Three-band Hubbard model for Na$_2$IrO$_3$: Topological insulator, zigzag antiferromagnet, and Kitaev-Heisenberg material

1.2 MB·

by  Manuel Laubach

Three-band Hubbard model for Na IrO :
2 3
Topological insulator, zigzag antiferromagnet, and Kitaev-Heisenberg material
Manuel Laubach,1 Johannes Reuther,2,3 Ronny Thomale,4 and Stephan Rachel1,5
1Institut fu¨r Theoretische Physik, Technische Universita¨t Dresden, 01062 Dresden, Germany
2Dahlem Center for Complex Quantum Systems and Fachbereich Physik, FreieUniversita¨t Berlin, 14195 Berlin,Germany
3Helmholtz-Zentrum Berlin fu¨r Materialien und Energie, D-14109 Berlin, Germany
4Institut fu¨r Theoretische Physik, Universita¨t Wu¨rzburg, 97074 Wu¨rzburg, Germany
5Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
Na IrO was one of the first materials proposed to feature the Kane-Mele type topological insu-
2 3
latorphase. ContemporaneouslyitwasclaimedthattheverysamematerialisinaMottinsulating
phase which is described by the Kitaev-Heisenberg (KH) model. First experiments indeed revealed
7
Mott insulating behavior in conjunction with antiferromagnetic long-range order. Further refined
1
experiments established antiferromagnetic order of zigzag type which is not captured by the KH
0
model. Since then several extensions and modifications of the KH model were proposed in order
2
to describe the experimental findings. Here we suggest that adding charge fluctuations to the KH

n modelrepresentsanalternativeexplanationofzigzagantiferromagnetism. Moreover,aphenomeno-
a logical three-band Hubbard model unifies all the pieces of the puzzle: topological insulator physics
J
forweakandKHmodelforstrongelectron–electroninteractionsaswellasazigzagantiferromagnet

7 at intermediate interaction strength.
1

] Introduction. The past decade of condensed matter netic ground state [11, 12] – as suggested earlier by ab
l
e physics has been strongly influenced by the rise of spin initio calculations[13] – which cannot be explained by
-
r orbit coupling. Although known from the early days of the original KH model. Several works proposed exten-
t
s quantummechanics,onlyrecentlyitstartedtounfoldits sions of this model[14–26] to account for the experimen-
.
t complexity and potential. Most notably, topological in- tal findings, ranging from longer-ranged Heisenberg ex-
a
sulators[1, 2] are spin orbit dominated band insulators change[14,15,26]and“negative”KHinteractions[17]to
m
with exotic surface properties. Amongst strongly cor- off-diagonalΓexchange[21,22]andlonger-rangedKitaev
-
d related systems, spin-orbital liquids[3] and spin orbit– interactions[19, 23, 25]. Furthermore, the formation of
n assisted Mott insulators[4] are fascinating states of mat- molecularorbitalcrystalshasbeenproposedasanexpla-
o ter which were proposed in the past years. nation for the phenomenology of Na IrO [16]. Despite
2 3
c
the variety of different models, it remains elusive until
[ Soon after the theoretical prediction of the quantum
today which might be the spin Hamiltonian to most ac-
1 spin Hall (QSH)[5] effect it was realized that graphene curately describe the phenomenology of Na IrO . More-
