, respectively, and is the spin-orbit coupling constant of the Ir ion.

In order to understand the electron occupation on each site, we collect the quadratic terms for each site in Eq. (22) and write down the onsite Hamiltonian as

 H(i)=∑mm′σσ′d†imσM(i)mσ,m′σ′dim′σ′ (23)

with

 M(i)mσ,m′σ′=ϵmδσσ′δmm′+λℓimm′⋅sσσ′, (24)

where is the Pauli matrix and is the matrix element of between the th and th orbital of the th Ir ion. It is useful to noteGoodenough (1967) that the vector of three-dimensional matrix orbital angular momentum operators projected into the manifold is actually proportional to the vector of orbital angular momentum operators for the 3 ordinary () states, but with a proportionality constant of ! That is, suppressing the indices,

 ℓi=−Li, (25)

where is a canonical angular momentum operator with . This effectively makes the spin-orbit coupling term directly analogous to the familiar one from an isolated atom with spherical symmetry in a shell, but with the sign of the spin-orbit coupling reversed.

### iii.4 Strong and weak spin orbit limits

Obviously the nature of the “spin” itself (i.e. the Kramer’s doublet ground state of the single whole in this multiplet) is crucially dependent upon the strength of the spin-orbit interaction, relative to the non-cubic splittings . This determines the nature of the wavefunctions of the Kramer’s pair, for instance the degree to which the “spin” carries true electron spin angular moment or instead orbital angular momentum. This is more fundamental than the exchange interaction, so we consider it first.

#### Strong spin orbit

In the strong spin-orbit limit, we can to a first approximation ignore the non-cubic splittings, and we have simply

 M(i)=λℓi⋅si=−λLi⋅si. (26)

This is of course diagonalized by constructing eigenstates of the “total angular momentum”

 Ji=Li+si. (27)

Because of the minus sign in Eq. (26), the highest energy doublet is simply the Kramer’s pair. This describes the wavefunction of the half-filled orbital. It is natural to define the effective spin operator in this case as

 Si=Ji. (28)

Clearly it is a strong mix of orbital and spin components. According to the Wigner-Eckart theorem, the matrix elements of , and are all proportional. This enables one, with a little Clebsch-Gordan algebra, to arrive at an expression for the magnetic moment operator (in the manifold)

 Mi=−μB(ℓi+2si)=+2μBSi, (29)

where is the Bohr magneton. Interestingly, this is the same magnitude but opposite sign as for a free electron! It will of course suffer corrections perturbative in , as one moves away from the strong spin orbit limit.

#### Weak spin orbit

Now consider the weak spin orbit limit. In this case, for , the half-filled doublet is simply the orbital, with two possible “true” spin orientations. Thus we approximately have

 Si≈si+O(λ/(ϵi−ϵj)). (30)

Now there is essentially no orbital angular momentum component to the spin (), and one obtains

 Mi=−2μBSi(1+O(λ|ϵ1,2−ϵ3|)). (31)

Note the important sign difference from Eq. (29). This is the most fundamental physical distinction between the weak and strong spin-orbit limits. However, the magnitude of the proportionality between the magnetization and spin – the -factor – is the same in both cases. This means that the simplest experimental measure, the Curie susceptibility, cannot distinguish the two possibilities. We will consider both cases below.

### iii.5 General exchange formulation

We now turn to the exchange calculations. Let us consider the general case first. We must deal with , which is a matrix. Diagonalize so that . Here, is a site-independent eigenvalue matrix, and is a unitary eigenvector matrix. has three different eigenvalues , and , each has a two-fold degeneracy due to Kramers’ degeneracy theorem. The effective spin operator will be defined to act in this doublet. In the strong and weak spin-orbit limits, we have explicitly Eq. (28) and Eq. (30), respectively. Furthermore, we define a new set of electron creation and annihilation operators

 aimσ=T(i)mσ,m′σ′dim′σ′ (32)

with annihilates an electron on the state with spin at site .

Without losing any generality, we assume that , then states are fully occupied and state is half-occupied, leading to a total spin- at every site. Accordingly, the magnetic momentum operator ( at each site should be projected onto the Kramers’ doublet ground states:

 MiμB = −Pi∑mnαβd†imα(ℓimnδαβ+δmnσαβ)dinβPi (33) = −G(i)3α,3βσβα⋅Si

with the vector of Pauli matrices. Also, and the effective spin operator are defined as

 G(i)lσ,jδ = ∑mnαβT(i)lσ,mα(ℓimnδαβ+δmnσαβ)T(i)∗jδ,nβ, Si = ∑α,β12a†i3ασαβai3β, (34)

and is the ground state projection operator:

 Pi=ai3↑∣ϕ⟩⟨ϕ∣a†i3↑+ai3↓∣ϕ⟩⟨ϕ∣a†i3↓. (35)

Here is the fully-occupied state. In the last step Eq. (33), has been used.

