Possible proximity of the Mott insulating Iridate Na2IrO3 to a topological phase: Phase diagram of the Heisenberg-Kitaev model in a magnetic field

0.45 MB·

by  Hong-Chen Jiang

PossibleproximityoftheMottinsulatingIridateNa IrO toatopologicalphase:
2 3
PhasediagramoftheHeisenberg-Kitaevmodelinamagneticfield
Hong-Chen Jiang,1 Zheng-Cheng Gu,2 Xiao-Liang Qi,1,3 and Simon Trebst1
1Microsoft Research, Station Q, University of California, Santa Barbara, CA 93106
2KavliInstituteforTheoreticalPhysics, UniversityofCalifornia, SantaBarbara, CA93106
3Department of Physics, Stanford University, Stanford, CA 94305
(Dated:January7,2011)
MotivatedbytherecentexperimentalobservationofaMottinsulatingstateforthelayeredIridateNa IrO ,
2 3
wediscusspossibleorderingstatesoftheeffectiveIridiummomentsinthepresenceofstrongspin-orbitcoupling
andamagneticfield.Forafieldpointinginthe(cid:104)111(cid:105)direction–perpendiculartothehexagonallatticeformed
bytheIridiummoments–wefindthatacombinationofHeisenbergandKitaevexchangeinteractionsgivesrise
toarichphasediagramwithbothsymmetrybreakingmagneticallyorderedphasesaswellasatopologically
1
ordered phase that is stable over a small range of coupling parameters. Our numerical simulations further
1
indicatetwoexoticmulticriticalpointsattheboundariesbetweentheseorderedphases.
0
2
PACSnumbers:71.20.Be,75.25.Dk,75.30.Et,75.10.Jm
n
a
J
In the realm of condensed matter physics, spin-orbit cou- ityprovideevidenceofeffectivespin-1/2momentsandmag-

6 pling has long been considered a residual, relativistic cor- neticcorrelationsbelowT ≈ 15KindicatingthatNa IrO
N 2 3

rection of minor relevance to the macroscopic properties of is indeed a Mott insulator [4]. Theoretically, it has been ar-
]
l a material. In recent years this perspective has dramatically gued[7,8]thattheinteractionsbetweentheeffectiveIridium
e
- changed,especiallyduetothetheoreticalpredictionandsub- moments in the Mott regime are captured by a combination
r sequentexperimentalobservationoffundamentallynewstates of isotropic and highly anisotropic exchanges, which can be
t
s of quantum matter, so-called topological insulators [1], that trackedbacktothespinandorbitalcomponentsoftheeffec-
.
t aresolelyduetotheeffectofspin-orbitcoupling. Thetopo- tive momenta. A microscopic Hamiltonian interpolating be-
a
m logicalinsulatorsexperimentallyrealizedsofararesemicon- tweenthesetwotypesofexchangesisgivenby
d- dbuanctdorths,eworhyoosefpnhoyns-iicnatelrparcotipnegrtieelsecctarnonbse.lIatrgiselayncainptteurreesdtibngy HHK =(1−α)(cid:88)(cid:126)σi·(cid:126)σj −2α (cid:88) σiγσjγ, (1)
n challenge,forboththeoryandexperiment,toidentifyaneven (cid:104)i,j(cid:105) γ−links
o
c broaderclassofmaterialswherethisphysicsplaysoutevenin wheretheσidenotetheeffectivespin-1/2momentoftheIr4+
[ thepresenceofinteractionsandstrongcorrelations[2]. Good ions, γ = x,y,z indicates the three different links of the

candidatematerialsforthelatteraretheIridates[3,4]. These hexagonal lattice, and 0 ≤ α ≤ 1 parametrizes the relative
1
5dtransitionmetaloxidesarepronetoexhibitelectroniccor-
v coupling strength of the isotropic and anisotropic exchange
relationsandform(weak)Mottinsulators,whiletherelatively
5 between the moments. For α = 0 the Hamiltonian reduces
4 largemassoftheIridiumions(Z = 77)givesrisetoacom- totheordinaryHeisenbergmodel,whileintheoppositelimit
1 parably strong spin-orbit coupling, which has been found to
of highly anisotropic exchanges (α = 1) the system corre-
1 be as large as λ ≈ 400 meV [5]. The most common va-
spondstotheKitaevmodel[9]. Thelatterisknowntoexhibit
.
