Spin-orbit-entangled electronic phases in 4d and 5d transition-metal compounds
Tomohiro Takayama, Jiri Chaloupka, Andrew Smerald, Giniyat Khaliullin, Hidenori Takagi
SSpin-orbit-entangled electronic phases in 4 d and 5 d transition-metal compounds Tomohiro Takayama,
1, 2
Jiˇr´ı Chaloupka,
3, 4
Andrew Smerald, Giniyat Khaliullin, and Hidenori Takagi
1, 2, 5 Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany Institute for Functional Matter and Quantum Technologies,University of Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany Department of Condensed Matter Physics, Masaryk University, Brno, Czech Republic Central European Institute of Technology, Masaryk University, Brno, Czech Republic Department of Physics, University of Tokyo, 7-3-1 Hongo, Tokyo 113-0033, Japan (Dated: February 5, 2021)Complex oxides with 4 d and 5 d transition-metal ions recently emerged as a new paradigm in cor-related electron physics, due to the interplay between spin-orbit coupling and electron interactions.For 4 d and 5 d ions, the spin-orbit coupling, ζ , can be as large as 0.2-0.4 eV, which is comparablewith and often exceeds other relevant parameters such as Hund’s coupling J H , noncubic crystalfield splitting ∆, and the electron hopping amplitude t . This gives rise to a variety of spin-orbit-entangled degrees of freedom and, crucially, non-trivial interactions between them that depend onthe d -electron configuration, the chemical bonding, and the lattice geometry. Exotic electronicphases often emerge, including spin-orbit assisted Mott insulators, quantum spin liquids, excitonicmagnetism, multipolar orderings and correlated topological semimetals. This paper provides a se-lective overview of some of the most interesting spin-orbit-entangled phases that arise in 4 d and 5 d transition-metal compounds. I. INTRODUCTION
In the 1960’s and 70’s, correlated-electron physics intransition-metal oxides was already an active field ofresearch, and a major topic in condensed matter sci-ence. The basic picture of spin and orbital ordering andthe interplay between them was unveiled during this pe-riod, and collected into the Kanamori-Goodenough rules.However, the exploration of exotic electronic phases be-yond conventional magnetic ordering was stymied by alack of materials and theoretical tools.In 1986, high- T c superconductivity was discovered inthe layered 3 d Cu oxides, which accelerated both ex-perimentally and theoretically the search for novel spin-charge-orbital coupled phenomena produced by electroncorrelations. The major arena of such exploration wascomplex oxides with 3 d transition metal ions from Tito Cu, which led to the discoveries of unconventionalsuperconductivity, colossal magneto-resistance, multifer-roics and exotic spin-charge-orbital orderings. 4 d and5 d transition metal oxides were also studied, but not asextensively as 3 d transition metal oxides, except for 4 d Sr RuO , where possible p -wave superconductivity wasdiscussed. This was at least partially due to the lessprominent effect of correlations in 4 d and 5 d , which arisesfrom the wavefunctions being more spatially extendedthan in 3 d .In the late 2000’s, the layered perovskite Sr IrO wasidentified as a weak Mott insulator, and the crucial roleof spin-orbit coupling (SOC) in stabilizing the Mott stateawoke a growing interest in 5 d Ir oxides and other 4 d and5 d compounds. For Ir ions, there are five 5 d -electrons,which reside in the t g manifold, and therefore have aneffective orbital moment L = 1 and form a spin-orbit-entangled J = 1/2 pseudospin. These J = 1/2 pseu-dospins behave in some ways like S = 1/2 spins, but have an internal spin-orbital texture where the up/down spinstates reside on different orbitals. The resulting spin-orbital entanglement gives rise to non-trivial interactionsbetween the J = 1/2 pseudospins, and this led to theproposal that the Kitaev model, with bond-dependentIsing interactions, can be realized on edge-shared honey-comb networks of J = 1/2 pseudospins. In consequence, J = 1/2 honeycomb magnets made out of 5 d Ir and4 d Ru ions have been extensively studied over the lastten years, in an effort to discover the expected quantumspin-liquid state and associated Majorana fermions.Despite the considerable excitement, the d J = 1/2Mott state is not the only spin-orbit-entangled state ofinterest among the many 4 d and 5 d transition metal com-pounds. With different filling of d -orbitals and differ-ent local structures, a rich variety of spin-orbit-entangledstates can be formed, which are characterized not onlyby dipolar moments but also by multipoles such asquadrupolar and octupolar moments. Exotic states ofsuch spin-orbit-entangled matter can be anticipated, re-flecting the internal spin-orbital texture and the latticesymmetry, and including excitonic magnetism and mul-tipolar liquids.4 d and 5 d transition metal compounds also form in-teresting itinerant states of matter, with one prominentexample being the topological semimetal. This arisesfrom the interplay of lattice symmetry and SOC, and,in contrast to typical topological semimetals, 4 d and 5 d semimetals tend to also have strong electron correlations.Thus they provide an arena in which to study the overlapbetween topological physics and strong correlation.Spin-orbit-entangled phases are also formed in 4 f and5 f electron systems, which have been extensively studied,and it is worth spelling out what makes 4 d and 5 d elec-tron systems distinct. One obvious difference is that theyinteract through exchange processes with a much larger a r X i v : . [ c ond - m a t . s t r- e l ] F e b energy scale, making them more accessible to experi-ment, and thus increasing the variety of phenomena thatcan be effectively probed. Also, the spin-orbit-entangled J states in 4 d and 5 d systems are much less localizedthan those of 4 f , and can often be itinerant, openingup, for example, the exploration of correlated topologi-cal semimetals, and SOC driven exotic states formed nearmetal-insulator transitions. The d - and the f - electronsystems thus clearly play a complementary role in theexploration of spin-orbit-entangled phases.This review is intended to provide readers with a broadperspective on the emerging plethora of 4 d and 5 d transi-tion metal oxides and related compounds. We would liketo address two basic questions: 1) What kind of exoticphases of spin-orbit-entangled matter are expected in 4 d and 5 d compounds? 2) To what extent are the proposedconcepts realized? We limit our discussion to the com-pounds with octahedrally coordinated 4 d and 5 d transi-tion metal ions accommodating less than 6 electrons intheir t g orbitals (low-spin configuration), where the ef-fect of the large SOC is prominent due to the smallercrystal field splitting of t g orbitals as compared to e g .As there are many reviews of pseudospin-1/2 d Mott in-sulators, here we will discuss spin-orbit-entangled statesin 4 d and 5 d transition metal compounds from a broaderperspective, covering, in addition to d compounds, d , d and d insulators, as well as itinerant systems withstrong SOC. II. CONCEPT OF SPIN-ORBIT-ENTANGLEDSTATES AND MATERIALS OVERVIEW
In Mott insulators, charge fluctuations are frozen, andthe low-energy physics is driven by the spin and orbitaldegrees of freedom of the constituent ions. In 3 d com-pounds, orbital degeneracy and hence orbital magnetismis largely quenched by noncubic crystal fields and theJahn-Teller (JT) mechanism, and magnetic moments arepredominantly of spin origin (with some exceptions men-tioned below). The large SOC in 4 d and 5 d transitionmetal ions competes with and may dominate over crys-tal field splitting and JT effects, and the revived orbitalmagnetism becomes a source of unusual interactions andexotic phases. We first discuss the spin-orbital structureof the low-energy states of transition metal ions in a mostcommon, cubic crystal field environment, and then pro-ceed to their interactions and collective behavior. A. Spin, orbital, and pseudospin moments in Mottinsulators
The valence wavefunctions of 4 d and 5 d ions are spa-tially extended and the Hund’s coupling is smaller thanthe cubic crystal field splitting, 10 Dq , between t g and e g orbitals. Low-spin ground states are therefore sta-bilized for 4 d and 5 d electron configurations more fre- quently than in the 3 d case, where the Hund’s couplingtypically overcomes the t g - e g crystal field splitting. Inthe low-spin state, electrons occupy t g orbitals formingtotal spin-1 / d , d ), spin-1 ( d , d ), and spin-3 / d ).In all of them (except d orbital singlet not discussedhere), the orbital sector is threefold degenerate, formallyisomorphic to the p -orbital degeneracy, and can thus bedescribed in terms of an effective orbital angular momen-tum L = 1. (The effective orbital angular momentum isoften distinguished by a special mark, see, e.g., the “ficti-cious angular momentum” ˜ l in the textbook by Abragamand Bleaney, but here we conveniently choose a simplernotation L .) By a direct calculation of matrix elementsof the physical orbital momentum L d = 2 of d -electronswithin the t g manifold, one finds a relation L d = − L between the angular momentum operators.By employing the effective L operator, the SOC readsas H = ∓ λ SL , where the negative (positive) sign refersto a less (more) than half-filled t g shell of d , d ( d , d ) configurations, and λ is related to the single-electronSOC strength, ζ , via λ = ζ/ S . The resulting spin-orbital levels are shown in Fig. 1(a). Apparently, SOCbreaks particle-hole symmetry: due to the above signchange, the levels are mutually inverted within the pairsof complementary electron/hole configurations such as d / d and d / d . The ground states thus have completelydifferent total angular momentum J = S + L . Its con-stituents S and L align in parallel (antiparallel) fashionfor the less (more) than half-filled case. The correspond-ing “shapes” of the ground-state electron densities aredepicted in Fig. 1(b). Their nonuniform spin polariza-tion with a coherent mixture of spin-up and down densi-ties clearly shows the coupling between the spin and theorbital motion of electrons.The observed variety of ionic ground states among d n configurations brings about a distinct physics for eachof the d n ions. d and d ions with J > / J = 0 d ions may developunusual magnetism due to condensation of the excited J = 1 level. d ions with Kramers doublet J = 1 / L , the magnetic momentoperator reads as M = 2 S − L . In principle, L comeswith the so-called covalency factor κ but we omit it forsimplicity. For d with parallel L = 1 and S = 1 /
2, onehas L = 2 S and thus M = 0, i.e. the J = 3 / g -factor and is thus nonmagnetic. In reality, κ < d with S = L = J/
2, one finds M = (1 / J , i.e. g = 1 / d with J = 0 is a nonmag-netic singlet. d with antiparallel S = 1 / L = 1has g -factor g = −
2, i.e. it is of opposite sign relative to J= J= J= L S ζ ζ J= L S ζ J= J=J=
23 23
L S ζ J= J= J= L S ζ ζ +2 +1 0 −1 −2 + − J = z + − spin down spin up d J =, d J = , J = d , d J =, (b)(a) d d d d FIG. 1. (a) Low-energy levels of d , d , d , and d ions in cubic crystal field. The degeneracy of the levels is shown bythe number of close lines. For less than half-filled t g shell, the SOC aligns the effective orbital angular momentum L andspin S to form larger total angular momentum: J = 3 / d case and J = 2 quintuplet in d case, respectively.In the case of more than half-filled t g shell, L and S are antialigned, leading to J = 0 singlet ground state for the d configuration while the d one hosts pseudospin J = 1 /
2. (b) Orbital shapes corresponding to the ground-state J -levels. Onlythe angular distribution of the electron density is considered. It is represented by a surface plot where the distance to theorigin is proportional to the integral density in the corresponding direction. The color of the surface indicates normalized spinpolarization ( ρ ↑ − ρ ↓ ) / ( ρ ↑ + ρ ↓ ) taking values in the range [ − , +1]. It is shown for electrons in the case of d and d statesand for the holes in the t g configuration in the case of d and d states. the electron g -factor. The above g -factors strongly devi-ating from pure spin g = 2 are the fingerprints of largeorbital contribution to magnetism. In the d case, e.g.,one finds that S = ( − / J only, while L = (4 / J ;that is, magnetism of d compounds is predominantly oforbital origin.Figure 2 focuses in detail on the particular case of the d configuration. The J = 2 ground state is special be-cause it is isomorphic to a d -electron with orbital moment L d = 2, and thus it has to split under cubic crystal fieldinto a triplet of T g and a doublet of E g symmetries (of-ten denoted as Γ and Γ states, respectively). The cor-responding wavefunctions in the basis of J z eigenstates | J z (cid:105) are: |± (cid:105) , and ( | +2 (cid:105)−|− (cid:105) ) / √ T g , and | (cid:105) and( | + 2 (cid:105) + | − (cid:105) ) / √ E g states, just like for single elec-tron d -orbitals of t g and e g symmetries, as required bythe above isomorphism. Physically, this splitting arises,e.g., due to the admixture of the t g e g configuration into t g by SOC. More specifically, second-order energy cor-rections to T g and E g levels are different, which gives asplitting of the J = 2 level by ∼ ζ / Dq . With SOCparameter ζ = 0 . Dq = 3 eV, typical for 4 d and 5 d ions, one obtains a sizable splitting of 20 meV, with the E g doublet being the lower one. It is evidentfrom the above wavefunctions that the E g state has nodipolar moment and is therefore magnetically silent. In-stead, the E g doublet hosts quadrupolar and octupolarmoments. This is again analogous to e g electrons, whichare quadrupole active, and it has been discussed in thecontext of manganites that they may host an octupolarmoment as well. While this effect was not observedin real e g orbital systems, the spin-orbit-entangled E g doublet may show octupolar order driven by intersite ex-change interactions, unless JT coupling to lattice stabi-lizes quadrupolar order instead.Concerning the JT activity of the ionic ground states,the d singlet and d Kramers doublet possess no orbitaldegeneracy and are thus “JT-silent” in the first approxi-mation. However, both d and d are JT active ions, andstructural phase transitions as in usual 3 d systems canbe expected. As shown in Fig. 1(b), the shapes of the ± / ± / d ion are different; there-fore, they will split under tetragonal lattice distortions.In fact, these two Kramers doublets can be regarded asan effective e g orbital, so JT coupling would read ex-actly as for the e g orbital, albeit with an effective JT t g - e g t g t g t g e g ζ /10 ∼ Dq E g T J= J= J= ζζ α+ i ββα− i E g (c)(b) E g mixing Dq (a) Octupolar orderQuadrupolar order βα FIG. 2. (a) Shifts and splitting of the J = 0 , , d ion when considering mixing of the ground state t g config-uration with t g e g states by virtue of SOC. Focusing on thelowest levels, we that find the originally five-fold degenerate J = 2 states split into an E g doublet and T g triplet. Eval-uated perturbatively for ζ (cid:28) Dq , the splitting comes outproportional to ζ / Dq . (b) Tetragonal compression leadsto an increased repulsion of d -electrons from apical oxygenions and further singles out “planar” states from E g and T g sets. The quadrupolar moment hosted by the E g doublet getspinned this way. (c) Complex combinations of the E g doubletstates that expose the octupolar moment of “cubic” shape. coupling constant reduced by 1 / √
