Table Of ContentKitaev Materials
Simon Trebst
Institute for Theoretical Physics
University of Cologne, 50937 Cologne, Germany
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4 Contents
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1.1 Bond-directionalinteractions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
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r 1.2 Kitaevmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
t
s
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a 2 HoneycombKitaevmaterials 9
m
2.1 Na IrO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 3
- 2.2 α-Li IrO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
d 2 3
n 2.3 α-RuCl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3
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[ 3 TriangularKitaevmaterials 15
3.1 Ba Ir Ti O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1 3 x 3−x 9
v 3.2 Othermaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
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4 Three-dimensionalKitaevmaterials 17
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7 4.1 Conceptualoverview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
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4.2 β-Li IrO andγ-Li IrO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
. 2 3 2 3
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4.3 Othermaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
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1 5 Outlook 24
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LectureNotesofthe48thIFFSpringSchool“TopologicalMatter–TopologicalInsulators,Skyrmions
andMajoranas”(ForschungszentrumJu¨lich,2017). Allrightsreserved.
2 SimonTrebst
1 Spin-orbit entangled Mott insulators
Transition-metal oxides with partially filled 4d and 5d shells exhibit an intricate interplay of
electronic, spin, and orbital degrees of freedom arising from a largely accidental balance of
electroniccorrelations,spin-orbitentanglement,andcrystal-fieldeffects[1]. Withdifferentma-
terialsexhibitingslighttiltstowardsoneofthethreeeffects,aremarkablybroadvarietyofnovel
forms of quantum matter can be explored. On the theoretical side, topology is found to play a
crucial role in these systems – an observation which, in the weakly correlated regime, has lead
tothediscoveryofthetopologicalbandinsulator[2,3]andsubsequentlyitsmetalliccousin,the
Weyl semi-metal [4, 5]. Upon increasing electronic correlations, Mott insulators with unusual
local moments such as spin-orbit entangled degrees of freedom can form and whose collective
behaviorgivesrisetounconventionaltypesofmagnetismincludingtheformationofquadrupo-
lar correlations or the emergence of so-called spin liquid states. A rough guide to these novel
typesofquantummatter,currentlywidelyexploredinmaterialswithsubstantialspin-orbitcou-
pling, is given by the general phase diagram of Fig. 1, adapted from an early review [6] of this
rapidlyevolvingfieldatthecurrentforefrontofcondensedmatterphysics.
U/t
Mott
spin-orbit entangled
insulator
s
Mott insulator
n
o
i
t
a
l
e
r Kitaev
r
o
materials
c
metal
band insulator
topological insulator
Weyl semi-metal
spin-orbit coupling �/t
Fig. 1: General form of the phase diagram in the presence of electronic correlations and spin-
orbitcoupling. Figureadaptedfromthereview[6].
Ourfocusherewillbeontheparticularlyintriguingscenariooftheformationofnoveltypesof
Mottinsulatorsinwhichthelocalmomentsarespin-orbitentangledj = 1/2Kramersdoublets
[7,8,9]. Thelatterareformedforionsinad5 electronicconfiguration–avastfamilyofsuchd5
materials exists that not only includes most iridates, which typically have a Ir4+ (5d5) valence,
but also some Ru-based materials with a Ru3+ (4d5) valence along with the (extremely toxic)
osmates and (so far largely unexplored) rhenates. The level scheme of such d5 ionic systems
is illustrated in Fig. 2. With the crystal field (e.g. of an octahedral oxygen cage) splitting off
the two e levels, this puts the five electrons with a total s = 1/2 magnetic moment into the
g
t orbitals with an effective (cid:96) = 1 orbital moment. Strong spin-orbit coupling then results in a
2g
system with a fully filled j = 3/2 band and a half-filled j = 1/2 band. The reduced bandwidth
KitaevMaterials 3
octahedral spin-orbit electronic
e
crystal field g coupling j = 1/2 correlations
j=1/2 Mott
~ 3eV ~ 400meV insulator
L~ ~� U
d-orbitals j = 3/2
·
t
2g
IrO cage
6
Fig. 2: Formation of spin-orbit entangled j = 1/2 moments for ions in a d5 electronic configu-
rationsuchasforthetypicaliridiumvalenceIr4+ ortherutheniumvalenceRu3+.