1 lence of the Iridium ions in these materials is Ir4+. The d-
a gapless spin-liquid ground state (for equal coupling along
0 orbitalsofthis5d5configurationaretypicallysplitbythesur-
thelinks)thatcanbegappedoutintoatopologicalphasewith
1
roundingcrystalfield,andfortheoctahedralgeometryofthe
1 non-Abelianquasiparticleexcitationsbycertaintime-reversal
IrO oxygencage,resultinanorbitalconfigurationwherefive
: 6 symmetry breaking perturbations [9]. One such perturbation
v electrons occupy the lowered, threefold degenerate t level.
2g isamagneticfieldpointinginthe(cid:104)111(cid:105)direction,perpendic-
i
X Spin-orbitcouplingwillfurtherliftthisdegeneracyofthet2g ulartothehoneycomblayer
orbitals and for strong coupling the effective l = 1 orbital
ar angular momentum [6] is combined with the s = 1/2 spin H =H −(cid:88)(cid:126)h·σ(cid:126) . (2)
HK+h HK i
degree of freedom carried by the hole of this partially filled i
t orbital configuration. This leaves us with two Kramers
2g The main result of our manuscript is the rich phase diagram
doublets of total angular momentum j = 3/2 and j = 1/2,
of this model, shown in Fig. 1. Besides two conventional,
ofwhichtheformerisoflowerenergyandfullyoccupiedby
magnetically ordered phases we find a topologically ordered
four electrons, while the partial filling of the latter gives rise
phase and two multicritical points, which we will discuss in
toaneffectivespin-1/2degreeoffreedom.
detailintheremainderofthemanuscript.
In this manuscript we focus on the Iridate Na IrO , in Numerical simulations.– We determine the ground-state
2 3
which NaIr O slabs are stacked along the crystallographic phase diagram of Hamiltonian (2) by extensive ‘quasi-2D’
2 6
c-axis, and the Ir4+ ions in the layers form a hexagonal lat- density-matrix renormalization group (DMRG) [10] calcula-
tice [4]. Recent measurements of the magnetic susceptibil- tions on systems with up to N = 64 sites. In particular,
2
2 2 senceofamagneticfield. Interpolatingtherelativecoupling
0.16
strength α between the isotropic Heisenberg limit (α = 0)
and the highly anisotropic Kitaev limit (α = 1) a sequence
1.5 h0.08 1.5 ofthreephaseshasbeenobserved[8]: TheNe´elorderedstate
h
d 0 of the Heisenberg limit is stable for α (cid:46) 0.4, when it gives
netic fiel 1 Necealn AteFdM polar0i.z75ed phase0α.8 0.85 1 wcpoaavryaemrtosetathe‘resrtcreoigpuiymp’leiNn0ge´.e8rleo(cid:46)gridmαeere≤0d.4s1ta(cid:46)thteeαicllou(cid:46)lslter0ca.tt8iev.deIingnrtoFhuiegn.ed2xstwteanhtdeiecidhs
g
a
m agaplessspinliquid. Nearα=1,perturbationtheoryreveals
0.5 0.5 thatthegaplessexcitationsofthisphaseareemergentMajo-
topological ranafermionsformingtwoDiracconesinmomentumspace.
canted phase
stripy AFM Including a magnetic field in the (cid:104)111(cid:105) direction a rich
0 0
phase diagram evolves out of this sequence of three phases.
0 0.2 0.4 0.6 0.8 1
Heisenberg coupling parameter α Kitaev Forthemagneticallyorderedstateswefindthattheorientation
model model
oftheorderintheNe´elandstripyAFMphasecantsalongthe
FIG. 1: (color online) Ground-state phase diagram of the (cid:104)111(cid:105) direction. To further characterize these canted states,
Heisenberg-Kitaev model (1) in a (cid:104)111(cid:105) magnetic field of strength itishelpfultoanalyzetheindependentsymmetriesofHamil-
h. Interpolating from the Heisenberg (α = 0) to Kitaev (α = 1) tonian (2). Besides the lattice translational symmetry T and
limitforsmallfieldstrength, asequenceofthreeorderedphasesis a reflection symmetry I around the centers of the hexagons,
observed:acantedNe´elstateforα(cid:46)0.4,acantedstripyNe´elstate there is an additional C∗ symmetry, which is a combination
illustratedinFig.2c)for0.4(cid:46)α(cid:46)0.8,andatopologicallyordered 3
of a three-fold rotation around an arbitrary lattice site and a
state for non-vanishing field around the Kitaev limit. All ordered
three-fold spin rotation along the (cid:104)111(cid:105) spin axis [12]. Both
phasesaredestroyedforsufficientlylargemagneticfieldgivingway
toapolarizedstate. canted phases break a subset ofthese discrete symmetries of
the Hamiltonian. The canted Ne´el order breaks the C∗ and
3
theI symmetries,whichthusleadstoasix-foldground-state
degeneracyinthisphase. Thecantedstripyphasebreaksboth
theC∗andtranslationalsymmetry(sincetheorderingpattern
3
doubles theunit cell). As aconsequence, wealso find asix-
foldground-statedegeneracyinthisphase.