3, as a result of SOCunification of the Hilbert space. Similarly, the E g dou-blet of the J = 2 manifold in the case of a d ion shouldexperience JT coupling. Overall, the JT effect (and re-lated structural transition) is still operative, but it affectsboth spin and orbital degrees of freedom simultaneouslyas a result of the spin-orbit transformation of the wave-functions, and conventional JT orbital ordering is con-verted into a magnetic quadrupolar ordering of J = 3 / J = 2 moments. An important consequence of spin- orbital entanglement is that, distinct from usual orbitalorder in 3 d systems, magnetic quadrupolar order breaksnot only the point-group symmetry of a crystal but alsothe rotational symmetry in magnetic space, resulting inanisotropic, non-Heisenberg-type magnetic interactions,such as XY or Ising models. In other words, JT couplingin spin-orbit-entangled systems has a direct and moreprofound influence on magnetism.The above discussion is based on the LS -couplingscheme, which is adequate for obtaining the groundstate quantum numbers. However, the correspondingwavefunctions, and hence effective g -factors, as well asexcited-state energy levels, may get some corrections to LS -coupling results. This is important for the interpre-tation of the experimental data. Similarly, the admix-ture of the t n − g e g configuration into the ground state t n g wavefunctions by SOC and multielectron Coulombinteractions is present for all d n , and may renormalizethe g -factors and wavefunctions. However, these effectscannot split the ground state J -levels, except the J = 2level of the d configuration, as discussed above.In general, the ground state manifold of transitionmetal ions in Mott insulators is conveniently describedin terms of effective spin (“pseudospin”) ˜ S , where 2 ˜ S + 1is the degeneracy of this manifold. For low-spin d n ionsin a cubic symmetry, pseudospin ˜ S formally correspondsto effective total angular momentum J (often called J eff ),with the exception of the d case with an E g doublet host-ing a pseudospin ˜ S = 1 /
2. One has to keep in mind how-ever, that even in the case of cubic symmetry, the pseu-dospin wavefunctions are different from pure J states be-cause of various corrections (deviations from LS scheme,admixture of e g states, etc.) discussed above. This iseven more so when noncubic crystal fields are presentand become comparable to SOC. We will occasionally useboth ˜ S and J pseudospin notations, depending on con-venience (e.g., reserving J for the Heisenberg exchangeconstant in some cases).In strong spin-orbit-entangled systems, the notion ofpseudospins remains useful even in doped systems, atleast at low doping where Mott correlations, and hencethe ionic spin-orbit multiplets, are still intact locally. Inhighly doped systems, a weakly-correlated regime, a con-ventional band picture emerges, where SOC operates ona single-electron level. B. Pseudospin interactions in Mott insulators
The key element when considering the interactionsamong pseudospins is the entanglement of spin and or-bital degrees of freedom. In the pseudospin state, various | L z , S z (cid:105) combinations are superposed, forming a compos-ite object. Figure 3 shows two important examples forthe d and d cases that will be extensively discussedlater. Mixing the spins and orbitals in a coherent way,pseudospins do experience all the interactions that op-erate both in the spin and orbital sectors, which have √ r ! √ J z = J = J = L z = S z = L z = − S z = + L z = + S z = − J z = + L z = S z = + L z = + S z = − = + − (a) − += (b) FIG. 3. (a) Decomposition of the J = 1 / d into | L z , S z (cid:105) components of the single hole in t g configuration. The effective angular momentum is indicatedby the rotating arrow, spin by the color following the conven-tion of Fig. 1. For both contributions, L z and S z sum up to J z = +1 /
2. (b) Similar decomposition of the J = 0 two-holeground state of d . The total orbital angular momentum L z and spin S z may be combined in three ways here. The inter-nal compensation in L and S creates a cubic-shaped objectshowing no spin polarization. very different symmetry properties. Electron exchangeprocesses conserve total spin, and hence the spin inter-actions are of isotropic Heisenberg ( S i S j ) form. Theorbital interactions are however far more complex – theyare anisotropic both in real and magnetic spaces. Inhigh-symmetry crystals, orbitals are strongly frustrated,because they are spatially anisotropic and hence can-not simultaneously satisfy all the interacting bond di-rections. Via the spin-orbital entanglement, the bond-directional and frustrating nature of the orbital interac-tions are transferred to the pseudospin interactions. Moreover, apart from the exchange interactions drivenby virtual electron hoppings, there are other contribu-tions to the orbital interactions, especially in the d and d cases. These are mediated by the orbital-lattice JTcoupling to the virtual JT-phonons, and by electrostaticmultipolar interactions between d -orbitals on differentsites. In low-energy effective Hamiltonians, theseinteractions transform into pseudospin multipolar cou-plings, driving both structural and “spin-nematic” tran-sitions breaking cubic symmetry in real and pseudospinspaces. The JT-driven interactions are also importantin d systems, as they split the excited J = 1 levels,and hence promote magnetic condensation. In the caseof d systems with Kramers-degenerate J = 1 / / d +1 d +1 p y p x t pd π t pd π t (a) zz yz xyy x xyzx (b) zx yz z FIG. 4. (a) Hopping via oxygen in the case of 180 ◦ M-O-Mbonds. The orbital label of the two active t g orbitals (here zx and yz ) is conserved. The third orbital ( xy , not shown)cannot connect to the oxygen p states. (b) Combining orbitalsinto L α eigenstates with α determined by the bond direction,the above rules lead to a conservation of L α = ± L α = 0 xy -orbital is inactive. tails of pseudospin dynamics, in the form of pseudospin-lattice coupling. In general, the low-energy pseudospin Hamiltoniansmay take various forms depending on the electron config-uration d n and symmetry of the crystal structure. Sen-sitivity of orbital interactions to bonding geometry is adecisive factor shaping the form of the pseudospin Hamil-tonians. We illustrate this by considering spin-orbital ex-change processes in two different cases – when metal(M)-oxygen(O) octahedra MO share the corners, and whenthey share the edges. These two cases are common intransition-metal compounds and are referred to as 180 ◦ and 90 ◦ bonding geometry, reflecting the approximateangle of the M-O-M bonds. For simplicity, we limitourselves to the case of d ions with wavefunctions [c.f.Fig. 3(a)]: | f ˜ ↑ (cid:105) = + sin ϑ | , ↑(cid:105) − cos ϑ | + 1 , ↓(cid:105) , (2.1) | f ˜ ↓ (cid:105) = − sin ϑ | , ↓(cid:105) + cos ϑ | − , ↑(cid:105) , (2.2)where we represent the pseudospin-1 / f -fermion that is associated with a hole in the full t g configuration. The spin-orbit mixing angle is determinedby tan 2 ϑ = 2 √ / (1+2∆ /λ ), where ∆ is tetragonal split-ting of the t g orbital level. In the cubic limit of ∆ = 0shown in Fig. 3(a), one has sin ϑ = 1 / √
3, cos ϑ = (cid:112) / ◦ case presented in Fig. 4, the nearest-neighbor (NN)hopping Hamiltonian takes the form H (180 ◦ ) = − t ( a † iσ a jσ + b † iσ b jσ + H . c . )= − t ( d † +1 iσ d +1 jσ + d †− iσ d − jσ + H . c . ) . (2.3) d +1 d −1 t pd π p z xyd xyd t pd π p z (a) (b) −it+it (c) (d) yzyxx y −t’x yzyzx yzxzzx FIG. 5. Hopping in the case of 90 ◦ M-O-M bonding geom-etry. (a), (b) Two t g orbitals zx and yz are interconnectedby the hopping via oxygen p orbitals. They are selected bythe orientation of the M O plaquette. (c) The complemen-tarity of the orbital labels connected in the M-O-M bridgeresults in orbital moment non-conserving hopping when con-sidering L α eigenstates. Here α = z so that L z = +1 isflipped to L z = − ± it . (d) Direct overlap of d orbitals opens an additional hopping channel where the re-maining L z = 0 xy -orbital is active. A summation over the spin index σ = ↑ , ↓ is assumed.Two of the three t g orbitals participate in oxygen-mediated hopping with the amplitude t = t pdπ / ∆ pd ;the active pair { a , b } is selected by the bond direction α and may be combined into effective orbital moment L α = ± | d ± (cid:105) . For example, the z bond con-sidered in Fig. 4 picks up | a (cid:105) ≡ | yz (cid:105) and | b (cid:105) ≡ | zx (cid:105) thatform L z = ± | d ± (cid:105) = ∓ ( | yz (cid:105) ± i | zx (cid:105) ) / √ | c (cid:105) ≡ | xy (cid:105) ≡ | d (cid:105) cannot couple tothe mediating p -orbitals of oxygen for symmetry rea-sons. Since the NN hopping preserves both spin and or-bital in this case, pseudospin is also a conserved quantity.Projecting H (180 ◦ ) onto the f -doublet subspace definedabove, one indeed observes pseudospin-conserving hop-ping H = − t f ( f † i ↑ f j ↑ + f † i ↓ f j ↓ +H . c . ), which should there-fore lead to isotropic Heisenberg exchange H = J ( ˜ S i ˜ S j )with J = 4 t f /U .The situation in the case of 90 ◦ bonding geometryis completely different. As shown in Fig. 5, two bond-selected t g orbitals { a , b } spanning the | d ± (cid:105) subspaceare again active in the oxygen-mediated hopping t = t pdπ / ∆ pd , but their labels get interchanged during thehopping: a ↔ b , i.e. yz ↔ zx for z bond. The thirdorbital c corresponding to | d (cid:105) participates in direct hop-ping t (cid:48) . The two hopping channels are captured by the NN Hamiltonian: H (90 ◦ ) = t ( a † iσ b jσ + b † iσ a jσ ) − t (cid:48) c † iσ c jσ + H . c . = it ( d † +1 iσ d − jσ − d †− iσ d +1 jσ ) − t (cid:48) d † iσ d jσ + H . c . (2.4)In contrast to the 180 ◦ case discussed above, the t -hopping term does not conserve L z , but changes it by∆ L z = ± | d +1 (cid:105) ↔ | d − (cid:105) . Due to spin conservation,the total angular momentum projection has to changeby the same amount, i.e. ∆ J z = ±
2. However, suchhopping cannot connect pseudospin-1 / J z = ± f † i ↑ f j ↓ + H . c . );indeed, projection of the t -term in H (90 ◦ ) onto pseu-dospin f -space gives simply zero. This implies thatthe pseudospin wavefunctions cannot form d - p - d bondingstates, and thus the conventional pseudospin exchangeterm 4 t /U due to hopping t via oxygen ions is com-pletely suppressed. The situation is similar to the e g orbital exchange in the 90 ◦ bonding geometry, where e g orbitals cannot form d - p - d bonding states and thusno spin-exchange process is possible. As in the e g case,the pseudospin interactions in the edge-shared geome-try are generated by various corrections to the abovepicture (a direct overlap of pseudospins due to the t (cid:48) -term, electron hopping to higher spin-orbital levels, cor-rections to pseudospin wavefunctions due to non-cubiccrystal fields, etc.). The resulting pseudospin Hamiltoni-ans are typically strongly anisotropic, and the most im-portant and actually leading term in real compounds isbond-dependent Ising coupling. Figure 6 illustrates howsuch an interaction emerges due to the t -hopping fromground state J = 1 / J = 3 / H = K ˜ S zi ˜ S zj , and the corresponding couplingconstant K ∝ − ( J H /U ) 4 t /U is of ferromagnetic sign. Considered on honeycomb lattices, this interaction gener-ates the famous Kitaev model where the Ising axis is notglobal but bond dependent, taking the mutually orthog-onal directions x , y , and z on three different NN bonds. This results in strong frustration and a spin-liquid groundstate. On the other hand, a direct hopping t (cid:48) , which con-serves both orbital and spin angular momentum, leads toconventional AF Heisenberg coupling ∝ t (cid:48) /U .We will later discuss the pseudospin interactions inmore detail in the context of some representative com-pounds, after a brief materials overview. C. Materials overview
As discussed above, the interactions between spin-orbit-entangled pseudospins critically depend on thebonding geometry, and the ground states are determinedby the network of each bonding unit, namely crystalstructures. Before discussing the properties of repre-sentative materials, it would be instructive to overview ↑↑↓ (cid:12)(cid:12)(cid:12)(cid:12) , + (cid:29) d − ↓ d ↓ d − ↑ d ↑ d + ↓ d ↑ (cid:12)(cid:12)(cid:12)(cid:12) , − (cid:29) d ↑ d + ↓ J H L ∆ = ± z L ∆ z = t’ t FIG. 6. Virtual processes generating the effective interactions among pseudospins J = 1 / d configurations in 90 ◦ bondinggeometry. An M O plaquette perpendicular to the z axis as in Fig. 5 is assumed. Compared to Fig. 3(a), here we scale the d , d ± orbitals to visually hint on their relative contributions to the pseudospin wavefunctions. (left part) L z -conserving directhopping t (cid:48) uses the d part of the hole wavefunction and leads to a conventional Heisenberg exchange ˜ S i ˜ S j following the Pauliexclusion principle for the d orbital. (right part) Hopping via oxygen t takes the d +1 part of the hole wavefunction and bythe L z flip creates a virtual d configuration combining the original d hole and J z = − / J = 1 / t hopping is to remove this J z = − / S zi ˜ S zj type. Hund’s exchange J H between the major d ± parts of the two holes in the virtual d configuration prefers alignedpseudospins on the two sites which results in ferromagnetic Kitaev interaction K ˜ S zi ˜ S zj with K < the crystal structures that are frequently seen in the 4 d and 5 d transition-metal compounds. We will introducecrystal structures comprising the corner-sharing or edge-sharing network of MO octahedra.