2 3
v has negligibly small spin orbit coupling such that the
over, it has even been questioned whether Coulomb in-
6 topologicalinsulatorphaseisnotobservable. Eversince,
teractionsareasstrongasbelievedandwhethertheMott
9
the field has been searching for other honeycomb lat-
8 limit is justified: ab initio results rather suggest a mod-
tice materials with possibly heavier elements and alter-
4 erateinteractionstrength[16,18]orevenastrong3DTI
native mechanisms to accomplish large gap QSH states
0 phase[27]; likewise, recent ARPES experiments[28] re-
. in honeycomb monolayers[6]. One of the first propos-
1 port the observation of metallic surface states which are
als to feature Kane-Mele-type QSH physics focussed on
0 notcompatiblewiththepictureofanantiferromagnetin
7 Na2IrO3 where the Ir atoms form an effective 2D honey- the deep Mott limit.
1 combnet[7]. Shitadeetal.arguedthatifCoulombinter-
: actionsarenottoostrongNa IrO mightexhibitacorre- Given the enormous interest in the KH model and
v 2 3
i latedtopologicalinsulator(TI)groundstate. Aroundthe its extensions, one might wonder how a corresponding
X
sametime,JackeliandKhaliullinproposedthatNa IrO Hubbard model could be reconstructed of which the KH
2 3
r mightbeaMottinsulatorwherethespinphysicsisdom- model is its strong coupling limit. A first guess might
a
inated by both Heisenberg and exotic Kitaev spin ex- be that the non-interacting TI Hamiltonian considered
change[8],referredtoasKitaev-Heisenberg(KH)model. by Shitade et al.[7] leads in the strong coupling limit to
Since the pure Kitaev model[9] is a rare example of an the KH model. A simple analysis reveals, however, that
exactlysolublequantumspinliquidsystem,theprospect in this case Kitaev spin exchange is generated on sec-
of approximately realizing it in an actual material has ondneighborlinks,whilenearestneighborexchangeisof
attracted enormous attention. The phase diagram of the pure Heisenberg type[29]. Likewise, one can construct a
KH model contains a N´eel ordered antiferromagnet, the single-orbital band structure which indeed leads to the
Kitaev spin liquid, and a stripy antiferromagnet as an KH Hamiltonian for large U[30]; it turns out, however,
intermediate phase[10]. The successful growth of sin- thatsuchabandstructureexplicitlybreakstime-reversal
gle crystals, however, established a zigzag antiferromag- symmetry since real spin-dependent hoppings on nearest
2
neighbor bonds are involved.
In this paper, we introduce a three-band Hubbard 1.0 0.8
model with a time-reversal invariant band structure
which is constructed such that its strong coupling limit 0.8 NI
0.6
isthepureKHspinmodel. Theminimalmodelweiden- SM
tified has three orbital degrees of freedom which can be
0.6
interpreted as the t2g manifold. Thus the phenomeno- α 0.4∆sp
logical model introduced here has a strong similarity to
the model studied earlier by Rau et al.[21]. The analy- 0.4 etal TI
m
sis of the non-interacting band structure reveals an ex- 0.2
0.2
tended topological insulator phase. Since there are only
metal
twotypesoftime-reversalinvariantinsulatingbandstruc- SM
0.0 0
tures in 2D[31], the weak-coupling limit of our three-
0.0 0.2 0.4 0.6 0.8 1.0
band model is topologically equivalent to the one pro-
λ
posed by Shitade et al.[7]. As we analyze the model in
the presence of increasing interaction strength we find
that it hosts a zigzag ordered phase at intermediate in- FIG. 1: Non-interacting λ-α phase diagram for t(cid:48) = t and
1 1
teraction strength U in consistency with experiments for C = 1. The single particle gap ∆ is indicated via color
sp
Na IrO . Inthissense,ourphenomenologicalmodeluni- codefortheinsulatingphases. Wefindnormalinsulator(NI),
2 3
fies the work by Shitadeet al.[7] at weak U, the work by topological insulator (TI), metallic phases, and semimetallic
lines (SM).
JackeliandKhaliullin[8]atstrongU,andtheexperimen-
tal results[11, 12] at intermediate U.