ofthelatterthenallowsfortheopeningofaMottgapevenfortherelativelymoderateelectronic
correlations of the 4d and 5d compounds. This latter point should again be emphasized. The
formation of a Mott insulator in these compounds is per se somewhat counterintuitive as the
larger atomic radii of their 4d and 5d constituents give rise to considerable atomic overlap,
whichresultsinalargeelectronicbandwidthandthusaneffectivesuppressionoftheelectronic
correlations. Assuch,conventionalwisdomhaslongconsideredthese“heavy”transitionmetal
compounds to generically form metallic states. It is only because of an enormously enhanced
spin-orbitcouplingthattheeffectiveelectronicbandwidthofthesematerialscanbereduced(via
theformationoftwoseparatej = 3/2andj = 1/2bands)toalevelthatthelargelysuppressed
electronic correlations can still drive the system into a Mott insulating state. These j = 1/2
Mott materials are therefore sometimes referred to as “spin-orbit assisted Mott insulators” and
arelocatedintheproximityofthemetal-insulatortransitionforlargespin-orbitcouplinginthe
generalphasediagramofFig.1.
Theformationofsuchaj = 1/2Mottinsulatorwasfirstobservedexperimentallyin2008/2009
[10,11]fortheperovskiteiridateSr IrO –a5dtransition-metaloxidewhichisanisostructural
2 4
analogue of La CuO , the parent compound of the cuprate superconductors. Remarkably, its
2 4
low-temperature physics indeed shares a striking resemblance to the phenomenology of the
cuprate superconductors including the formation of long-range antiferromagnetic order of the
j = 1/2 pseudospins for the undoped material and the emergence of a pseudogap phase and
associated Fermi arcs for electron-doped systems [12, 13, 14]. There is an intense ongoing
search[15,16]forsuperconductivityinSr IrO andotherperovskiteiridates.
2 4
In parallel, much attention has been drawn towards j = 1/2 Mott insulators that exhibit bond-
directional exchange interactions and are thought to exhibit unconventional forms of mag-
netism, such as the emergence of spin liquids [17, 18] or the formation of non-trivial spin
textures. We refer to these systems as Kitaev materials. Of particular interest here are the
sister compounds Na IrO and α-Li IrO and more recently α-RuCl that form Mott insula-
2 3 2 3 3
tors, in which local j = 1/2 moments are aligned in (almost decoupled) hexagonal layers.
As such they are perfect candidate materials for a solid-state realization of the Kitaev hon-
eycomb model [19] as envisioned by Jackeli and Khaliullin in 2009 [9]. Over the last few
years, materials synthesis of novel j = 1/2 Mott insulators has been thriving at a remarkable
pace and has further broadened our view on Kitaev materials beyond the original honeycomb
structure. As will be outlined in the remainder of this chapter, j = 1/2 Mott insulators with
strong bond-directional exchange interactions have also been synthesized for triangular lattice
geometries, e.g. in the family of hexagonal perovskites Ba Ir Ti O , and three-dimensional,
3 x 3−x 9
tricoordinated lattice generalizations of the honeycomb lattice (dubbed the hyper-honeycomb
4 SimonTrebst
and stripy-honeycomb) in polymorphs of α-Li IrO , i.e. the iridates β-Li IrO and γ-Li IrO ,
2 3 2 3 2 3
respectively. Inaddition,theoreticalprogresshasbeenmadeincomprehensivelyclassifyingthe
gapless spin liquid ground states of three-dimensional Kitaev models. It will be the purpose of
this chapter to provide what is hopefully a concise review of the current status of these Kitaev
materialsandtheirconceptualunderstanding.
1.1 Bond-directional interactions
At the heart of all Kitaev materials are bond-directional interactions, i.e. Ising-like interactions
where the exchange easy axis depends on the spatial orientation of an exchange bond, and
which dominate in coupling strength over all other exchange types. The microscopic origin
of such bond-directional interactions in d5 transition metal compounds has been worked out
in a pioneering 2005 paper by Khaliullin [7] and later refined to the context of Kitaev-type
interactions in joint work of Jackeli and Khaliullin [9]. These papers have undoubtedly shaped
the formation of what is now a burgeoning field of experimental and theoretical exploration of
Kitaevmaterials.
I: corner-sharing II: edge-sharing “parallel edge”-sharing
Fig. 3: Illustration of possible geometric orientations of neighboring IrO octahedra that give
6
rise to different types of (dominant) exchange interactions between the magnetic moments lo-
cated on the iridium ion at the center of these octahedra. For the corner-sharing geometry (I)
one finds a dominant symmetric Heisenberg exchange, while for the edge-sharing geometries
(II)onefindsadominantbond-directional,Kitaev-typeexchange.