1. Corner-sharing network of MO octahedra The most representative structure with corner-sharingMO octahedra is the perovskite structure with a chem-ical formula of ABO (A and B are cations). Smalltransition-metal ions are generally accommodated intothe B-site, and the BO octahedra form a three-dimensional corner-sharing network. The stability of theperovskite structure is empirically evaluated by the Gold-schmidt tolerance factor t = ( r A + r O )/ √ r B + r O ) where r A , r B and r O are the ionic radius of A, B and oxy-gen ions, respectively. Note that the perovskite structureconsists of alternate stacking of AO layer and BO layer. t = 1 means that the ionic radii are ideal to form a cubicperovskite structure [Fig. 7(a)] with the perfect match-ing of the spacings of constituent ions for AO and BO layers. As the ionic radius r B for 4 d and 5 d transition-metal ions is relatively large, the tolerance factor t of4 d and 5 d perovskites are normally less than 1, givingrise to lattice distortions to compensate the mismatchbetween AO and BO layers. A distorted perovskitestructure frequently found in 4 d and 5 d transition-metaloxides is the orthorhombic GdFeO -type (Space group P bnm ) [Fig. 7(b)]. The BO octahedra rotate about the c -axis and tilt around the [110] direction (Glazer nota-tion: a − a − c +16 ). Because of this distortion, the B-O-Bangle is smaller than 180 ◦ . Many 4 d and 5 d transition-metal perovskites such as CaRuO , NaOsO and SrIrO crystallize in the GdFeO -type structure. In addition to the three-dimensional network, thequasi-two-dimensional analogue with 180 ◦ bonding ge-ometry is realized in the layered derivatives of perovskitestructure. The square lattice of octahedrally-coordinatedtransition-metal ions is seen in the K NiF -type (A BO )layered structure, where the alternate stacking of (AO) -BO layers along the c -axis is formed. Generally, theBO octahedra are tetragonally distorted in the layeredperovskites such as Sr VO . As in the ABO -type per-ovskite, the mismatch of ionic radii of A and B cationsresults in the rotation and the tilting distortion of BO octahedra, making the B-O-B angle less than 180 ◦ . Forexample, the layered iridate Sr IrO possesses the ro-tation of IrO octahedra about the c -axis, whereasCa RuO hosts both a rotation and tilting distortion ofRuO octahedra. The K NiF (A BO )-type perovskite is an end mem-ber of a Ruddlesden-Popper series with a general chemi-cal formula of A n+1 B n O . This formula can be rewrit-ten as AO(ABO ) n [= AO(AO-BO ) n ], which makes iteasier to view the crystal structures; there are n-layers ofBO octahedra, sandwiched by the double rock-salt-typeAO layers as illustrated in Fig. 7(c). Generally, with in-creasing number of layers, n , the electronic structure be-comes more three-dimensional and hence the bandwidthincreases, which may induce a metal-insulator transition FIG. 7. Crystal structures of perovskite and its derivatives.(a) Cubic perovskite ABO . (b) Orthorhombic perovskitewith the GdFeO -type distortion. (c) Ruddlesden-Popper se-ries layered perovskite A n+1 B n O . (d) Double-perovskiteA B (cid:48) B (cid:48)(cid:48) O (left) and A MX -type halides (right). The crys-tal structures are visualized by using VESTA software. as a function of n .The double-perovskite structure is a derivative of per-ovskite, and also called rock-salt-ordered perovskite. Inthe double-perovskites, two different cations, B (cid:48) and B (cid:48)(cid:48) ,occupy the octahedral site alternately and form a rock-salt sublattice. The ordered arrangement of two differentB cations is usually seen when the difference of valencestate of the two cations is more than 2. Both B (cid:48) andB (cid:48)(cid:48) ions comprise a face-center-cubic (FCC) sublattice.Note that there is neither a direct B (cid:48) -O-B (cid:48) bond nor aB (cid:48)(cid:48) -O-B (cid:48)(cid:48) bond. The FCC lattice with d or d ions hasbeen proposed to be a possible realization of multipolarordering of d electrons, and the double-perovskite ox-ides with magnetic B (cid:48) and nonmagnetic B (cid:48)(cid:48) ions havebeen studied intensively as will be discussed in SectionV. The FCC lattice of d or d ions is also realized inthe series of transition-metal halides with a chemical for-mula A MX (A + : alkali ion, M : transition-metal ion,and X − : halogen ion), where MX − octahedra and A + ions form anti-fluorite-like arrangement. This structure can be viewed as a B (cid:48)(cid:48) -site deficient double-perovskiteA M (cid:3) X , where (cid:3) denotes a vacancy. A MX crys-tallizes in a cubic structure with a large A ion such asCs + . M ions with an ideal cubic environment form aFCC sublattice, but also form a distorted structure whenthe size of A ion is small. A wide variety of 4 d and 5 d transition-metal elements can be accommodated into thisstructure.Another important class of materials with corner-sharing MO octahedra is the pyrochlore oxide with ageneral formula A B O (more specifically A B O O (cid:48) where O and O (cid:48) represent two different oxygen sites)[Fig. 8(a)]. Transition-metal ions are accommodatedinto the B-cation site, which forms a BO octahedron,whereas the A-cation is surrounded by six O and two O (cid:48) atoms in a distorted cubic-like environment. The sub-lattice of the B-cations, as well as that of A-cations, is anetwork of corner-shared tetrahedra called the pyrochlorelattice [Fig. 8(c)]. The pyrochlore lattice is known toprovide geometrical frustration when magnetic momentsof constituent ions interact antiferromagnetically. Thereare many ways to view the pyrochlore structure as de-scribed in Ref. [24]. Most conventionally, the B atom islocated at the origin of unit cell (Wyckoff position 16 c )for the space group F d m (No. 227, origin choice 2). Inthis setting, the only tuneable parameters are the latticeconstant and the x coordinate of the O site. With x = 0.3125, the BO forms an ideal octahedron and theB-O-B angle is ∼ ◦ . In 4 d and 5 d transition-metal py-rochlore oxides, x is usually larger than 0.3125, and theBO octahedra are compressed along the [111] directionpointing to the center of B-tetrahedra. The compres-sive distortion gives rise to a trigonal crystal field on Bions and decreases the B-O-B angle between the neigh-boring octahedra from 141 ◦ , which reduces the hoppingamplitude and thus bandwidth. When the ionic radiusof the A atom becomes smaller, the trigonal distortionis enhanced. A metal-insulator transition is seen as afunction of the size of A ions in pyrochlore oxides such asmolybdates A Mo O and iridates A Ir O (A: trivalentions such as rare-earth or Y ). The trigonal crystalfield which splits the t g manifold potentially competeswith SOC.
2. Edge-sharing network of MO As discussed above, the edge-sharing, namely 90 ◦ M-O-M, bonding geometry provides magnetic interactionsdistinct from those in 180 ◦ bonds. With the edge-sharingnetwork of MO octahedra, one can realize a variety oflattice structures of interest, such as the triangular lat-tice in ABO , the honeycomb lattice in A BO and thepyrochlore lattice in AB O spinels. They can be con-structed from the rock-salt structure.To derive the layered triangular and honeycomb struc-tures from the rock-salt-type B (cid:48)(cid:48) O − (B (cid:48)(cid:48) : transition-metal atom), first consider the rock-salt structure viewed FIG. 8. Crystal structures of (a) pyrochlore oxide A B O and (b) spinel oxide AB O . (c) Pyrochlore sublattice com-prised by B (or A) atoms of pyrochlore oxide or by B atomsof spinel oxide. (d) Hyperkagome sublattice of Ir atoms foundin Na Ir O . The pyrochlore sublattice is shared by 3:1 ratioof Ir and Na atoms in an ordered manner. along the cubic [111] direction [Fig. 9(a)]. It consists ofan alternating stack of the triangular B (cid:48)(cid:48) planes and thetriangular O − planes. By replacing every pair of adja-cent B (cid:48)(cid:48) planes with an A + plane and B (cid:48) plane, wehave the layered AB (cid:48) O -type structure with triangularlayers of A + and B (cid:48) [Fig. 9(b)]. The B (cid:48) O octahedraform the edge-shared triangular lattice. The trivalentB (cid:48) can be replaced by a 2:1 ratio of B and A + ions.The large difference of valence states between A + andB cations facilitates the ordered arrangement of twocations in the triangular plane. As a result, the A / B / layers contain a honeycomb network of BO octahedraconnected by three of their six edges [Fig. 9(c)]. The al-ternate stacking of an A + -cation layer and an A +1 / B / layer corresponds to the chemical formula A BO [=A(A / B / )O ] as found in Na IrO and Li RuO . The three-dimensional honeycomb structure of β -Li IrO and γ -Li IrO can be derived from the rock-salt structureas well, but the ordering pattern of Li + and Ir ionsare different from the [111] ordering described above. Those 4 d and 5 d transition-metal oxides with a honey-comb network are attracting attention as a realization ofexotic quantum magnetism.In the ordered rock-salt structures described above,all octahedral voids created by oxygen atoms are filledby cations. The rock-salt structure also has tetrahedralvoids that can be occupied by cations. By partially fill-ing the tetrahedral and octahedral voids, a spinel struc-ture AB O can be constructed [Fig. 8(b)]. In the spinelstructure, B cations form a network of corner-sharedtetrahedra as in the pyrochlore oxides. The crucial dif-ference from A B O pyrochlore oxides is that the BO FIG. 9. (a) Rock-salt structure of B (cid:48)(cid:48)
O. (b) AB (cid:48) O -type struc-ture formed by replacing B (cid:48)(cid:48) with A + and B (cid:48) ions whichstack alternately along the c -axis. (c) Layered honeycombstructure of A BO . The triangular layer of B (cid:48) ions in (b)is substituted by the 2:1 ratio of B and A + ions forming ahoneycomb lattice. octahedra in the spinel structure are connected by edge-sharing bonds. The number of spinel oxides containing4 d or 5 d transition-metal atoms is rather limited. Whenmultiple cations occupy the pyrochlore B-sublattice, theymay form an ordered arrangement. The prominent ex-ample is hyperkagome iridate Na Ir O . In Na Ir O ,the B-site pyrochlore lattice of the spinel is shared in a3:1 ratio of Ir and Na atoms. The Ir sublattice is viewedas corner-shared triangles in three dimensions, which hasbeen dubbed the hyperkagome lattice [Fig. 8(d)]. Theproperties of hyperkagome iridate will be discussed inSections III.C and VI.C. III. PSEUDOSPIN-1/2 MAGNETISM IN d COMPOUNDS
The collective behavior of d ions with Kramers dou-blet ground states can be described in terms of apseudospin-1 / In Sec. II, we emphasized the differ-ence between the 180 ◦ and 90 ◦ bonding geometry thatlead to either conventional Heisenberg interaction orstrongly frustrated bond-selective interactions of the Ki-taev type. Focusing on these two cases, we now considera few representative examples of d compounds realizingpseudospin-1 / A. 180 ◦ M-O-M bonding, perovskites