For sufficiently large magnetic field, the order of both
cantedphasesisdestroyedandtheygivewaytoasimplepo-
larizedstate. Ournumericalsimulationsstronglysuggestthat
the transitions between the polarized state and these canted
FIG. 2: (color online) a) The hone√ycomb lattice spanned by unit statesarecontinuous, whichisinagreementwiththeirspon-
vectors(cid:126)a1 =(1,0)and(cid:126)a2 =(1/2, 3/2).Illustrationofmagnetic taneoussymmetrybreaking. Ontheotherhand,thetransition
stateswithb)Neelorderandc)stripyNeelorder.
betweenthetwocantedstatesatfinitefieldstrength(indicated
bytheboldlineinFig.1)isfoundtobefirst-order.Inoursim-
ulationsthisisindicatedbyasharpdropofthefirstderivative
we consider clusters of size N = 2×N ×N , which are
1 2 of the energy dE/dα as a function of the coupling parame-
spannedbymultiplesN (cid:126)a andN (cid:126)a oftheunitcellvectors
1 1 √ 2 2 terαacrossthistransition–asshowninFig.3forincreasing
(cid:126)a = (1,0) and(cid:126)a = (1/2, 3/2) as illustrated in Fig. 2.
1 2 strength of the magnetic field h. Approaching the endpoint
ItshouldbenotedthatthenumericalanalysisofHamiltonian
(2)isachallengingendeavor,sincenotonlytheentireHilbert
spaceneedstobeconsidered(duetothelackofSU(2)invari-
70
)
ance),butonealsohastoworkwithcomplexdatatypes(due α
d 60
to the (cid:104)111(cid:105) orientation of the magnetic field). Our DMRG E/
d 50
calculations keep up to m = 2048 states, which is found δ(
to give excellent convergence with typical truncation errors p 40
m 30
of less than 10−8. We further use periodic boundary condi- u
tions in both lattice directions, which reduces finite-size ef- gy j 20
fects. WehavedeterminedthephaseboundariesinFig.1by ner 10
e 0
extensivescansoftheground-stateenergy,magnetization,and
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
theirderivativesinthe(α,h)-parameterspace[11]. magnetic field h
Magnetically ordered states.– We start our discussion of
thephasediagramshowninFig.1byfirstrecapitulatingprevi- FIG. 3: (color online) Energy jump along the first-order transition
betweenthecantedNe´elandstripyAFM.
ousresults[8]fortheHeisenberg-Kitaevmodel(1)intheab-
3
of this first-order line around h (cid:39) 0.7 this drop smoothly 8 8
c
vanishes, which indicates that this endpoint possibly is a tri-
κ 6 6
critical point at which the two canted magnetically ordered m t o p noolong-iAcable lpiahnase polarized phase
phases and the polarized phase meet. The existence of such
er4 4
atricriticalpointcanbeunderstoodwithinaLandaudescrip- g t
n
tionwithtwodistinctorderparameters–correspondingtothe ru2 2
discretesymmetrybreakingofthetwodifferentmagnetically
orderedphases–openingagaptomagnonexcitations. 0 0
0 0.08 0.16 0.24 0.32
Topologicalphase.– Wenowturntothespinliquidphase magnetic field h
foundforcouplingparameters0.8 (cid:46) α ≤ 1. FortheKitaev
limit (α = 1) it has previously been argued [9] that an in- h)0.5 50
flfiornogimitceastlihlmeyagolarfidpeellerdsesdalsoppnhingastlehiq.euA(cid:104)i1ds1iw1n(cid:105)teodwiarieglclatdipoipsnecdwusinslolidnnr-iAtvhebeethlfioealnsloytwsotpienomg- zation M(000...243 κκκκ ==== 0063.09 234000M / dh
our numerical simulations allow us to confirm the existence eti d
gn0.1 10
ofsuchatopologicallyorderedstateforsmallmagneticfield a
m
0 0
strengths not only in the Kitaev limit, but for the full extent
0 0.08 0.16 0.24 0.32 0.4 0.48 0 0.08 0.16 0.24 0.32 0.4 0.48
h h
ofthegaplessspinliquidphase,asindicatedinthephasedi-
agram of Fig. 1. We will further present an independent and
non-perturbative way to determine the topological nature of
FIG.4:(coloronline)Toppanel:Ground-statephasediagramofthe
this phase. This complements the original argument by Ki- Kitaevmodel(α=1)intheh−κplane,wherehisthestrengthof
taev[9],whichwasprimarilybasedonaperturbationexpan- amagneticfieldpointinginthe(cid:104)111(cid:105)directionandκisthestrength
sionshowingthattheleadingordereffectofasmallmagnetic ofatime-reversalsymmetrybreakingthree-siteterm. Lowerpanel:
field h is to introduce a topological mass term for the Majo- Magnetizationsweepsanditsderivativeforvariousκ.