What Khaliullin and Jackeli have realized is that the geometric orientation of neighboring IrO
6
octahedra plays a crucial role in determining the microscopic exchange of the magnetic mo-
mentslocatedontheiridiumionatthecenteroftheseoctahedra. Theydistinguishtwoelemen-
tary scenarios, which are illustrated (along with an extension) in Fig. 3. First, for perovskite
iridates such as Sr IrO two neighboring IrO octahedra share a corner. For the coupling be-
2 4 6
tween the two neighboring iridium ions this means that there is a single Ir-O-Ir exchange path,
which is also referred to as 180◦ bond. The dominant coupling along this type of bond is –
despite the presence of a strong spin-orbit coupling – a symmetric Heisenberg exchange be-
tween the spin-orbit entangled j = 1/2 moments. The second scenario plays out in materials,
in which two neighboring IrO octahedra share an edge. Here there are two Ir-O-Ir exchange
6
paths, which exhibit a 90◦ bonding geometry. The fact that there are two exchange paths turns
outtobecrucial,asthetwoalternativepathsleadtoadestructiveinterferenceofthesymmetric
Heisenbergexchangewhenrestrictingthecouplingtoariseexclusivelyfromthej = 1/2bands,
or alternatively to a significant suppression to some residual Heisenberg exchange when con-
sidering the full multi-orbital model also including the (virtual) j = 3/2 bands. In lieu of the
highlysuppressedisotropicexchange,itisabond-directionalcoupling,stemmingfromHund’s
KitaevMaterials 5
coupling and mediated through the multiplet structure of the excited levels, that takes center
stage and becomes the dominant coupling. The bond-directionality of the coupling arises since
the pair of d-orbitals linked for two neighboring octahedra depends on the type of edge shared
at their intersection. This in turn gives rise to a spatially oriented Ising-type coupling between
thespin-orbitentangledj = 1/2momentswherethemagneticeasy-axisisperpendiculartothe
plane spanned by the two exchange paths [7, 9]. The strength of this bond-directional coupling
isgivenby
8t2J
HSγSγ, (1)
− 3U2 1 2
wheretistheeffectiveinterorbitalhoppingmediatedbytheoxygenions,J istheHund’scou-
H
plingandU theelectroniccorrelationstrength. Thetwospin-orbitentangledj = 1/2moments,
represented by the usual vectors of SU(2) spin-1/2 Pauli matrices S and S , are coupled only
1 2
viaasinglespincomponentγ = x,y,z. Assuchtheedge-sharing90◦ bondgeometrynaturally
gives rise to a quantum compass model [20], originally introduced by Kugel and Khomskii for
theorbitaldegreesoffreedominJahn-Tellersystems[21]andrealizedhereforthefirsttimein
a truly magnetic variant. It is precisely this edge-sharing exchange coupling, which has been
envisioned [9] to lead to a realization of Kitaev couplings in the honeycomb iridates Na IrO ,
2 3
α-Li IrO , and α-RuCl and which has later also been found to be realized in the 3D hyper-
2 3 3
honeycomb and strip-honeycomb materials β-Li IrO and γ-Li IrO . This second scenario of
2 3 2 3
edge-sharing exchange can be further expanded by considering octahedral geometries where
neighboring IrO octahedra do not share an edge, but have two parallel edges as illustrated in
6
theright-mostpanelofFig.3. Followingtheexactsamelineofarguments,onealsoarrivesata
destructiveinterferencefortheisotropicHeisenbergexchangeinthissettingandtheemergence
of a bond-directional exchange. This latter scenario is realized in triangular Kitaev materials
suchasBa IrTi O .
3 2 9
In passing we note that recently also the microscopics of face-sharing ocahedra [22] has been
discussed,whichisrelevant,forinstance,tothequantummagnetismofBa AlIr O [23].