The perovskite iridate Sr IrO has emerged as amodel system for understanding the spin-orbit-entangledmagnetism of 5 d electrons. The first experimentalevidence for the J = 1 / and later by resonant elastic x-ray scattering, which confirmed the complex structureof the d hole.The relevant pseudospin-1 / octahedra in the per-ovskite structure with approximately 180 ◦ Ir-O-Ir bonds(the bonds are not completely straight due to 11 ◦ in-plane octahedra rotations ). Since the pseudospin wave-functions overlap well in the d - p - d hopping channel, andhopping is pseudospin conserving as discussed in Sec.II.B,the dominant interaction is represented by Heisenbergcoupling. However, there are additional terms whicharise due to hoppings to higher level orbitals, tetrago-nal distortions and octahedral rotations, and these leadto the following NN-interaction Hamiltonian: H = J ( ˜ S i ˜ S j ) + D ( ˜ S i × ˜ S j ) + A ( ˜ S i r ij )( ˜ S j r ij ) + J z ˜ S zi ˜ S zj . (3.1)Here, the D term is an antisymmetric Dzyaloshinskii-Moriya (DM) interaction caused by octahedral rotationsand A and J z represent symmetric anisotropy terms.The J z term is derived from the combined effect of theoctahedral rotations and tetragonal fields, while the Aterm with dipole-dipole-coupling type bond-directionalstructure is symmetry allowed even in an ideal cubicstructure. The coupling constants have been calcu-lated in Ref. [14], and vary as a function of tetrago-nal crystal field, rotation angle etc. This Hamiltoniannicely accounts for a number of properties of Sr IrO ,including nearly Heisenberg spin dynamics akin to thecuprates. A closer look in magnon data reveals that the modelabove has to be extended, including longer-range interac-tions J and J , which turn out to be much larger than incuprates. This might be related to the fact that 5 d elec-trons are more extended spatially, and to the relativelysmall Mott gap. More recently, it has been shown thatthe pseudospins in iridates couple to lattice degrees offreedom via a dynamical admixture of higher-lying spin-orbital levels to the ground state wavefunctions, whichexplains the observed in-plane magnon gaps, and pre-dicts sizable magnetostriction effects breaking tetragonalsymmetry below T N .In addition to elastic x-ray scattering, the spin-orbit-entangled nature of the d ions in Sr IrO was de-tected by resonant inelastic x-ray scattering (RIXS) ex-periments that observed transitions from J = 1 / J = 3 / similar to doped holes inantiferromagnetic cuprates. Also like in the hole-dopedcuprates, spin-excitation spectra obtained by RIXS onLa-doped Sr IrO revealed paramagnons persistentwell into the metallic phase.Encouraged by the above analogies with cuprates,doped Sr IrO samples have been studied in the searchfor unconventional superconductivity. However, experi-ments on doped Sr IrO are severely impeded by difficul-ties in obtaining clean samples. Techniques beyond con-ventional chemical doping such as La-substitution pro-ducing electron doped Sr − x La x IrO have to be em-ployed. For example, surface electron doping achieved bypotassium deposition on the surface of parent Sr IrO en-abled ARPES and scanning tunneling microscopy (STM)studies; another promising route is ionic liquidgating. Although no clear evidence for superconduc-tivity was so far detected, Fermi surface and pseudo-gap phenomena as in cuprates have been observed inARPES and STM experiments. For more a de-tailed account on doped Sr IrO , we recommend the re-cent review. Next, we briefly discuss the bilayer iridate Sr Ir O .Being “in-between” quasi-two-dimensional insulatorSr IrO and three-dimensional metal SrIrO , this com-pound is close to the Mott transition, with a small insu-lating gap. Nonetheless, pseudospin-1 / J = 3 / showing that the spin-orbit-entangled nature of low-energy states remains largely in-tact. A remarkable observation is that the magnetic mo-ment direction and magnon spectra in this compoundare radically different from those in the sister compoundSr IrO . Possible explanations for this have been offered– based on enhanced anisotropic pseudospin couplings and on dimer formation on the links connecting the twolayers. B. 90 ◦ M-O-M bonding, honeycomb lattice
The case of J = 1 / IrO (see Fig. 9), may realize a Kitaev honeycombmodel. Since there is already a vast literature on thistopic, including several review articles, we will makejust a few remarks concerning the “unwanted” (i.e. non-Kitaev) exchange terms that are present in the Kitaev-model candidate materials studied so far.As explained in Sec.II.B above, the pseudospin-1 / ◦ case) do originate from higher order pro-1cesses involving spin-orbit J = 3/2 virtual states, or fromcommunication of the pseudospins via a direct t (cid:48) hopping,as illustrated in Fig. 6. Hoppings to higher lying e g stateswith subsequent Hund’s coupling, as well as pd charge-transfer excitations do also contribute. Phenomenologi-cally, symmetry considerations dictate the following gen-eral form of NN interactions H = K ˜ S zi ˜ S zj + J ( ˜ S i ˜ S j ) + Γ( ˜ S xi ˜ S yj + ˜ S yi ˜ S xj )+ Γ (cid:48) ( ˜ S xi ˜ S zj + ˜ S zi ˜ S xj + ˜ S yi ˜ S zj + ˜ S zi ˜ S yj ) , (3.2)which is expressed here for an M O plaquette perpen-dicular to the cubic z axis, as shown in Fig. 5. Thefirst term represents the Kitaev interaction. Accord-ing to perturbative calculations, the off-diagonal ex-change Γ arises from combined t and t (cid:48) hoppings, whilethe Γ (cid:48) term is generated by trigonal crystal fields thatmodify the pseudospin wavefunctions. Trigonal field alsosuppresses the bond-dependent nature of the pseudospininteractions, so the cubic limit is desired to supportthe Kitaev coupling K and to suppress the Γ (cid:48) term. How-ever, the J and Γ terms, generated by a direct hopping t (cid:48) and other possible hopping channels, remain finite evenin the cubic limit. The crucial parameter here is the M-Mdistance that controls the magnitude of the direct overlapof the d -wavefunctions.Apart from NN J , Γ, and Γ (cid:48) terms, the longer-range(second/third NN J / ) interactions are likely present inmany Kitaev materials. These couplings, which are detri-mental to the Kitaev spin liquid, are also related to thespatial extension of the 4 d and 5 d orbitals, and to thelattice structure “details” such as the presence/absenceof cations (e.g. Li or Na) within or near the honeycombplanes [see Fig. 9(c)], opening additional hopping chan-nels.Nevertheless, the Kitaev-type couplings appear to bedominant in 5 d -iridates and also 4 d -ruthenium chloride,as evidenced by a number of experiments (see, e.g., thereview [57]), although they are not yet strong enoughto overcome the destructive effects of non-Kitaev termsdiscussed above. So the efforts to design materials withsuppressed “unwanted” interactions have to be contin-ued. An interesting perpective in this context may bethe employment of J = 1 / ions, as has beenproposed recently. Although the energy scales forthe pseudospin interactions are smaller in this case, theless-extended nature of 3 d wavefunctions may reduce thelonger-range couplings, improving thus the conditions forthe realization of the Kitaev model.The non-Heisenberg, bond-dependent nature of thepseudospin interactions is expected to survive in weaklydoped compounds, and affect their metallic properties.In particular, it has been suggested that they should leadto an unconventional p -wave pairing. However, ex-perimental data on doped d compounds with 90 ◦ bond-ing geometry is scarce, because a “clean” doping of Mottinsulators is a challenge in general. C. Pseudospins-1/2 on frustrated lattices
A combination of geometrical frustration with spin-orbital frustration may open an interesting pathwayto exotic magnetism. In fact, the bond-dependentpseudospin interactions have been first discussed inthe context of geometrically frustrated triangular andhyperkagome lattices. Here we discuss some spin-orbit-entangled pseudospin-1/2 systems with geometri-cally frustrated lattices.
1. Hyperkagome iridate Na Ir O Among complex iridium oxides, Na Ir O appears tobe the first example of an exotic quantum magnet. InNa Ir O , Ir atoms share the B-site pyrochlore lattice ofthe spinel with Na atoms, and form a network of corner-shared triangles dubbed the hyperkagome lattice, as de-scribed in Sec.II.C.2. The hyperkagome lattice is geo-metrically frustrated if the Ir moments couple antiferro-magnetically.Indeed, Na Ir O displays a strong antiferromagneticinteraction inferred from the large negative Weiss tem-perature ( | θ CW | ∼
650 K) in the magnetic susceptibility χ ( T ). Nevertheless, no sign of magnetic ordering hasbeen seen down to 2 K both in χ ( T ) and the specificheat C ( T ) as shown in Fig. 10. Na Ir O thus appearedas the first candidate for a three-dimensional quantumspin liquid.In the χ ( T ) of Na Ir O , a small bifurcation wasseen at around 6 K [the inset to Fig. 10(a)]. It wasoriginally interpreted as a glassy behavior due to asmall amount of impurity/defects. However, it has beenpointed out from the Na-NMR and µ SR measurementsthat Na Ir O exhibits a spin-glassy frozen state or qua-sistatic spin correlation. We note that the presenceof such a glassy state may be associated with the disor-der of Na ions in the octahedral A-site. Since the suc-cessful growth of Na Ir O single crystals was reportedrecently, the understanding of its magnetic groundstate is advancing.After the discovery of the spin-liquid behavior inNa Ir O , a plethora of theoretical studies have beenput forward. In the early days, most of the modelshave treated the Ir moments as S = 1/2 and consideredthe antiferromagnetic Heisenberg model on the frustratedhyperkagome lattice. The classical model predicted thepresence of nematic order. For the quantum limit, theground state has been discussed to be either a spin-liquid with spinon Fermi surface or a topological Z spinliquid .Since the IrO octahedra form an edge-sharing net-work, as in honeycomb-based iridates, the presence ofanisotropic magnetic exchanges such as Kitaev-type cou-pling is anticipated. The electronic structure calculationshowed that SOC of Ir gives rise to a split of the t g orbitals into spin-orbit-entangled states with J = 1/22 FIG. 10. Temperature dependence of (a) inverse magneticsusceptibility χ − ( T ), and (b) magnetic specific heat C m di-vided by temperature. Insets: (a) Temperature dependenceof χ ( T ) at various magnetic fields. (b) C m /T at various mag-netic fields, showing a power law behavior C m ( T ) ∼ T n (2 < n <
3) at low temperatures. The figure is reproducedwith permission from Ref. [31] ( © and 3/2 characters. In the pure Kitaev limit on a hy-perkagome lattice, a nonmagnetic ground state, possiblyspin-liquid or valence-bond-solid, has been postulated. In reality, as in honeycomb iridates, other magnetic ex-changes are present, and the antiferromagnetic Heisen-berg coupling seems to be the leading term. A micro-scopic model that takes into account both the Heisenbergterm and anisotropic exchanges, such as the DM interac-tion, predicts the emergence of q = 0 noncoplanar orderor incommensurate order depending on the magnitudeof the anisotropic terms. The spin-glass state ofNa Ir O may be associated with the presence of suchcompeting magnetic phases.