ranafermions–however,suchaperturbativeargumentshould
be carefully tested when applied to a gapless state. For this
purpose, weconsideranadditionalthree-spinexchangeterm behavior of the gap for the Majorana fermions in the exact
κ,indicatedbythebluebondsinFig. 2a),inourHamiltonian solution for h = 0. The dispersion of the Majorana fermion
(cid:114)
(cid:12) (cid:12)2
HHK+h+κ =HHK+h−κ(cid:88)σixσjyσkz. (3) isgivenbyEk =2 (cid:12)(cid:12)1+ei(cid:126)k·(cid:126)a1 +ei(cid:126)k·(cid:126)a2(cid:12)(cid:12) +κ2sin2((cid:126)k√·(cid:126)a1).
ijk Forsmallκ(cid:28)1,theMajoranafermionhasagapE (cid:39) 3κ.
g
IntheKitaevlimit(α=1,h=0)thisHamiltonianisexactly However, for large κ (cid:29) 1 the gap of Majorana fermion re-
solvableinthesameMajoranafermionrepresentationusedin mainsfiniteandindependentfromκ,givenbyEg (cid:39)2. Since
thesolutionoftheunperturbedKitaevmodel[9]. Inparticu- the magnetic field strength hc required to destroy the topo-
lar,onecanprovethatthethree-spinexchangeκbreakstime- logical phase is determined by the Majorana fermion gap at
reversal symmetry and gaps out the spin liquid phase into a h = 0, thecriticalfieldhc thusalsoincreasesandthensatu-
topologically ordered state with non-Abelian excitations, so- ratesatlargeκ.
called Ising anyons. To demonstrate that a small magnetic Field-driventransitionoutofthetopologicalphase.– We
field in the (cid:104)111(cid:105) direction drives the system into the same now return to the phase diagram of the Heisenberg-Kitaev
phase, we have numerically calculated the phase diagram in modelinFig.1andfocusonthetransitionbetweenthetopo-
the presence of both perturbations as shown in Fig. 4. The logically ordered state and the polarized state for large field
phaseboundarieswereagainobtainedbyscanningthederiva- strength. For the Kitaev limit (α = 1) this transition occurs
tivesofground-stateenergyandmagnetizationinthe(h,κ)- at a critical field strength of hc ≈ 0.072 and remains almost
parameterspace. Inparticular,thisphasediagramshowsthat constant as the coupling parameter α is decreased. Interest-
one can adiabatically connect the phase for large κ and van- ingly,ournumericssuggestthatthisfield-drivenphasetransi-
ishingmagneticfieldwiththephaseforsmall,non-vanishing tion might be continuous or weakly first-order. In particular,
magneticfieldandκ=0,thusprovingthatthemagneticfield wefind thatthe second-derivativeof theground stateenergy
gapsoutthespinliquidintothesamenon-Abeliantopological −d2E/dh2 atthistransitiondivergeswithincreasingsystem
phasestabilizedbythethree-spinexchange. Theonlyfeature size, while the magnetization M(h) does not show any dis-
in the diagram is a single phase transition line which sepa- continuity,asshowninFig.5a)andb),respectively.