5 2 11
Goingbeyondthesesymmetry-guidedmicroscopicconsiderationsandperformingabinitiocal-
culations [24, 25], one finds that the generic Hamiltonian describing the interactions between
j = 1/2 spin-orbit entangled moments of spin-orbit assisted Mott insulators takes the general
form
(cid:88) (cid:16) (cid:17)
H = J S S +K SγSγ +Γ SαSβ +SβSα , (2)
− i j i j i j i j
γ−bonds
where the sum runs over nearest-neighbor spins at sites coupled by a bond i,j along the
(cid:104) (cid:105)
γ = (x,y,z)direction. ThestrengthoftheisotropicHeisenbergcouplingisgivenbyJ,andthe
bond-directional couplings include (i) a Kitaev term of stength K that couples the component
γ of the spins along a γ-bond and and (ii) a symmetric off-diagonal exchange Γ that couples
the two orthogonal spin components α,β γ for a bond along the γ = x,y,z direction. The
⊥
relativestrengthandcouplingsignofthevariouscouplingsvariesfrommaterialtomaterial,but
a common thread in all Kitaev materials is that the Kitaev coupling is the dominant exchange
coupling,i.e.K > J,Γ,withasimilarratiooftheKitaevtoHeisenbergexchangeof K/J 4
| | ≈
formanyj = 1/2Mottinsulators(foramoredetaileddiscussion,seebelow). Themicroscopic
model (2), which is also referred to as the JKΓ model, is often simplified to the Heisenberg-
Kitaev model (Γ = 0), which has first been conceptualized and studied in the context of the
6 SimonTrebst
honeycomb iridates by Chaloupka, Jackeli, and Khaliullin [26]. We will return to this model
afteradiscussionofthepureKitaevmodel(J,Γ = 0)inthenextsection.
Theeffectofbond-directionalinteractionsisastrongexchangefrustrationarisingfromthesim-
ple fact that these interactions cannot be simultaneously minimized, as illustrated for Kitaev-
type couplings in Fig. 4. Like geometric frustration, its more widely known counterpart that
arises when the lattice geometry gives rise to constraints that cannot be simultaneously satis-
fied,theeffectofexchangefrustrationistoinhibitmagneticorderingandgiverisetoaresidual
ground-state entropy. This is true already on the classical level. For instance, the classical
Kitaev honeycomb model does not exhibit a finite-temperature phase transition [27, 28], but
undergoes a thermal crossover to an extensively degenerate Coulomb phase [29] at zero tem-
perature.
y
?
z SySy
i j
SizSjz SxSx
i j
x
Fig. 4: Exchange frustration arising from spin-orbit induced bond-directional interactions,
i.e. Ising-like couplings where the exchange easy axis depends on the spatial orientation of
an exchange bond. Spins subject to these bond-directional interactions cannot simultaneously
minimizeallcouplings,whichholdsbothforquantumandclassicalmoments.
1.2 Kitaev model
ThepureKitaevmodel,whichcouplesSU(2)spin-1/2degreesoffreedomwithbond-directional
interactions – Ising-like couplings with a magnetic easy axis sensitive to the spatial orientation
γ ofthebond(seeFigs.4and5foranillustration)
(cid:88)
H = K SγSγ, (3)
Kitaev − γ i j
γ−bonds
is of central interest in condensed matter physics (and beyond). For one, it is famously known
for harboring both gapped and gapless quantum spin-liquid ground states. At the same time, it
is one of the rare microscopic models that can be solved exactly, as demonstrated in seminal
work[19]byAlexeiKitaevin2006. Thisanalyticaltractabilityallowsonetopreciselydescribe
and track the fractionalization of the original spin-orbit entangled degrees of freedom S into a
i
fermionic degree of freedom, a so-called Majorana fermion, and a Z gauge field as illustrated
2
in Fig. 5. The Z gauge field turns out to be static, it orders at zero temperature and its ele-
2
mentary vison excitations are found to be massive. The Majorana fermions, on the other hand,
KitaevMaterials 7
K
z
gapped spin liquid
Kx + Kz + Ky = const.
K
x
K K
y z
K gapless spin liquid
x
Majorana metal
K
y
Fig.5: Left: ThehoneycombKitaevmodelwithbond-directionalcouplingsK ,K andK . The
x y z
model can be analytically solved by introducing four flavors of Majorana fermions (indicated
by the yellow, blue, green and brown circles) and recombining them into a static Z gauge field
2
(indicated by the blue, green and brown ovals) and a remaining itinerant Majorana fermion
(yellowcircle). Right: PhasediagramoftheKitaevmodelplottedforaplaneK +K +K =
x y z
const. If one of the three couplings dominates, the system forms a gapped spin liquid indicated
bytheblueshading. AroundthepointofisotropiccouplingstrengthsK = K = K (indicated
x y z
bythereddot)agaplessspinliquidemerges,whichcanbebestcharacterizedasa(semi-)metal
oftheMajoranafermions.