2. Pseudospin-1/2 on pyrochlore lattice
In addition to the hyperkagome lattice, the pyrochlorelattice of pseudospin-1/2 states with an edge-sharedbonding geometry is found in an A-site deficient spinelIr O . Theoretically, Ir O is discussed to host spin-ice-type “2-in-2-out” magnetic correlation and a U(1)quantum spin-liquid state is predicted under tetragonalstrain. Ir O has been obtained only in a thin-film form,and its magnetic properties remain yet to be investigated.The frustrated magnetism of Ir moments is also re-alized in pyrochlore oxides A Ir O (A: trivalent cation). The electronic ground state of A Ir O depends on thesize of the A-ion (ionic radius r A ), which likely controlsthe bandwidth of Ir 5 d electrons. With the largest r A inthe family of A Ir O , Pr Ir O exhibits a metallic be-havior down to the lowest temperature measured. With aslightly smaller r A such as Nd , Sm or Eu , A Ir O shows a metal-to-insulator transition as a function oftemperature, accompanied by a magnetic order. Foran even smaller r A than that of Eu , A Ir O remainsinsulating up to well above room temperature while mag-netic ordering takes place only at a low temperature,pointing to the Mott insulating state. We focus hereon the magnetism of Ir pseudospin-1/2 moments in theMott insulating ground state. The metallic states of thepyrochlore iridates will be discussed in Sec.VI.B.For the pyrochlore iridates in the insulating limit, thelocal electronic state of Ir 5 d electrons is primarily of J = 1/2 character, but a sizable mixing of the J = 3/2components is present. The mixing was attributed tothe presence of a trigonal crystal field, which is generatednot only by the oxygen cage but also by the surround-ing cations. Recently, it turned out that the inter-sitehopping plays a dominant role in the J = 3/2 mixing. Infact, by suppressing the hopping, a nearly pure J = 1/2state can be realized. The magnetic interaction between the J = 1/2 pseu-dospins is predominantly attributed to the antiferromag-netic superexchange interaction via oxygen ions, wherethe Ir-O-Ir angle is approximately 130 ◦ . Although theantiferromagnetic Heisenberg model on the pyrochlorelattice is predicted to show no magnetic ordering down to0 K, a DM interaction is present in the pyrochlore ox-ides as there is no inversion symmetry between the NN Iratoms. It was shown in the spin Hamiltonian includingHeisenberg and DM interaction on a pyrochlore latticethat the positive DM term gives rise to the all-in-all-out(AIAO) magnetic order, where all the magnetic momentson a tetrahedron of pyrochlore lattice are pointing inwardor outward along the local [111] direction [Fig. 11(a)]. On the other hand, when the DM term is negative, non-coplanar XY-type magnetic order appears where the mo-ments lie in the plane perpendicular to the local [111]direction.The magnetic structure of pyrochlore iridates has beenstudied by resonant x-ray scattering, and the q = 0magnetic order of Ir moments was revealed in Eu Ir O [Fig. 11(b)] . The q = 0 propagation vector sug-gests the formation of the AIAO magnetic ordering orthe non-coplanar XY antiferromagnetic order, as ex-pected for the presence of DM interactions. In Sm Ir O and Eu Ir O , the gapped magnon dispersion revealedby RIXS [Fig. 11(c)] supports the AIAO order of Irmoments. The AIAO magnetic order is discussed togive rise to a Weyl semimetallic state in the vicinity ofthe metal-insulator transition. FIG. 11. (a) All-in-all-out (AIAO) magnetic ordering on apyrochlore lattice. (b) The q = 0 magnetic order revealed byresonant x-ray scattering in Eu Ir O at the Ir L absorp-tion edge. The peaks at A(B) correspond to Ir 2 p / → t g (2 p / → e g ) excitation, respectively. The strong resonantenhancement at A in the σ - π (cid:48) channel points to a magneticscattering. The resonant enhancements in the σ - σ (cid:48) channel atboth A and B originate from anisotropic tensor susceptibility(ATS) scattering. The figure is reproduced with permissionfrom Ref. [93] ( © Ir O ob-tained from RIXS. The black dots are the experimental datapoints and the blue dotted lines show the calculated magnondispersion assuming AIAO magnetic order. The figure is re-produced with permission from Ref. [94] ( © IV. J = 0 SYSTEMS: EXCITONIC MAGNETISM Perhaps the most radical impact of SOC on magnetismis realized in compounds of 4 d and 5 d ions with d con-figuration, such as Re , Ru , Os , and Ir . Forthese ions with spatially extended d -orbitals, Hund’s cou-pling is smaller than the octahedral crystal field splitting10 Dq , so all four electrons occupy t g levels. The result-ing t g configuration has total spin S = 1 and threefoldorbital degeneracy described by an effective orbital mo-ment L = 1. Despite having well defined spin and or-bital moments on every lattice site, some d compoundslack any magnetic order. This is because SOC λ SL with λ > S and L moments into a local singlet statewith zero total angular momentum J = 0, as shown inFig. 1.Nonetheless, these nominally “nonmagnetic” ionsmay develop a collective magnetism due to interactioneffects. Although there are no preexisting local mo-ments in the ionic ground state, the J = 1 excitationsbecome dispersive modes in a crystal, and these mobile spin-orbit excitons may condense into a magnetically or-dered state. For this to happen, the exchange interac-tions should exceed a critical value sufficient to over-come the energy gap λ between J = 0 and J = 1 ionicstates. The condensate wavefunction comprises a coher-ent superposition of singlet and triplet states and carriesa magnetic moment, whose length is determined by thedegree of admixture of triplets in the wavefunction. Nearthe quantum critical point (QCP), the ordered momentcan be very small, and the magnetic condensate stronglyfluctuates both in phase and amplitude (i.e. rotation ofmoments and their length oscillations). Formally, this isanalogous to magnon condensation phenomenon in quan-tum dimer systems, but the underlying physics andenergy scales involved here are different. While the spingap in dimer models originates from antiferromagneticexchange of two spins forming a dimer, the magnetic gapin d systems is of intraionic nature and given by SOC.Spin-orbit-entangled J = 0 compounds are interest-ing for possible novel phases near the magnetic QCP,which can be driven by doping, pressure, and lattice con-trol. Here, the new element is that J = 1 excitons arespin-orbit-entangled objects, and, as we will see shortly,their interactions can be anisotropic and highly frustrat-ing. Thus, J = 0 systems with “soft” moments cannaturally realize interplay between the two phenomena– frustration and quantum criticality – a topic of currentinterest. As a general property of orbitally degeneratesystems, the symmetry and low-energy behavior of spin-orbit excitonic models is dictated by chemical bondinggeometry, and we discuss two representative cases below.
A. 180 ◦ M-O-M bonding, perovskites
Similarly to the pseudospin-1 / d case, the straight180 ◦ bond geometry leads to a nearly isotropic model; inthe following we thus first focus on the isotropic model of O (3) symmetry. When approaching the excitonic mag-net formally, we need to properly reflect the ionic levelstructure with nonmagnetic J = 0 ground state andlow-lying J = 1 excitations. The most natural wayis to introduce hardcore bosons associated with the lo-cal excitation J = 0 →
1. These bosons, called heretriplons T , come with the energy cost λ reflected bya local term λn T = λT † T and experience various pro-cesses corresponding to the exchange interactions be-tween different sites. In the second order in T opera-tors, they include triplon hopping and creation or anni-hilation of triplon pairs in the symmetry-allowed com-bination ∝ ( T +1 T − − T T + T − T +1 ) ij , where the in-dices ( ± , J z of the triplons. Instead of J z eigenstates, it is convenient to use the basis consisting ofthree triplon operators T α of Cartesian flavors (“colors”) α = x, y, z defined as T x = i √ ( T − T − ) , T y = √ ( T + T − ) , T z = iT , (4.1)4 J= J= d exchange / λ triplongas J= J= d ω q ω q (a) E (b) 0 QCP λ SOC triplon condensate o r d e r e d m o m e n t (c) exchange (0,0) (π,π) q triplon (0,0) (π,π) q magnonHiggs x T y T FIG. 12. (a) In the singlet-triplet model for d systems withlarge SOC, each site is supposed to host nonmagnetic J = 0ground state and low-lying J = 1 triplet excitations at theenergy λ . Competing with the intrasite SOC gap λ are vari-ous intersite exchange processes such as a transfer of a J = 1excitation to a neighboring site or their pairwise creation andannihilation. (b) Ordered moment in an excitonic magnetdepending on the ratio of exchange strength versus the localSOC gap λ . The quantum critical point (QCP) separates thelarge- λ phase where “costly” triplet excitations move in an in-coherent way and the phase where the condensate of tripletsis established. (c) Schematic dispersions of the elementaryexcitations. Before the condensation, the elementary excita-tions are carried by triplons whose dispersion softens near theAF momentum as the QCP is approached. Once the triploncondensate is formed, the oscillations of its amplitude andthe moment direction become the new fundamental modes –“Higgs” mode and magnons, respectively. The red spot at ( π , π ) represents a magnetic Bragg point. which form a vector T . The main contribution to theexchange J -Hamiltonian derived in Ref. [98] then takesa manifestly O (3) symmetric form: H = λ (cid:88) i T † i T i + (cid:88) (cid:104) ij (cid:105) J ij (cid:16) T † i T j − T † i T † j + H . c . (cid:17) . (4.2) E / ζ ∆ / ζ J z = J z = ± J z = ± J = z J= ∆ = 0 J= ∆ = +2ζ J = z J = z ± (b)(a) J = z ∆ = −2ζ FIG. 13. (a) Splitting of d levels in tetragonal crystalfield (measured in units of ζ = 2 λ ). Out of the three tripletexcitations, MeO octahedra elongation/compression selectssingle J z = 0 state or the pair of J z = ± J z = 0 ionic ground state (evolving from cubic J = 0state), they form a local basis for an effective spin-1/2 (at∆ <
0) or spin-1 (at ∆ >
0) low-energy models. (b) Shapesof the relevant low-energy states represented in the same wayas in Fig. 1 (electron density is used, not the hole one).
The phase diagram of this model is determined by thecompetition of the triplon cost λ and superexchange cou-pling J as schematically shown in Fig. 12(b). At suf-ficient strength of the superexchange, triplons undergoBose-Einstein condensation like in spin-dimer systems. However, the physical meaning of triplons is very dif-ferent here. Since the magnetic moment of a d ionresides primarily on the transition between the J = 0and J = 1 states, as described by T , the presence of atriplon condensate with (cid:104) T (cid:105) ∝ i e i QR , where Q is theordering vector, directly translates to long-range mag-netic order. The resulting magnetic order is character-ized also by an unusual excitation spectra, see Fig. 12(c).The condensation is preceded by softening of the three-fold degenerate triplon modes near Q . After condensa-tion, the modes split, giving rise to a two-fold degener-ate magnon branch with XY -type dispersion (i.e. withmaximum at q = 0) and the amplitude (Higgs) mode ofthe condensate. These two hallmarks of soft-spin mag-netism can be probed experimentally, as was done in the J = 0 model system Ca RuO with d Ru ions. RuO was identified asa Mott insulator showing a metal-insulator transi-tion around 360 K and antiferromagnetic order below T N ≈ Early experiments revealed thatSOC induces a substantial orbital angular momentum inRu 4 d levels, supporting the above J = 0 picture. InRu ions, the SOC strength is roughly ζ ≈
150 meV,giving the magnetic gap between J = 0 and 1 states ofthe order of λ = ζ/ ≈
75 meV. Compared to λ , ex-change interactions are somewhat smaller in ruthenatesand would not be able to overcome such a gap. How-ever, as shown in Fig. 13(a), tetragonal distortion andthe associated crystal field ∆ splits the J = 1 excitationand may reduce the gap significantly. The lower doublet T x/y (for the ∆ > RuO ) can thencondense, giving rise to magnetic order, with the orderedmoment in the RuO plane. This scenario points to aninteresting possibility of a lattice-controlled QCP (e.g.by strain) instead of by magnetic field or high-pressureas in the dimer system TlCuCl . This also suggeststhe importance of the Jahn-Teller effect even in J = 0systems, which acts through the splitting of triplon levelsand renormalization of the ground state wavefunction. In Ca RuO with ∆ >
0, one of the J = 1 states islifted up by the crystal field, and we are left with threelow-energy states: ground state singlet and the exciteddoublet, evolving from cubic J = 0 and J z = ± T x/y )states, respectively. These three states, whose wavefunc-tions are shown in the right hand side of Fig. 13(b), canbe used as a local basis for an effective spin-1. The result-ing ˜ S = 1 Hamiltonian obtained by mapping Eq. (4.2)onto this basis has the form of the XY model with a largesingle-ion anisotropy: H = E (cid:88) i ( ˜ S zi ) + J (cid:88) (cid:104) ij (cid:105) (cid:16) ˜ S xi ˜ S xj + ˜ S yi ˜ S yj (cid:17) , (4.3)where E denotes a singlet-doublet excitation gap that issmaller than the singlet-triplet splitting λ in Eq. (4.2)due to a crystal field effect. As a result, the exchangeinteraction J may overcome the reduced spin gap E andinduce magnetic order.The expected XY -type of magnon dispersion wasindeed observed by inelastic neutron scattering onCa RuO . The experimental dispersion presented inFig. 14(a) additionally features a large magnon gap dueto orthorhombicity, which is not included in the abovesimplified model. The observation of the amplitudeHiggs mode is to some extent hindered by its strongdecay into a 2D two-magnon continuum (as predictedtheoretically ), which makes it a very broad featurein the INS spectra near the AF wavevector Q = ( π, π ),see Fig. 14(c). On the other hand, the mode is relativelywell defined away from Q as visible in Figs. 14(a),(b).A more direct probe of the Higgs mode that enters INSspectra at momentum Q is the Raman scattering in theusually magnetically silent A g channel. In Ref. [103], aHiggs mode in the scalar channel, “unspoiled” by thetwo-magnon continuum, was identified in Raman spec- ω ( m e V ) wavevector q π /2, π /2) ( π ,0) ( π , π ) (0,0) ( π ,0) L T T ′ i n t e n s it y ( a . u . ) ω (meV) q =( π , π ) T L T ′ i n t e n s it y ( a . u . ) q =(0,0) polarized INS, spin-flip channel abc 0 2 4 6 8 T LT ′ (a)(c)(b) FIG. 14. (a) Magnetic excitations of Ca RuO mapped byinelastic neutron scattering. The lines show model dispersionsobtained within the model of Ref. [102] (its simpler versionis described in the text). The red line indicates longitudi-nal mode L corresponding to the Higgs mode, and the bluelines represent the in-plane magnon T (solid line) and out-of-plane magnon T’ (dashed line). Upper and lower insets arepictorial representations of the Higgs mode (condensate am-plitude oscillations) and magnons (rotations of magnetic mo-ments), respectively. (b) Magnetic response at zero wavevec-tor q = (0 ,