rates the topologically ordered state from the fully polarized While the limited system sizes in our study do not allow
stateexpectedforlargemagneticfieldstrengths. FortheKi- to unambiguously determine the continuous nature of this
taevlimit(κ = 0)thistransitionoccursforh (cid:39) 0.072. Itis field-driven phase transition, our numerics nevertheless pro-
c
interestingtonotethatthecriticalfieldh initiallygrowswith videsomefurtherinsightswhatmightcausesuchacontinuous
c
increasing κ, but then saturates to some finite value around transition[13]. Tothisend,weplotthenumberofvorticesin
κ (cid:38) 6. Physically, this saturation can be understood by the thegroundstateasafunctionofmagneticfield,i.e. thenum-
4
8 700
2 6 2α600 h = 0.06
E / dh4 2E / d500 NN == 6448 == 22xx86xx44
2-d2 e -d400 NN == 3224 == 22xx44xx43
v
0 0.5 80 vati300
60 eri200
h d
M / d M0.25 40 2nd 100
d 0 20 0
0 0.08 0.16 0.24
h 0.785 0.79 0.795 0.8 0.805 0.81 0.815 0.82
5 0 coupling α
r
be 4
um 3 N = 64 = 2x8x4 FIG.6:(coloronline)Constantfieldscaninthevicinityofthe(multi-
n N = 32 = 2x4x4 critical)pointseparatingthestripyAFMfromthetopologicalphase.
ex 2 N = 36 = 2x6x3
rt 1 N = 24 = 2x4x3
o
v
0
Outlook.– Having established the rich phase diagram of
0 0.1 0.2 0.3 0.4
magnetic field h theHeisenberg-Kitaevmodelinamagneticfield,itisinterest-
ing to speculate where one would place the Iridate Na IrO .
2 3
FIG.5:(coloronline)PhasetransitionfortheKitaevmodel(α=1) Whileexperiments[4]reportindicationsofanAFMordered
ina(cid:104)111(cid:105)magneticfieldofstrengthh. a)Secondderivativeofthe ground state below T ≈ 15 K, the precise nature of the or-
N
groundstateenergy−d2E/dh2,b)MagnetizationM(h)anditsfirst derremainsopen. Giventheconsiderablesuppressionofthe
derivativedM(h)/dhfordifferentsystemsizes.c)Vortexnumber.
orderingtemperatureT incomparisonwiththeCurie-Weiss
N
temperatureΘ ≈116K[4],whichistypicallyinterpreted
CW
asanindicatoroffrustration,analternativeexplanationwould
ber of plaquettes with a non-trivial flux, in Fig. 5c). Below betheproximitytoaquantumcriticalpoint,suchasthemul-
the critical magnetic field, i.e. h < h , there are no vortices ticritical point α ≈ 0.8 in the context of our phase diagram.
c
indicating a deconfined phase as expected in the presence of This would bring the material in close proximity to the spin
a vortex gap. At the phase transition, however, the vortices liquidphaseforα (cid:38) 0.8andthetopologicalphasefoundfor
appeartocondenseandthenumberofvorticesintheground a magnetic field pointing in the (cid:104)111(cid:105) direction. To further
state quickly increase above the critical field strength. The substantiatethispossibility, itisdesirabletostudythefinite-
nature of the phase transition might thus be framed in terms temperature phase diagram of our model system and to con-
of a confinement-deconfinement transition of a non-Abelian sider the effectsof disorder, such as sitemixing between the
gaugefield,akintotheconfinement-deconfinementtransition IrandNasites[4]. Finally,itwouldbeinterestingtobringthe
intheAbeliandiscretegaugetheory[14]. Anexampleofthe Mottphysicsdiscussedinthismanuscriptincompetitionwith
latteristheZ gaugetheory,e.g. thetoriccodeinamagnetic thetopologicalinsulatorphasesuggestedinRef.17.
2
field [15], for which it is well known that flux condensation
We acknowledge discussions with G. Jackeli, R. Kaul, D.
leadstoaconfinementtransition[16].
N. Sheng, and R. Thomale. XLQ is supported partly by the
Asecondmulticriticalpoint.– Finally, wenotethatthere AlfredP.SloanFoundation.
appearstobeasecondmulticriticalpointinourphasediagram
aroundα ≈ 0.8andh = 0,wherethestripyAFMphaseand
the gapless spin liquid meet. We find that in the presence of
themagneticfieldthetransitionlinesofthefield-drivenphase
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5
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2
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x y y z z x
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similaritiesfortopologicalstateswithAbelianandnon-Abelian
topologicalorder,therealsoexistimportantdifferences.Forthe