remain itinerant and form a gapless state – a Majorana metal – around the point of equal cou-
pling K = K = K . For the honeycomb lattice, this Majorana metal is a semi-metal with a
x y z
Diracconedispersion(wellknownfromtheanalogouscalculationoffreecomplexfermionsfor
graphene-likeelectronsystems). Ifoneofthethreecouplingsdominates,thesystemundergoes
a phase transition (e.g. for dominant K coupling along the line K = K +K ) into a gapped
z z x y
spin liquid. This latter state exhibits Abelian (Z ) topological order akin to the well-known
2
toriccodemodel[30]andmacroscopicentanglement. Applyingamagneticfieldalongthe111-
direction,i.e.couplingthemagneticfieldtoallthreespincomponents,gapsoutthegaplessspin
liquidintoanevenmoreexoticspinliquidwithnon-Abelian(Ising-type)topologicalorder[19].
Thenon-Abeliancharacterofthelatterisidenticaltothatofap +ip superconductor[31],the
x y
Moore-Readstate[32]proposedfortheν = 5/2fractionalquantumHallstate,heterostructures
ofsuperconductorsandtopologicalbandinsulators[33]orsemicoductors[34],aswellasthatof
a network [35] of Majorana wires [36, 37] – all physical systems, which have gathered consid-
erable interest in the context of proposals for fault-taulerant topological quantum computation
[30, 38]. Despite this similarity, the search for Kitaev materials and a solid-state realization
of the Kitaev model is probably less driven by a potential application in quantum computing
technologies, but deeply inspired by the fundamental pursuit of (i) the synthesis of spin liquid
materials, (ii) the experimental discovery of Majorana fermions, and (iii) a direct experimental
probe of the underlying (Z ) gauge physics – such experimental evidence for gauge physics
2
in a condensed-matter context has long been lacking, despite theorists using the concept of Z
2
gauge theories in the classical statistical mechanics of nematics [39] and to capture the physics
offractionalizationinquantummany-bodysystems[40,41,42]fordecades.
8 SimonTrebst
The conceptual understanding of the physics of the Kitaev model has been steadily growing
since its initial description and analytical solution [19]. This includes the fundamental role of
week ending
vacancies [43,P4R4]L, d1e1p0le,ti0o9n7[2405]4, a(2nd01im3)purities [46], dPisHordYerSeIffCecAts L[43,R47E, 4V8]I,EanWd moLreETTERS 1 MARCH 2013
exoticphenomenasuchasthestrain-inducedformationofLandaulevels[49]ortopologicalliq-
uid nucleationc[o50m,p5o1n]eanritssindgepfoernmdivnogrtoexn-vthoretesxubinltaetrtaiccteioinnsdeinx.thTehniosnt-rAanbse-lian pIhnaspea[r5t2ic]u. lar, the ’ 3! zigzag state is identical to the
The emergence of p-wave superconductiv~ity upon doping has been discussed [53, 54, 55, 56]. ¼ 4
formation results in the S Hamiltonian of the same form as Heisenberg-AF state of the fictitious spins [24].
Withaneyetow(1a)r,dbsuatmwaittehriaelffreeacltiizvaetiocnouopfltihneg(spuK~re)KKitaevJmoadnedlsJ~everaleJx,perimenTtaolsoibgt-ain the phase boundaries, we have diagonalized
natures have been discussed including the dynamical r¼esponþse [57, 58], w¼ho#se distinct spectral
revealing a hidden SU 2 symmetry of the model at the model numerically, using a hexagonal 24-site cluster
gap can be proKbed via Jine(lawshtiecrneeutthreonKsciaðtatteÞevrintger(mINSK)~orvaelneicsthroens)s.piFnorrestohneance (wEiStRh)p,ethreiodic boundary conditions. The cluster is compat-
Raman response [¼59,#60], the inelastic light scattering response [61], which can be probed via
angles, the mapping reads as tan’~ tan’ 1. iblewith the above 4-sublattice transformation and ’ ’~
resonant Raman scattering, as well as the resonant ine¼las#tic X-ray#scattering (RIXS) response $
Phase diagram.—In its full parameter space, the KH mapping. As seen in Fig. 1(b), the second derivative of the
[62].