0) obtained by polarized INS. In-plane polariza-tion ( ab , squares) and out-of-plane polarization ( c , circles)were resolved in the experiment. The experimental data areoverplotted on top of the theoretical magnetic response thatis decomposed according to the polarization of the modes. (c)The same as in (b) for the ordering wavevector q = ( π, π ).All the data are taken from Ref. [102] ( © tra of Ca RuO , and found to couple to phonons, givingthem pronounced Fano-like lineshapes. Such an interac-tion with lattice modes is natural for triplons, since theyhave a “shape” inherited from orbitals, and hence cou-ple to lattice vibrations via the Jahn-Teller mechanismas mentioned above.Spin-orbit exciton condensation and related magnetic6QCP are more likely realized in 4 d compounds such asruthenates, where SOC and exchange interactions are ofcomparable scale and their competition can be tunedexperimentally. On the other hand, the 5 d ion (Ir or Os ) compounds are typically nonmagnetic, sincespin-orbit J = 1 excitations are too high in energy, asevidenced by RIXS experiments in 5 d -electron doubleperovskites. Weak magnetism detected in the 5 d iridate has been explained as originating from the Ir and Ir magnetic defects, while the regular Ir sitesremain indeed nonmagnetic. B. 90 ◦ M-O-M bonding, honeycomb lattice
When contrasting the 180 ◦ and 90 ◦ bonding geome-tries, we encounter a situation analogous to the J = 1 / ◦ bonding geometry gen-erates (in leading order) the isotropic O (3) model ofEq. (4.2), discussed above, the bonds with 90 ◦ oxy-gen bridges are highly selective in terms of the activeflavors for the triplon interactions. Roughly speaking,when the oxygen-mediated hopping t dominates, eachbond allows exchange processes of the type containedin Eq. (4.2) for two triplon flavors only, depending onthe bond direction. For instance, the T x and T y bosonsare equally active in z -type bonds, while the T z bosoncannot move in that direction. For the honeycomb lat-tice, the resulting pattern of active triplon pairs is pre-sented in Fig. 15(a). On the other hand, the dominantdirect hopping t (cid:48) leads to the complementary Kitaev-likepattern of Fig. 15(b), with a direct correspondence be-tween the bond direction α and triplon flavor T α active onthat bond. Each of these two cases is strongly frus-trated; however, the nature of the corresponding groundstates is very different.In the interaction pattern of Fig. 15(a), each bondshows an O (2) symmetry of the triplon exchange Hamil-tonian, that is, H ( z ) ij = J (cid:88) α = x,y ( T † αi T αj − T † αi T † αj + H . c . ) (4.4)for a z -bond (cid:104) ij (cid:105) . However, the global symmetry ofthe model is only the discrete C one. Namely, thereare three zigzag chains (colored differently), along whichthe individual triplon flavors can move. This arrange-ment does not support 2D long-range order but insteadleads to effective dimensionality reduction like in compassmodels: C symmetry is broken by selecting one par-ticular triplon component, with antiferromagnetic corre-lations along the corresponding 1D zigzag. Zigzag chainsinteract via the hard-core constraint only (triplon densitychannel), so there are no phase relations and magneticorder between different chains. The resulting magneticcorrelations are highly anisotropic, both in real and spinspaces. This is a combination of an orbital ordered andspin-nematic state, made possible due to spin-orbital en-tanglement. (a) (b) x zy xyyz zx FIG. 15. Patterns indicating active flavors (colors) for bond-selective triplon interactions on honeycomb lattice: (a) Oxy-gen mediated t -hopping case – XY -type interactions – twoflavors are active for a given bond directions (e.g., T x and T y on z -type vertical bonds). Each color forms a system of sep-arate zigzag chains (one of the zigzag chains for T z boson ismarked by shading). The symmetry resembles famous com-pass models where each spin component interacts within itsown 1D chain. (b) Direct t (cid:48) -hopping case – bond dependentIsing-type of interactions. There is one-to-one correspondencebetween the active triplon color and the bond direction ( T z on z bond, etc), establishing a bosonic analog of the Kitaevmodel. The other limit illustrated by Fig. 15(b) may be calleda bosonic Kitaev model, following the formal similar-ity of the triplon exchange Hamiltonian, i.e. H ( α ) ij = J ( T † αi T αj − T † αi T † αj +H . c . ) for α -type bonds, to the Kitaevinteraction KS αi S αj . As found in Ref. [119], the strongfrustration of Kitaev-type prevents a magnetic conden-sation at any strength of the exchange coupling J rel-ative to spin gap λ . Interestingly, the model shares anumber of other features with the spin-1 / Z conserved quanti-ties, magnetic correlations are strictly short-ranged andconfined to nearest-neighbor sites, and the excitationspectrum has a spin gap. However, the strongly cor-related triplon “liquid” ground state found in the largeexchange limit J (cid:29) λ is smoothly connected to dilutetriplon gas and hence misses the defining character-istics (long-range entanglement and emergent nonlocalexcitations) of genuine spin liquids. Consequently, noquasiparticle modes (like Majorana bands in spin-1/2 Ki-taev model) are present within the spin gap. Nonethe-less, this strongly correlated paramagnet is far from beingtrivial – in contrast to what is conventional in pure spinsystems, magnetic correlations are highly anisotropic andstrictly short-ranged even in the limit where the spin gapis very small and the QCP is close by. Magnetic order canbe induced by subdominant (non-Kitaev type) triplon in-teractions, as well as by doping, which suppresses the spingap. Also, it has been found that triplon excitations ac-quire nontrivial band topology and protected edge statesin a magnetic field. By mixing the above two complementary anisotropiclimits with the corresponding couplings J ∝ t /U and7 J ∝ t (cid:48) /U in one-to-one ratio, we recover an isotropictriplon model of Eq. (4.2). Since the honeycomb lat-tice is not geometrically frustrated, the model shows thesame quantum critical behavior as in the square-latticecase, i.e. dispersing triplons condense at a QCP and giverise to long-range antiferromagnetic order. In this con-text, the ratio of the oxygen-mediated and direct hoppingamplitudes t/t (cid:48) turns out to be an important “handle”determining the degree of frustration (as well as its type)of a singlet-triplet system with 90 ◦ bonding geometry.On the materials side, the Ru-based honeycomblattice compounds are potential candidates to realizefrustrated spin-orbit exciton models. In particular,Ag LiRu O , which is derived from Li RuO bysubstituting Ag ions for Li ions between the honeycombplanes, is of interest. While hexagonal symmetry is heav-ily broken by the structural and spin-orbital dimeriza-tion in Li RuO , Ag LiRu O avoids this tran-sition and thus may serve as a model system to study J = 0 physics in a nearly ideal honeycomb lattice. Thiscompound shows no magnetic order, which im-plies that the triplon interactions are either too weak toovercome the spin-orbit gap, or they are dominated byKitaev-type couplings and thus highly frustrated.To summarize this section, we note that physics ofspin-orbit-entangled J = 0 compounds is still in its in-fancy, and indicate below a few directions for future stud-ies.(i) Frustrated spin-orbit exciton models, possible ex-otic phases and magnetic QCP in these models; topo-logical properties of spin-orbit excitations. Experimentsin various edge-shared structures and geometrically frus-trated lattices, pressure and strain control of magneticand structural transitions.(ii) The nature of metallic states induced by elec-tron doping, which injects J = 1 / J = 0 states. Fermion hoppingis accompanied by creation and annihilation of spin-orbit excitons, which should give rise to a strongly cor-related metal. In the case of perovskite lattices with180 ◦ bonding geometry, Ref. [127] suggested that elec-tron doping induces ferromagnetic correlations, and pos-sible triplet pairing mediated by spin-orbit excitations.On the experimental side, several studies founddoping driven ferromagnetic state in Ca RuO ; interest-ingly, the recent work has reported also on signaturesof superconductivity. In compounds with 90 ◦ bondinggeometry, the hopping rules are different and interactionsare frustrated; studies of doping effects in such systemsmay bring some surprises. V. MULTIPOLAR PHYSICS IN d -ELECTRONSYSTEMS WITH STRONG SPIN-ORBITCOUPLING Multipolar ordering in Mott insulators covers a wholehost of phenomena, ranging from the relatively standard quadrupole ordering of e g electrons due to a cooperativeJahn-Teller effect to the formation of bond multipolesin highly quantum-entangled frustrated magnets. In the absence of significant SOC, the orbital and spindegrees of freedom typically order at different tempera-tures. At high temperature the cooperative Jahn-Tellereffect drives both a structural distortion of the lattice andan associated orbital quadrupole order, while at lowertemperature the exchange interaction causes the spins toorder. Strong SOC ties together the spin and orbital de-grees of freedom, negating the simple picture of separatetransitions. As a consequence, the intermediate phasepicks up a spin contribution to the quadrupole order, andat the same time the two transitions tend to get pushedcloser together in temperature.The introduction of strong SOC also opens up thepossibility of unusual types of interactions, in particularhigher-order biquadratic and triatic terms in the Hamil-tonian. These interactions can drive more unusual typesof multipolar order, such as an octupolar ground statesimilar to those found in f -electron systems, or,when combined with frustration, cause the multipolar or-der to melt away, leaving behind a multipolar spin-liquid. A. Quadrupole ordering
The idea of tensor order parameters, familiar from thetheory of classical multipole ordering, can be readily gen-eralised to the quantum case. Just as dipolar order is as-sociated with a non-zero expectation value of the vector (cid:104) J (cid:105) , quadrupole order is associated with a finite expecta-tion value of the rank-2 tensor, Q µνi = 12 (cid:104) J µi J νj + J νi J µj (cid:105) − (cid:104) J i · J j (cid:105) δ µν , µ, ν ∈ { x, y, z } (5.1)where the site indices i, j can refer to the same or differentsites.
1. Quadrupoles in d systems Quadrupole ordering is very common in strongly spin-orbit-entangled d systems, since d ions with a J = 3/2ground state are Jahn-Teller active, as discussed in Sec-tion II.A. The onset of quadrupole order occurs whenthe degeneracy of the J = 3 / J z = ± / J z = ± / J = 3 / d ions is reduced.The driving force for the quadrupole-ordering transi-tion comes predominantly from electrostatic and Jahn-Teller interactions, with a helping hand from the ex-change interaction. A good way to see this theoretically8 FIG. 16. Schematic phase diagram proposed for stronglyspin-orbit coupled d ions. For a large enough interaction V there are two phase transitions as a function of temper-ature, with a high-temperature transition into a quadrupoleordered (QO) phase followed by a low-temperature transitioninto one of various dipolar phases, including antiferromagnetic(AF) and canted antiferrogmanetic (CAF[100], CAF[110]) or-ders. The figure is reproduced with permission from Ref. [137]( © is to project the well-known interactions for the 6-fold de-generate t g manifold of electron configurations into the J = 3 / In addition to the usual bilinearinteractions, the resulting effective Hamiltonian also con-tains large biquadratic interactions, such as ( J zi ) ( J zj ) ,between neighbouring sites. It has been known for along time that these can be rewritten as quadrupole-quadrupole interactions, and so it is not surprisingthat they favour quadrupole ordering.Double-perovskite oxides [Fig. 7(d)] provide some ofthe cleanest material realisations of spin-orbit-entangled d Mott insulators. The wide spacing of the magneticions makes them good Mott insulators with small inter-site interactions, and allows a cubic ionic environmentto be retained to low temperature. Figure 16 illustratesa generic phase diagram proposed for d systems withdouble-perovskite structure. While none of the known double-perovskite materi-als have a completely vanishing dipolar magnetic mo-ment, as would be expected for isolated J = 3 / NaOsO has an effective mo-ment of approximately 0 . µ B . It also shows two tran-sitions, with a higher temperature structural transitionat T q = 9 . T m ≈ . However, sincethe symmetry above T q is likely tetragonal rather thancubic, as suggested by the approximately R ln2 entropyrecovery above T q , it is not clear how effectively thesystem explores the full J = 3 / MgReO , where there is (a)(b) FIG. 17. Evidence for the formation of a strongly spin-orbit-entangled J = 3 / R ln4 entropyin Ba MgReO at high temperatures, taken with permissionfrom Ref. [137] ( © L -edge RIXS spectrum showing the splitting of the t g level by SOC. Reproduced with permission from Ref. [142]( © an effective moment of about 0 . µ B , a high-temperaturetransition at T q ≈
33 K and a low-temperature tran-sition at T m ≈
18 K to a similar magnetically orderedstate to Ba NaOsO . However, unlike Ba NaOsO the high temperature structure is cubic, and heat capac-ity measurements reveal that the full R ln 4 entropy ofthe J = 3 / A small distortion of ReO oc-tahedra was observed below T q , which is consistent withquadrupole ordering. The closely related A TaCl (A = Cs, Rb) family ap-pears to provide a particularly good realisation of the J = 3 / . − . µ B . The suitability ofthe J = 3 / R ln 4 entropy at high temperature. As with the doubleperovskite oxides, two transitions are observed, with theupper transition at T q ≈
30 K for Cs and T q ≈
45 Kfor Rb and the lower transition at T m ≈ T m ≈
10 K for Rb. The upper transition is associated9with a structural transition from cubic to compressedtetragonal, and is suggestive of a ferro-quadrupolar phaseforming via selection of the J z = ± /
2. Quadrupoles in d systems Quadrupolar order for d ions can be expected eitherfrom ordering of the low-lying nonmagnetic E g doublet(see Fig. 2), or driven by a combination of electrostatic,Jahn-Teller and exchange interactions acting within thefull J = 2 quintuplet. However, there is currently alack of materials showing the type of double quadrupo-lar and magnetic transitions observed in many d com-pounds. B. Octupole ordering
Octupole phases involve the ordering of the rank-3 ten-sor, O µνξi = (cid:104) J µi J νj J ξk (cid:105) , µ, ν, ξ ∈ { x, y, z } , (5.2)in the absence of dipolar or quadrupolar order, where thebar indicates symmetrisation over the superscripts.
1. Octupoles in d systems A candidate to realise octupolar order in the absenceof any concomitant dipolar order is the material Sr VO with perovskite structure. Although V (3 d ) is notusually thought of as a strongly spin-orbit coupled ion,the combination of SOC and a tetragonal elongation ofthe oxygen octahedra conspire to select a J z = ± / t g manifold, as can beseen in Fig. 18(a). Projection of the usual exchangeHamiltonian for t g electrons into this ground state dou-blet reveals a checkerboard ground state of alternating | ψ (cid:105) = ( | / (cid:105)±|− / (cid:105) ) / √
2. Octupoles in d systems Octupolar order has been suggested to be realised inthe d double perovskite family Ba M OsO ( M = Zn,Mg, Ca). While phase transitions are observed atapproximately 30 K (Zn) and 50 K (Ca, Mg), there is noassociated development of dipolar magnetic order.