Let’s go back to Eq. (22), and express the microscopic Hamiltonian in terms of and . Given the Hamiltonian in Eq. (19), which includes the largest Coulomb energy but neglects the smaller Hunds-rule exchange coupling between electrons in different orbitals on the same atom (and other similar interactions), only hopping through the half-filled orbital contributes to the super-exchange interaction. This is in accord with the “Goodenough-Kanamori” rules, which state that the exchange coupling contributed from a half-occupied orbital and a fully-occupied orbitals is much weaker than the one from two half-occupied orbitals. Thus, we only need to focus on the hopping between the orbitals, as half-occupied orbital. The microscopic Hamiltonian is written as

 H = ∑knσϵpnp†knσpknσ+Up2∑knn′σσ′p†knσp†kn′σ′pkn′σ′pknσ+∑jmσEma†jmσajmσ+Ud2∑jmm′σσ′a†jmσa†jm′σ′ajm′σ′ajmσ (36) + ∑jk(j)n∑αβ[(~tj3,knδαβ+Cj,kn⋅σαβ)a†j3αpknβ+H.c.]+∑⟨jj′⟩∑αβ[(~tdj3,j′3δαβ+Cdjj′⋅σαβ)a†j3αaj′3β+H.c.],

with

 ~tj3,kn = ∑mσ12tjm,knT(j)3σ,mσ Cj,kn = ∑m,αβ12tjm,knT(j)3α,mβσβα, (37)

and

 ~tdj3,j′3 = ∑mm′,ασ12tdjm,j′m′T(j)3α,mσT(j′)†m′σ,3α Cdjj′ = ∑mm′,σ,αβ12tdjm,j′m′T(j)3α,mσT(j′)†m′σ,3βσβα, (38)

where is vector of the three Pauli matrices. Now we may follow the standard perturbative treatment of superexchange. We consider separately the superexchange through the intermediate O ions, and the direct exchange contributions.

#### Superexchange through oxygen ions

In this case the leading contribution is fourth order in hopping, i.e. a result of fourth order degenerate perturbation theory. We must include four “hops” between Ir and O ions, which consist of “hops” described by spin-isotropic matrix elements, and “hops” given by anisotropic matrix elements. One thereby obtains the exchange Hamiltonian as

 Hex=∑⟨ij⟩[JSi⋅Sj+Dij⋅(Si×Sj)+Si⋅↔Γij⋅Sj] (39)

with the first two terms the Heisenberg and DM interactions precisely as in Eq. (2), and the third term the anisotropic exchange. The explicit formulae for the coupling constants are:

 J = 4∑kn,k′n′sij,kngkn,k′n′sji,k′n′ (40) Dij = −4i∑kn,k′n′(vij,kngkn,k′n′sji,k′n′−sij,kngkn,k′n′vji,k′n′) (41) ↔Γij = 4∑kn,k′n′[(←vij,kngkn,k′n′→vji,k′n′+←vji,kngkn,k′n′→vij,k′n′)−↔1(vij,kn⋅gkn,k′n′vji,k′n′)]. (42)

The vector with arrow or indicates that inner product is taken with the spin operator put in the direction of the arrow. is a unit matrix. , and are given by

 sij,kn = ~ti3,kn~tkn,j3+Ci,kn⋅Ckn,j (43) vij,kn = Ci,kn~tkn,j3+~ti3,knCkn,j+i(Ci,kn×Ckn,j) (44) gkn,k′n′ = (1−12δkk′δnn′)(~ϵ−1pkn+~ϵ−1pk′n′)2~ϵpkn+~ϵpk′n′+Upδkk′+(~ϵpkn~ϵpk′n′Ud)−1 (45)

with . In the following subsections, we will try to estimate these exchange couplings in both the strong and weak spin-orbit interaction cases.

#### Direct exchange

Here we require only second order perturbation theory in the direct matrix elements. One obtains the resultsMoriya (1960):

 J = 2∣∣~tdij∣∣2Ud, (46) Dij = −4iUd(Cdij~tdji−~tdijCdji), (47) ↔Γij = 4Ud(←Cdij→Cdji+←Cdji→Cdij−↔1(Cdij⋅Cdji)). (48)

### iii.6 Strong spin-orbit interaction

As discussed above in Sec. III.4.1, in the strong spin-orbit limit, , one can obtain effective total angular momentum eigenstates with . Choosing Eq. (28), and rewriting the corresponding eigenstates in the canonical basis, Eq. (32) becomes

 ai3↑ = 1√3((−i)di,xz↓+di,yz↓+di,xy↑) (49) ai3↓ = 1√3((i)di,xz↑+di,yz↑−di,xy↓), (50)

in which, we have expressed / in terms of the annihilation operator to avoid the position dependence of the coefficients.

#### Superexchange through oxygen ions

The complicated expression of Eq. (42) requires simplification if we want to have a quantative understanding of the exchange coupling. However, some information can be immediately obtained from Eq. (50), in particular that all , which makes , and only remain terms with . To simplify further, we need some explicit form for the transfer integrals . Hence, we will make further approximation that the surrounding octahedra of Ir are perfect so that we can apply the cubic symmetry to find out the nonvanishing transfer integrals and also the relation between them, which is listed in Table. 2 for Ir A and B in Fig. 4. Deviations from these forms should presumably be small, since the non-cubic distortion is.