model accommodates 6 different phases, best visualized GSenergyE with respect to the ’well detectsthe phase
GoingbeyondthepureKitaevmodel,considerableattentionhasbeendevotedtotheHeisenberg- GS
using the phase-angle ’ as in Fig. 1(a). In addition to the transitions. Three pairs of linked transition points are
Kitaev model. Its principal phase diagram [63] is shown in Fig. 6 for a parametrization of the
previously discussed [16,22,23] Ne´el-AF, stripy-AF, and found: 88 ;92 and ( 76 , 108 ) for the spin
coupling strength J = cosφ, K = sinφ. Besides extended spin liquid phases around the & & & &
SL states near ’ 0, !, and !, respectively, we liquid-or’deðr transitiÞons arou#nd !,#and (162 , 34 ) or
pure Kitaev limit (with either ferro-¼or anti#fer4romagnet#ic2coupling) there are four magnetically ’ 2 & # &
ordered phaseso.bTsherisveinc3lumdeosrecosntvaetnesti.onFailrsfteroronmeaigsn‘e‘tAicFa’’nd(KNe´>el0o)rdKeriitnagevarountdhtehetrpaunrseitions between ordered phases.
Heisenberg limsiptsi,na-sliwqueildl asneaasrtri’py an!d.a zSiegc-zoangdo,rdtheeredFpMhaspehsa.sTehebororiagdinlyof the laTttehrectarnansitions from zigzag-AF to FM, and from stripy-
¼ 2
be readily undeerxstteonoddinbyg aodvuearlitthye[7t]h(isrodmqeutiamdersanretfeorfredthteo a’s Kcilrecinled.uTalhitey [64]A)FofttoheNsee´el-AF are expected to be of first order by sym-
phases to the tFwMo caonnvdensttiroipnyal-lAyFordstearetedsoanrees.cIonnnpaerctticeudla[rs,eteheFKigle.i1n(dau)]albityy allowmsettorym;tahpecorrespondingpeaksinFig.1(b)areindeedvery
the left- and ritghhet-h4a-nsdubsliadtetsicoef ttrhaencsifrocurmlaratpiohans,ewdhiaigcrhamimopflFieigs.t6heoinrtoidoenne-anothsherarbpy.tThhee spin liquid-order transitions near ’ ! lead
relation tical dynamics. Finally, near ’ 3!, the most wanted to wider and much less pronounced peaks, su¼gg#est2ing a
¼ 4
phase, zigJzag AFJ, appearasndoccupyiKng al2mJo+stKa,quarter of second(-4)(or weakly first-) order transition [16]. On the
→ − →
and a concurretnhtefopuhra-ssuebslaptaticcee.rTothaatinokns[7to].tThheisabdouvaleitmy tahpepreinbyg,ciotninseuctnsdtehre-ferrocmoangtnraertyto, liquid-order transitions around ’ ! show up
¼ 2
stood that the zigzag and Ne´el states are isomorphic, too.
the stripy phase, and the Ne´el ordered state to the zig-zag ordered phase (as well aassthveersypinnarrow peaks; on the finite cluster studied, they
liquidphasesontothemselves). correspond to real level crossings. The nature of these
phase transitions remains to be clarified [25].
(a) liquid WhileatJ 0(i.e.,’ !)thesignofKisirrelevant
¼ ¼ ’ 2
zigzag
[21], the stability of the AF- and FM-type Kitaev spin
liquids against J perturbation is very different: the SL
phase near ! ( !) is less (more) robust. This phase
2 # 2
Neel behavior is related to a different nature of the competing
ϕ
ordered phases: for the ! SL, these are highly quantum
2
zigzag and Ne´el states, while the SL near ! is sand-
FM # 2
wichedbymoreclassical(FMand‘‘fluctuationfree’’stripy
[16]) states which are energetically less favorable than the
quantum SL state.
stripy
Exchange interactions in Na IrO .—Having fixed the
liquid 2 3
parameter space (K >0, J <0) for the zigzag phase, we
turn now to the physical processes behind the model (1).
Fig.6: Phasedi(abg)ramoftNhee´eHl eilsiqeunidbezrigg-zKagitaevFmModel,lirqeupidrosdturicpeydNfe´reolmRef.[63].