FIG. 18. Local states and collective excitations in Sr VO .(a) Splitting of the V t g levels by a tetragonal crystal field∆ cf and spin-orbit coupling λ results in a J z = ± / M = ( π, π ), reflecting the absence of dipolarorder in the ground state. The energy ω is in units of J = t /U . The figures are taken with permission from Ref. [149]( © Furthermore, the development of quadrupolar order isincompatible with the absence of detectable lattice dis-tortion. At the same time the recovery of only R ln 2 ofentropy at temperatures considerably above the transi-tion is indicative of a low-lying doublet, and matches theexpected E g - T g splitting shown in Fig. 2.From a theoretical point of view, projection of the in-teractions between t g electrons into the J = 2 quintu-plet shows the importance of bitriatic interactions, suchas ( J zi ) ( J zj ) . These can be rewritten as interac-tions between octupoles, and, if large enough comparedto competing bilinear and biquadratic interactions, candrive the formation of octupolar order. This may providea mechanism for selecting octupolar order with a ferro-octupolar ground-state wavefunction that is a complexmix of the E g states as shown in Fig. 2(c), and in termsof J z states is given by | ψ (cid:105) = | (cid:105) + i √ | (cid:105) + |− (cid:105) . The breaking of time-reversal symmetry at the transitionsupports this scenario.
C. Multipoles and frustration
Often more interesting than those systems that showrobust multipolar order, are those that combine multipo-lar order with spin-liquid behaviour, or those that avoidmultipole ordering and instead form spin liquids withmultipolar correlations. This type of behaviour is as-sociated with frustration, which arises in myriad ways instrongly spin-orbit-entangled systems due to the inter-play of lattice geometry with directional-dependent ex-change and higher-order biquadratic and bitriatic inter-0actions. d on the FCC lattice While not a spin liquid, the double perovskiteBa YMoO does form a valence-bond glass, in whicha disordered pattern of spin-singlet dimers freezes attemperatures below about 50 K, as can be seen inFig. 19. One suggestion is that this could be associated with ahidden SU (2) symmetry that can emerge from the com-plicated and apparently unsymmetric Hamiltonian be-tween J = 3 / Solving this Hamiltonianfor 2 sites, i and j , gives a lowest-energy singlet state | ψ (cid:105) = ( | / (cid:105) i |− / (cid:105) j − |− / (cid:105) i | / (cid:105) j ) / √
2, and, extend-ing this to the FCC lattice, results in a degenerate setof singlet dimer coverings with lower energy than anymagnetically ordered state.
The idea is that in thematerial a small disorder is responsible for selecting oneof the many degenerate dimer configurations, resultingin a dimer glass. This idea is appealing, and, due to thenature of the excitations above the dimer states, gives anexplanation for the experimentally determined soft gap,but there remains the question of whether Jahn-Teller in-teractions, active in d systems, play an important role. FIG. 19. Valence-bond glass formation in Ba YMoO . Heatcapacity measurements show no evidence of a phase transi-tion, but do show evidence for a gradual freezing, centred ona broad maximum at about 50 K. This suggests the forma-tion of an amorphous valence-bond state, with a distributionof triplet excitation energies. The figure is reproduced withpermission from Ref. [158] ( © d on the pyrochlore lattice The material Y Mo O provides an example of howspin-glass and potentially spin-liquid physics can emergeout of a quadrupolar phase. The Mo ions form a pyrochlore sublattice and sit inoxygen octahedra that have a large trigonal distortion atall temperatures, with band structure calculations sug-gesting that the characteristic energy scale of the trig-onal splitting is more than 100 meV [see Fig. 20(a)].When combined with SOC this results in a low energy J z = ± z axis ori-entated along the local in/out axes of the Mo tetrahedra,as shown in Fig. 20(b). Since there are no interactionsthat can transform J z = ± This would suggest thateither an all-in-all-out ordered state or a spin-ice-like dis-ordered 2-in-2-out configuration should be realised. How-ever, neutron scattering experiments suggest that spindegrees of freedom alone are insufficient to describe thelow-temperature behaviour of the system.
Evidence for what else needs to be taken into accountcomes from x-ray and neutron pair distribution analyses,which show that the Mo ions are not forming a perfectpyrochlore lattice, but instead their positions are shiftedtowards or away from the tetrahedral centres in a dis-ordered 2-in-2-out pattern [see Fig. 20(c)].
The ex-periments further show that the oxygen octahedra aredragged along by the Mo ions, resulting in very littlechange in the local crystal-field environment, but largevariations in the Mo-O-Mo bond angles, with individualbond angles dependent on the details of the 2-in-2-outlattice displacements. Deviations from the average Mo-O-Mo bond angle are expected to result in large changesto both the strength and sign of the exchange interactions[see Fig. 20(d)], resulting in a large coupling between thelattice and spin degrees of freedom and a resulting distri-bution in the exchange interactions.
As such, thesematerials are nice examples of the interplay of SOC withstrong magneto-elastic coupling.At low temperatures Y Mo O shows spin-glassbehaviour, and a number of explanations have beenput forward to explain this. One possibility isthat the low-temperature spin-glass state freezes out ofan intermediate-temperature spin-lattice-liquid state, inwhich the strong magneto-elastic coupling ties togetherthe spin and lattice degrees of freedom, but the systemremains dynamic and explores an extensive set of low-energy configurations.
VI. SPIN-ORBIT-COUPLED EXOTIC METALSAND NON-TRIVIAL TOPOLOGICAL PHASES
In the previous sections, we discussed the spin-orbit-entangled electronic phases in Mott insulators. However,the Mott insulating state of 4 d and 5 d transition-metaloxides is not so robust and often close to a metal-insulatortransition. In fact, metallic ground states are also fre-quently observed. In the itinerant limit, the spin-orbit-entangled states form bands which may be understood1 long short medium (a) Mo Mo O O (b) (c) (d) FIG. 20. The interplay of spin and lattice degrees of freedom in Y Mo O . (a) Average positions of Mo and O ions, showingthe pyrochlore lattice of Mo ions. (b) J z = ± z ) axes ofthe Mo tetrahedra. (c) Mo ions displace into or away from tetrahedral centres, creating long, short and medium length Mo-Mo separations. (d) Superexchange paths in the neighbouring MoO octahedra: (upper part) “ π -type” superexchange paththat dominates when the Mo’s form an undistorted pyrochlore lattice; (lower part) additional “ σ -type” superexchange paththat becomes increasingly important the more the Mo-O-Mo bond angle is changed from its average value. The figures arereproduced with permission from Ref. [165] ( © in the framework of jj -coupling. The strong SOC of 4 d and 5 d electrons can drastically modify the band struc-ture and may give rise to exotic metallic states, poten-tially with nontrivial topological character. The emer-gence of exotic phases such as nodal-line semimetals,Weyl semimetals, and topological Mott insulators hasbeen theoretically discussed. Compared to typical topo-logical semimetals composed of s and p electrons, thepresence of electron correlations in these oxides with d -electrons is expected to provide a distinct physics of cor-related topological materials. We review in this sectionthe exotic metallic states in perovskite and pyrochloreiridates, as well as in the doped hyperkagome. In addi-tion to iridates, the recently verified hidden multipolarphase and possible unconventional superconductivity inthe pyrochlore rhenate will be discussed. A. Orthorhombic perovskite AIrO (A = Ca, Sr)with Dirac line node As discussed in Sec.III.A, the layered perovskitesSr IrO and Sr Ir O are Mott insulators with local-ized J = 1/2 pseudospins. In this series of Ruddlesden-Popper perovskite Sr n+1 Ir n O (n = 1, 2, . . . ), the Ir5 d bandwidth is expected to increase as a function of thenumber of IrO planes, n. SrIrO , which corresponds tothe limit of n = ∞ , crystallizes in an orthorhombic per-ovskite with rotation and tilting distortion of IrO oc-tahedra (space group P bnm ), illustrated in Fig. 7(b). This orthorhombic perovskite is a metastable phase sta-bilized under high-pressure or in a thin-film form; at am-bient pressure, SrIrO crystallizes in a distorted 6H-typeperovskite structure. The orthorhombic perovskite SrIrO was shown to bemetallic from the transport and optical properties. It is in fact a semimetal with a small carrier density,which is produced by an interplay of crystalline symme-try and strong SOC. If there were no rotations and tilts of IrO octahedra, cubic SrIrO would have a half-filled J = 1/2 band with a moderate bandwidth. When the rota-tions and the tiltings of IrO are incorporated, the bandsare back-folded, and many crossing points in the J = 1/2bands show up. The incorporation of SOC opens a gapat many of the crossing points, which makes the systemclose to a band insulator with 20 d -electrons per unit cellwith four Ir atoms. In reality, the presence of symmetry-protected band crossing and the overlap of split bandsgive rise to a semimetallic state. The semimetallic band structure of SrIrO hosts theDirac bands near the Fermi energy E F , which preventsa gap opening. A density functional theory calculationand a tight-binding analysis showed the two interpene-trating Dirac dispersions around the U-point of the Bril-louin zone, which yield a nodal-line [Fig. 21(b)]. The Dirac nodes are protected by the nonsymmorphicsymmetry of the space group
P bnm , which contains twoglide symmetries, in addition to space- and time-reversalsymmetries.
The Dirac points are located slightlybelow E F , and there are other heavy hole bands cross-ing E F to retain the charge neutrality. The presence oflinearly dispersive electron bands was confirmed by anARPES measurement of thin-films. We note that theambient pressure phase of SrIrO , crystallizing in a mon-oclinic C /c structure, is also a Dirac semimetal pro-tected by the nonsymmorphic symmetry ( c -glide). One of the characteristic features of Dirac semimet-als is the presence of highly mobile carriers, which havebeen indeed identified in the perovskite CaIrO , isostruc-tural to SrIrO . A carrier mobility as large as 60,000cm /V · s is observed at low temperatures, as shown inFig. 22. The remarkably high mobility is discussed tobe attributed to the proximity of Dirac nodes to E F . Because of the smaller ionic radius of Ca as comparedto that of Sr , CaIrO inherits larger rotation and tilt-ing of IrO octahedra, which reduces the bandwidth andenhances electron correlations. The strong correlationrenormalizes the band structure and places the Dirac2 FIG. 21. Band structures of SrIrO obtained from LDA + U calculation with Hubbard U = 2 eV: (a) without SOC, and (b)with SOC ζ = 2 ζ at ( ζ at is atomic spin-orbit coupling). Thefigure is reproduced with permission from Ref. [173] ( © nodes near E F .It is important to unravel the key factor determiningthe evolution from a 3D Dirac semimetal to a magneticinsulator in the series of Sr n+1 Ir n O . In bulk form,Sr n+1 Ir n O with n ≥ and nonmagnetic SrTiO layers,i.e. [(SrIrO ) m /SrTiO ], has been designed. By in-creasing the number of SrIrO layers m, the dimension-ality, and thus the bandwidth, of SrIrO layers can becontrolled. The metal-insulator transition takes place ataround m = 3 as shown in Fig. 23(a). The insulating sam-ples with m = 1 and 2 show a magnetic transition withweak-ferromagnetic moments, which are induced by therotations of IrO octahedra about the [001] axis and theresultant DM interaction [Fig. 23(d)]. The intimate cor-relation between the metal-insulator transition and theappearance of magnetic order suggests that magnetismis essential for the occurrence of a metal-insulator tran-sition with reducing m.The nodal-line Dirac semimetallic state of SrIrO can be potentially exploited as a platform for othercorrelated topological phases by the application ofsymmetry-breaking perturbations such as magnetic fieldand strain. In particular, a variety of superlatticestructures has been proposed to realize novel topologi-
FIG. 22. Transport properties of orthorhombic perovskiteCaIrO . (a) Temperature dependence of longitudinal resis-tivity ρ xx (b) Hall conductivity σ xy as a function of magneticfield at several temperatures. (c), (d) Temperature depen-dence of carrier mobility µ tr and carrier density n / , respec-tively. The huge mobility as large as 60,000 cm /V · s is seenbelow 1 K. The figure is reproduced from Ref. [178], CC-BY-4.0 (http://creativecommons.org/licenses/by/4.0/).FIG. 23. Temperature dependent (a) resistivity ρ ( T ), (b) − d (ln ρ ) /dT , (c) Hall constant R H , and (d) in-plane magneti-zation M ( T ) of (001) superlattice [(SrIrO ) m /SrTiO ] with m = 1, 2, 3, 4 and ∞ . The figure is reproduced with permissionfrom Ref. [179] ( © cal phases. By introducing a staggered potential thatbreaks the mirror-symmetry, for example the (001) su-perlattice of [(SrIrO )/(SrRhO )], the appearance of atopological insulator phase is anticipated. The super-lattice of [(SrIrO ) /(CaIrO ) ] has been predicted to bea topological semimetal hosting a double-helicoid surfacestate. In addition to the (001) superlattices, a topologicalinsulator phase was also predicted from fabricating a bi-layer of SrIrO along the [111] direction. In a bi-layer of SrIrO , the IrO octahedra form a buckled hon-3eycomb lattice. As in the celebrated graphene, electronhopping on a honeycomb lattice gives rise to Dirac bands.When a trigonal crystal field is incorporated, it opens agap at the Dirac points, giving rise to a Z topologicalinsulator. In fact, the fabrication of a [111] oriented thin-film istechnically challenging in perovskite oxides A B O ,since the (111) surfaces, AO or B planes, are polar, incontrast to the (001) surfaces of AO or BO . Addi-tional difficulties arise from a size mismatch of SrIrO with a standard substrate like SrTiO and the stabilityof monoclinic SrIrO with a hexagonal motif on a [111]substrate. The stabilization of the orthorhombic phaseby optimizing the A-site ion through Ca substition for Sris quite useful to overcome this difficulty. (111) superlat-tices of [(Ca . Sr . IrO ) /(SrTiO )] with m = 1, 2 and3 have been successfully fabricated. In contrast to theprediction of a topological insulator, the (111) superlat-tices with m ≤ B. Potential topological semimetallic state inpyrochlore iridates
Since soon after the discovery of spin-orbit-entangledphases in iridates, the pyrochlore iridates A Ir O (A:trivalent cation) have been attracting tremendous inter-est, as they provide a unique interplay between SOC,electron correlation and frustration. There have been aplethora of theoretical proposals for non-trivial topolog-ical phases, including Z topological insulators, topo-logical Mott insulators, Weyl semimetals, and axioninsulators. The general trend for the electronic structure of py-rochlore iridates has been understood as follows.