Phenomenological model. The spinfull three-band
the Pauli vector. When λ > 0 the spin-orbit coupling
modelinvestigatedinthisworkconsistsofakinetic,spin-
generatesalowenergyJ =1/2doublet,representingthe
orbit and interaction term
subspaceinwhichthespinphysicsoftheKHmodeltakes
H =H +H +H (1) place.
kin SO I
The simplest type of interaction term is an onsite
with 1/6 electron filling. The kinetic part takes the form Hubbard repulsion H = U (cid:80) (N 1)2 with N =
I 2 i i − i
(cid:80) d† d which energetically favors single site oc-
Hkin = (cid:88) (cid:88) d†inαTnγn(i(cid:48),j)din(cid:48)α+h.c., (2) cupn,aαnciyn.αPeinrαforming a strong coupling expansion in sec-
(cid:104)i,j(cid:105)n,n(cid:48),α ond order in t /U as described in Ref. [21] and pro-
1/2
jecting the result on the low energy J = 1/2 subspace
(†)
where dinα denotes a fermionic annihilation (creation) (whichimpliesU (cid:29)t1/2 andλ(cid:29)t21/2/U),yieldanusual
soppienraαtor on,sit.e Ti,hweiothrbiotrablitinadlinces∈ar{eyzre,mxzin,xisyc}entanodf spin-isotropic Heisenberg coupling H = J(cid:80)(cid:104)i,j(cid:105)S˜i ·S˜j
the com∈mo{n↑ l↓a}belling of a t2g manifold which our model with J = 4Ut21 where S˜i are local spin-1/2 operators act-
is closely related to. The sum is over pairs of near- ing in the J = 1/2 doublet. We find that the minimal
est neighbor sites i,j on the honeycomb lattice and SU(2) symmetric extension of the plain Hubbard repul-
(cid:104) (cid:105)
γ(i,j) x,y,z indicates the type of inequivalent Ki- sion which generates anisotropic spin interactions in this
∈ { }
taev bond the pair i,j belongs to. For a z-bond the subspace has the form
(cid:104) (cid:105)
hopping matrix Tz is given by
nn(cid:48) H = U −3JH (cid:88)(N 1)2 2J (cid:88)S2 (5)
t t 0  I 2 i− − H i
1 2 i i
Tz =t2 t1 0  , (3)
0 0 t(cid:48) with
1
and the matrices Tnxn(cid:48) and Tnyn(cid:48) follow by cyclic permu- Si = 21 (cid:88) d†inασαα(cid:48)dinα(cid:48) (6)
tationsoftherowsandcolumns. Unlessexplicitlystated n,α,α(cid:48)
otherwise we will assume t = t(cid:48) in the following. The
1 1
spin-orbit coupling has the usual form and the Hund’s coupling JH. The prefactors have been
chosentoresembletheKanamoriHamiltonianwhichhas
HSO = λ2 (cid:88) (cid:88) d†inαLnn(cid:48) ·σαα(cid:48)din(cid:48)α(cid:48) , (4) [b2e1e,n3u2s,ed33t]o. de(sNcoritbeetmhautltitphleetfiunltlerKacatniaomnsoirni tH2gamshiletlols-
i n,n(cid:48),α,α(cid:48)
nian also contains additional L2 terms.) Together with
i
whereLdenotesanangularmomentumoperatorin3 3 Eqs. (2) and (4) this type of minimal interaction will be
×
matrix representation with L2 = l(l+1) = 2 and σ is investigated for different coupling strengths below.