Exchange interactions in Mott insulators arise due to vir-
-6 500 tual hoppings of electrons. This may happen in many
The conceptual understanding of this extended model has been furthered by a discussion of its
finite-temperature ph-y8sics [65] and dynamical response [66] along with the effectsdoifffaenreanpt-ways, depending sensitively on chemical bond-
400
plied magnetic field-1[067], lattice distortions [28], as well as the inclusion o2fϕfurtherinnge,igihnbtroar-ionic electron structure, etc. The case of present
d
Heisenberg [68]Aand Kitaev [69] interactions. The potentially multicri t3i0c0al A point atitnhteerpehsats(ei.e., strong spin-orbit coupling, t5 configuration,
E/GS -12 200 / GS and 90&-bonding geometry) has been add2rgessed in several
-14 E
2 papers [8,11,16,26]. There are the following four physical
100 d
-16 − processes that contribute to K and J couplings.
-18 0 Process 1: Direct hopping t between NN t orbitals.
0 2g
0 π/2 π 3π/2 2π Since no oxygen orbital is involved, 90 bonding is irrele-
&
ϕ
vant; the resulting Hamiltonian is H I S S with I
1 1 i j 1
¼ ( ’
2t 2=U [16]. Here, U is the Coulomb repulsion between
FIG. 1 (color online). (a) Phase diagram of the Kitaev- ð3 0Þ
Heisenberg model containing 2 spin-liquid and 4 spin-ordered t electrons. Typically, one has t =t<1, when compared
2g 0
phases. The transition points (open dots on the ’ circle) are to the indirect hopping t of t orbitals via oxygen ions.
2g
obtained by an exact diagonalization. The gray lines inside the Process 2: Interorbital NN t e hopping ~t. This is
2g g
circle connect the points related by the exact mapping (see text). #
the dominant pathway in 90 bonding geometry since
Open and solid circles in the insets indicate up and down spins. &
it involves strong t overlap between oxygen-2p
The rectangular box in the zigzag pattern (top-left) shows the pd"
and e orbitals; typically, ~t=t 2. The corresponding
magnetic unit cell. (b) Ground-state energy E and its second g
GS )
derivative d2E =d’2 revealing the phase transitions. Hamiltonian is [11]
GS
#
097204-2
KitaevMaterials 9
transition between the spin liquid and the magnetically ordered phases has been investigated
both numerically [67] and analytically [70]. The dynamical spin structure factor beyond the
Kitaev limit has recently been discussed [71] as have been liquid-liquid transitions upon intro-
ducing an additional Ising coupling [72]. Extensions including charge fluctuations have also
been discussed [73]. Finally, a classical variant of the Heisenberg-Kitaev model has also been
explored[74,75,28],includingtheformationofspintexturesinamagneticfield[76,77].
2 Honeycomb Kitaev materials
The quest to synthesize and explore Kitaev materials has been kick-started in 2009 by the pro-
found theoretical vision of Jackeli and Khaliullin [9] that laid out in remarkably precise terms
whattheelementaryingredients(initsliteralsense)forasuccessfulmaterialsearchstrategyare.
In particular, they not only explained the microscopic origin of Kitaev-type bond-directional
exchange interactions in certain (edge sharing) 4d5 and 5d5 transition metal compounds (as re-
viewedintheprevioussection). Theyalsolaidoutadetailedproposalforamaterialrealization
of the honeycomb Kitaev model in the iridate α-Li IrO and, more generally, iridates of the
2 3
form A IrO with a crystal structure as illustrated in Fig. 7 – a proposal that was quickly re-
2 3
fined[26]toalsoincludeNa IrO which,atthetime,hadalreadybeensynthesized[78,79]and
2 3
scrutinizedasapotentialtopologicalinsulator[80].
Fig. 7: Crystal structure of the honeycomb Kitaev materials A IrO such as Na IrO and α-
2 3 2 3
Li IrO .
2 3
2.1 Na IrO
2 3
Na IrO wasindependentlysynthesizedtoexploreitspotentialforKitaevphysicsin2009/2010
2 3
by the groups of Takagi [78] and Gegenwart [79]. Since then samples have been grown in a
number of labs around the world, including large single-crystals of diameters up to 10 mm in
the group of Cao [81]. With samples readily available, numerous experimental probes have
been taken that have revealed that Na IrO is indeed a Mott insulator with an insulating gap
2 3
of ∆ = 340 meV (opening at around 300 K) measured in optical transmission experiments
[82]. Fits of the magnetic susceptibility confirm the predominant j = 1/2 nature of the local
moments indicating magnetic moments of 1.79(2)µ [79, 83], rather close to the value of
B
1.74µ expected for spin 1/2 moments, while X-ray absorption spectroscopy [84] points to a
B
10 SimonTrebst
small admixture of j = 1/2 and j = 3/2 states. Resonant inelastic x-ray scattering (RIXS)
[85,86]findsevidenceforasmalltrigonaldistortionoftheIrO octahedraresultinginacrystal
6
field splitting of the j = 3/2 states of about 110 meV, which, however, is considerably smaller
thanthetypicalstrengthofthespin-orbitcouplingoftheiridatesλ 400 500meV[87,88].