When the on-site electron correlation U is weak, theyshow a semimetallic electronic structure. The semimetal-lic state may contain small pocket Fermi surfaces[Fig. 24(a)] or a quadratic band touching point at theΓ point near the Fermi energy [Fig. 24(b)], dependingon the hopping parameters. By increasing the elec-tron correlation, AIAO order of Ir magnetic momentstakes place. When the original nonmagnetic state is asemimetal with quadratic band touching, the AIAO or-der splits the degenerate bands and gives rise to crossingsof linearly-dispersing non-degenerate bands. The resul-tant semimetallic phase is a Weyl semimetal with nodesof opposite chiralities. There are 4 pairs of Weyl nodesalong the [111] or equivalent directions in the Brillouinzone. When U is increased further, the Weyl nodes moveto the high-symmetry point of the Brillouin zone and thedistance between the pair of nodes increases. Eventually,the pair of Weyl nodes with different chiralities meets atthe zone boundary and annihilates, which renders a gapover the whole Brillouin zone and makes the system atrivial AIAO antiferromagnetic insulator. FIG. 24. Calculated band structures of pyrochlore iridatenear Fermi energy with different on-site Hubbard repulsion U .The left and right columns show the results for the differentmagnitude of hopping parameters. The figure is taken withpermission from Ref. [190] ( © In real materials, the relative strength of U can betuned effectively by changing the bandwidth of Ir 5 d states. As described in Sec.III.C.2, the bandwidth is re-duced by decreasing the ionic radius of the A-cation, r A ,i.e. changing the degree of trigonal distortion. Amongthe family of pyrochlore iridates A Ir O , Pr Ir O ,which has the largest r A , remains metallic down tothe lowest temperature measured. Pr Ir O shows apoor metallic behavior with a small carrier density of ∼ cm − . The ARPES measurement revealed thatPr Ir O has a quadratic band-touching at the Γ point asshown in Fig. 25. In this nodal semimetallic state, thedensity of states near E F increases steeply since DOS( E ) ∝ √ E , which results in pronounced electron correlationsand potentially leads to a non-Fermi liquid behavior. Another interesting behavior of Pr Ir O is that Pr f moments do not show a long-range magnetic order, butinstead a spin-liquid-like behavior. A finite Hall con-ductivity was observed at zero magnetic field despite theabsence of hysteresis in the magnetization curve, whichhas been discussed to originate from the chirality of thespin-liquid state.
With decreasing r A , a temperature-driven metal-insulator transition is observed for A = Nd, Sm, andEu. The metal-insulator transition accompanies theAIAO magnetic order of Ir 5 d moments as discussed inSec.III.C, and thus the low-temperature phase was ex-pected as a possible realization of a Weyl semimetal.However, the presence of a charge gap has been seenat temperatures well below the magnetic ordering tem-4 FIG. 25. Quadratic Fermi node of Pr Ir O revealed by theARPES measurement. (a) Energy dispersion along k x direc-tion measured with different incidence photon energies. (b)The ARPES data in the k x − k (111) sheet superposed on thecalculated band dispersion. The figure is reproduced fromRef. [192], CC-BY-4.0 (http://creativecommons.org/licenses/by/4.0/). perature T N for Ir 5 d moments even in Nd Ir O , which is right next to Pr Ir O . This is incompatiblewith the Weyl semimetallic state. A Weyl semimetalphase might be realized only in the critical vicinity ofa metal-insulator transition and therefore hidden. Finetuning of the metal-insulator transition using pressureor doping may help approaching a Weyl semimetal. In-deed, suppression of the metal-insulator transition wasobserved by the application of pressure or by dop-ing a small amount of Rh atoms onto the Ir site, whichmay stabilize the Weyl semimetallic state.Although the ground state of Nd Ir O is unlikelyto be a Weyl semimetal at ambient conditions, a dras-tic magnetic-field-induced change of transport propertieswas discovered, reflecting the modification of Nd mag-netic order. When a magnetic field is applied alongthe [001] direction, the AIAO order of Nd f momentsis switched into the 2-in-2-out configuration above ∼ d electrons via the f - d magnetic exchange. The high-field semimetallic state has been proposed to be a nodal-line semimetal. On the other hand, an applicationof magnetic field along the [111] direction induces the 3-in-1-out order of Nd moments, which is discussed torealize another Weyl semimetallic phase. The putative Weyl semimetallic state in pyrochlore iri-dates is expected to show characteristic features such assurface Fermi arcs and anomalous Hall effect (AHE). TheAHE is associated with the fact that the Weyl nodes canbe regarded as a source/sink of Berry curvature. In abulk pyrochlore iridate, the anomalous Hall conductivityis canceled because of the cubic symmetry.
However,in a strained thin-film, the cubic symmetry is broken andthe emergence of an AHE has been predicted.
Experi-mentally, such an AHE was indeed observed in thin-film
FIG. 26. Angle-dependent magnetoresistance in Nd Ir O .The tables on the top indicate the magnetic configuration ofNd and Ir sublattices such as AIAO (AOAI) order (0-4 and 4-0), 2-in-2-out configuration (2-2) and 3-in-1-out state (3-1 or1-3), respectively. The figure is reproduced with permissionfrom Ref. [200] ( © pyrochlore iridates. For the Pr Ir O thin-film, this wasargued to arise from the strain-induced Weyl semimetal-lic state with a magnetic order at the surface/interfaceas well as the breaking of cubic-symmetry. The AHEwas observed also in the insulating pyrochlores such asEu Ir O and Nd Ir O , but was attributed to spin-chirality or domain walls of AIAO magnetic or-der, rather than the anomalous conductivity from Weylnodes. C. Spin-orbit-coupled semimetal out of thecompetition with molecular orbital formation
A metallic state is realized also by carrier dopinginto spin-orbit-entangled Mott insulators. In partic-ular, carrier-doping into Sr IrO has been attemptedintensively in the search for superconductivity, moti-vated by the cuprate physics, as discussed in Sec.III.A.A spin-orbit-coupled metallic state induced by carrier-doping was found also in the doped hyperkagome iri-date Na Ir O . A sister compound Na Ir O , whichshares the same hyperkagome sublattice of Ir atoms wassynthesized. The chemical formula indicates that Irhas a valence state of Ir . , i.e. 1/3-hole doped stateof the Na Ir O Mott insulator.Naively, we would expect the 1/3-hole-doped Mottinsulator to be a correlated metal with a large Fermisurface. Na Ir O , as well as the sister compoundLi Ir O , shows a metallic behavior, but turned outto be a semimetal with a small number of electrons andholes, rather than a large Fermi-surface metal. Thefirst-principle calculations indicate that the semimetallic5
FIG. 27. Calculated band structure of Na Ir O . (a) Scalarrelativistic band structure showing a band insulating state.The right panel illustrates the molecular orbital formation onthe Ir hyperkagome lattice. (b) Relativistic band structure in-cluding SOC. The bands which form hole and electron pocketsare colored in red and magenta, respectively. The right panelschematically represents the suppression of molecular orbitalformation by SOC. The figures are reproduced from Ref. [208],CC-BY-4.0 (http://creativecommons.org/licenses/by/4.0/). electronic structure is produced by an interplay of molec-ular orbital formation and SOC. The calculation with-out SOC yields a band insulator as the ground state ofNa Ir O , despite the non-integer number of d -electronsper Ir atom. The band insulating state can be under-stood as the formation of Ir trimer molecules with 14 d -electrons on the triangular unit of the hyperkagomelattice [Fig. 27(a)]. The incorporation of SOC sup-presses the formation of molecular orbitals by orbitalmixing. The conduction and valence bands made outof the molecular orbitals get broader and overlap, givingrise to a semimetallic state with small pockets of Fermisurface. Such a competition between molecular orbitalformation and SOC is likely a common feature of 4 d and 5 d transition-metal oxides with spatially extended d -orbitals. Indeed, J = 1/2 magnets often switch intoa dimerized state of transition-metal ions, which can beviewed as a molecular orbital formation, for example, inhoneycomb-based iridates and in the ruthenium chlorideunder high pressure. D. Spin-orbit-coupled metallic state in Cd Re O The pyrochlore material Cd Re O has received muchattention in recent years due to the spontaneous breakingof inversion symmetry, and its impact on superconduc-tivity below T c ∼ Re O is that of strongly spin-orbit-coupled met-als with relatively weak electron-electron interactions. Starting from the high-temperature metal with intacttime-reversal and inversion symmetries, one can considerthe possible Fermi-surface instabilities. These includephases in which inversion symmetry is spontaneously bro-ken, while time-reversal symmetry remains intact, andthe result is a deformation and splitting of the Fermi-surface into spin polarised bands, with momentum-dependent spin orientation (see Fig. 28). Many of theseelectronic order parameters couple to the lattice, andshould therefore drive a structural phase transition. Theinstability that may be relevant to Cd Re O results ina quadrupolar order parameter, and can be thought of asthe electron analog of chiral nematic liquid crystals. The inversion symmetry breaking instability may openup the possibility of unconventional, odd-parity, topolog-ical superconductivity in the vicinity of the associatedquantum critical point.
The superconductivity me-diated by the fluctuations of the inversion-symmetry-breaking order parameter can be either pure p -wave ormixed s - and p -wave, where the s - and p -mixed statecomes from distinct superconducting channels developingfrom the weakly-coupled, SOC-split bands (see Fig. 28).In the case that the p -wave channel is dominant, a topo-logically non-trivial state is expected, and the topologicaltransition between the s - and p -wave dominated regionsis particularly interesting due to the presence of unusualvortex defects associated with the enlarged symmetry. Experimentally, an inversion-symmetry-breakingstructural transition has been observed in Cd Re O at T s1 ≈
200 K, while superconductivity sets in at T c ≈ Analysis of second-harmonic generationexperiments has been used to tease apart the latticeand electronic changes at T s1 , and suggests that aninversion-symmetry-breaking electronic nematic phasesis formed at the transition. For lower temperatures,while at ambient pressure the superconductivity appearsto be essentially s -wave, pressure can be used to tune thesystem, increasing both T c and the upper critical field, B c2 , with the significant increase of the latter takento indicate the enhancement of the p -wave channel. While the agreement between theory and experiment isvery encouraging, the experimental phase diagram as afunction of both temperature and pressure is consider-ably more complicated than the theoretical predictions,and much work remains on both the theoretical andexperimental fronts.
VII. CONCLUSION
Correlated electrons in the presence of strong SOCform a rich variety of localized and itinerant spin-orbit-6
FIG. 28. Schematic theoretical phase diagram for spin-orbit-coupled metals. The breaking of inversion symmetry drivesan itinerant multipolar-ordered phase with spin split bands,where the spin orientation is tied to the momentum.
As afunction of a control parameter, such as pressure, a quantumcritical point for inversion-symmetry breaking emerges. Ataround the quantum critical point, a dome-like superconduct-ing phase may be anticipated. The superconducting domeconsists of pure p -wave region and mixed s - and p -wave re-gion. Exotic topological properties are expected when the p -wave pairing dominates. The figure is reproduced withpermission from Ref. [214] ( © entangled phases in 4 d and 5 d transition metal com-pounds. Localized 4 d and 5 d systems with J = 1/2pseudospins have been explored extensively in the last decade, which has established the 4 d and 5 d transi-tion metal oxides and related compounds as an emer-gent paradigm in the search for unprecedented quantumphases. The Kitaev model has been shown to be relevantin a family of d J = 1/2 honeycomb magnets. Partlymotivated by the J = 1/2 physics in the insulating d systems, the research effort on d - d and itinerant sys-tems has become quite active recently. Many attractivespin-orbit-entangled states are anticipated to emerge, in-cluding multipolar orderings, excitonic magnetism, a cor-related topological insulator, and a topological supercon-ductor. As seen in this review, their potential as a mineof novel electronic phases has not yet been explored fully,particularly for d - d and itinerant systems. Conceptshave been put forward, but their realization requires thedevelopment of novel materials and approaches. Unusualbehaviors have been observed in experiments, but under-standing the physics behind them requires more elabo-rate and realistic theories. Besides, many yet unknownexotic phases likely remain hidden, and are waiting to beunveiled both theoretically and experimentally. We areconvinced that the whole family of 4 d and 5 d correlatedoxides and related compounds with strong SOC consti-tutes a rich mine of novel quantum phases and is worthyof further exploration. ACKNOWLEDGMENT
T.T., A.S. and H.T. were supported by Alexander vonHumboldt Foundation. J.Ch. acknowledges supportby Czech Science Foundation (GA ˇCR) under ProjectNo. GA19-16937S. G.Kh. acknowledges support bythe European Research Council under Advanced Grant669550 (Com4Com). A. Abragam and B. Bleaney,
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