≈ −
The system, however, does not exhibit the sought after spin liquid ground state, but is found
to magnetically order at around T = 15 K, a temperature significantly below the Curie-Weiss
N
temperature of Θ = 125 K [83]. The suppression of the ordering temperature with regard
CW
−
to the Curie-Weiss temperature indicates magnetic frustration which, quantified by the ratio
f = Θ /T 8, is unusually high for a quantum magnet on a bipartite lattice. It has
CW N
| | ≈
been rationalized [65, 83] to arise from geometric frustration arising from next-nearest neigh-
borcouplingswithintheelementaryhexagonsofthehoneycombstructureandisnotindicative
of a close proximity to the Kitaev spin liquid. The exact form of the magnetic ordering was
subsequently determined to be of zig-zag type by resonant x-ray magnetic scattering [89] and
inelastic neutron scattering [81, 90]. The microscopic origin of this zig-zag ordering has been
scrutinizedinlightoftheHeisenberg-Kitaevmodel[63]supplementedwithnext-nearestneigh-
bor Heisenberg [83] and Kitaev [91, 92] couplings along with a flurry of ab initio calculations
[93,94,95,84]and,mostrecently,dynamicalmean-fieldstudies[96].
ProbablythebestexperimentalevidencethatNa IrO shouldindeedbeconsideredtobethefirst
2 3
KitaevmaterialtobesynthesizedcomesfromdiffusemagneticX-rayscattering[97],whichhas
providedadirectexperimentalobservationofbond-directionalexchangeandadominantKitaev
coupling.
Several ideas have been put forward to bring Na IrO closer to the Kitaev spin liquid regime,
2 3
such as the making of thin films [98] or heterostructures [99], but probably the most practical
schemehasbeentoreplacetheNaatomsbythesmallerLiatomsandinsteadexplorethephysics
ofLi IrO .
2 3
2.2 α-Li IrO
2 3
The exploration of Li IrO as candidate Kitaev material has indeed lent further impetus to the
2 3
field. Probably the biggest surprise came when its synthesis lead to the discovery of several
polymorphs[83,100,101]whichhavebeendubbedα-Li IrO ,β-Li IrO ,andγ-Li IrO inthe
2 3 2 3 2 3
sequence of their discovery. Only α-Li IrO is isostructural to Na IrO and exhibits iridium
2 3 2 3
honeycomb layers (as depicted in Fig. 7), while the other two polymorphs, β-Li IrO and γ-
2 3
Li IrO , have been found to exhibit three-dimensional, tricoordinated networks of the iridium
2 3
ions. Wewilldiscussthelatterinthecontextofthree-dimensionalKitaevmaterialsbelow.
Thefirsthigh-qualitysamplesofα-Li IrO havebeensynthesizedbytheGegenwartgroup[83]
2 3
in 2012. Magnetic susceptibility fits reveal magnetic moment of 1.83(5)µ , a value slightly
B
largerthanexpectedforspin-1/2degreesoffreedom[83]. LikeitssistercompoundNa IrO ,α-
2 3
Li IrO undergoesamagneticorderingtransitionatT = 15KwithaCurie-Weisstemperature
2 3 N
ofΘ = 33K.ThinfilmsofLi IrO createdbypulsedlaserdepositiononZrO :Y(001)single
2 3 2
−
crystalline substrates reveal [102] a small optical gap of 300 meV indicative of a Mott gap of
similarsizeasinitssistercompoundNa IrO .
2 3
The further experimental analysis of α-Li IrO has been largely hampered by the fact that (in
2 3
contrast to Na IrO ) it has remained extremely challenging to synthesize bulk single-crystals
2 3
– an essential ingredient e.g. for neutron scattering studies, in particular to compensate for the
large neutron absorption cross-section of the iridium ions. The early powder samples were
found to exhibit single-crystalline ordering only on length scales smaller than 10 µm. Re-