3
At strong coupling (assuming U t and λ AF stripy QSL
1/2
(cid:29) (cid:29)
t2 /U) the Hamiltonian (1) reduces to the KH model ∞
1/2 50 1
HKH =J (cid:88)S˜i·S˜j −K(cid:88)S˜γi(i,j)·S˜γj(i,j) (7) 30 stripy
(cid:104)i,j(cid:105) (cid:104)i,j(cid:105)
10
with parameters 3
J = 4t21(3U −4JH) , K = 16JHt22 . (8) U AF NMI ∆
2
3U(U 4J ) 3U(U 4J )
H H
− −
zigzag
When U > 4J the model features antiferromagnetic
H
1
Heisenberg and ferromagnetic Kitaev interactions (i.e.,
TI
J,K >0) as has been originally proposed by Chaloupka SM
NI
et al.[10]. Casting this Hamiltonian into the conve- 0 0
nientformwheretheexchangeinteractionsarewrittenas
0 0.2 0.4 0.6 0.8 1
J =1 α,K =2α,Eq.(8)yieldsthehoppingamplitudes
− α
(cid:115) (cid:114)
1 α α
t1 =C − , t2 =C (9) FIG. 2: Interacting phase diagram in the U-α plane for
3U/J 4 2
H − λ=C =1 and t(cid:48)1 =t1. Below the black horizontal line, the
range from weak to intermediate interactions 0 < U < 3 is
where the parameter C sets the energy scale of the ki-
shown,whileaboveit,intermediatetostronginteractions3<
netic Hamiltonian. In total, α, U, J , λ, C define a five
H U <50. Inaddition,thespinlimitU →∞isshown. FiniteU
parameter model, for which we will study the effects of resultsareobtainedwithinVCAandthestrongcouplingspin
charge fluctuations away from the strong coupling limit modelissolvedusingEDfor24sites. Wefindtopological(TI)
(notethatoneparametercanbeabsorbedintoanoverall and normal insulating (NI) phases, semi-metallic (SM) lines
energy scale). This three band system can be considered (brown),zigzag,N´eel(AF),andstripyantiferromagnets,and
a non-magnetic insulator (NMI) phase. For details see main
as a simplified (and particle-hole transformed) model for
text.
Na IrO as it has been studied in Ref. [21]. A signifi-
2 3
cant simplification comes from setting t =t(cid:48) in Eq. (3).
1 1
Giventheorbitalstructureofthet states,t <t(cid:48) would
2g 1 1
stricted to approximations. Below, we will present nu-
certainly better account for the microscopic situation in
mericalevidencethatthereisaphasetransitionfromthe
Na IrO . However,thisgeneratesadditionaloff-diagonal
2 3
topologicalinsulatorphaseintoazigzagantiferromagnet
and symmetric Γ-exchange, which complicates the effec-
(AFM)atintermediateinteractionstrengthbasedonthe
tive spin model and may drive spiral types of magnetic
order. We, hence, mostly consider t = t(cid:48) but also dis- Variational Cluster Approach (VCA). VCA is a quan-
1 1
cuss the qualitative changes of our results when t <t(cid:48). tum cluster approach very similar to Cluster Perturba-
1 1
tionTheory[34]andClusterDynamicalMean-FieldThe-
We first focus on the properties of the non-interacting
ory[35] where the interacting Green’s function is com-
model, U = J = 0. Instead of t we use α and
H 1/2
puted on a small cluster which is then used to con-
the spin orbit amplitude λ as parameters (here we keep
struct the full Green’s function for the infinitely large
U/J =5inordertobeconsistentwiththediscussionof
H
system within a self-consistent, variational scheme. Here
the interacting case below). Calculating the band struc-
we use a C-shaped four-site cluster with three orbitals
ture, we identify (semi-)metallic and insulating phases
where the Z topological invariant reveals both topolog- per site corresponding to an effective 12-site cluster.
2
For details about the method we refer the interested
ically trivial and non-trivial insulating regimes. These
reader to Refs.[36–39]. The VCA method[36, 40] has
findings are summarized in the non-interacting phase di-
been successfully applied to study interacting topologi-
agram in Fig.1 where we also show the single-particle
calphases[30,37,39,41]aswellasmagneticallyordered
gap for the TI and normal insulator (NI) phases. Since
systems[37, 38, 42].
there are only two types of time-reversal invariant insu-
latingbandstructuresin2D(trivialandtopological)[31] Using VCA, we obtain the U-α phase diagram for
the TI phase is topologically equivalent to the one pro- 0 < U 50, see Fig.2. To illustrate different coupling
≤
posed by Shitadeet al. Hence, together with the above regimes, we divided the range of interactions into two
resultsfromastrongcouplingexpansion, ourthree-band parts: weak to intermediate interactions 0 < U < 3 and
modelnaturallyunifiestheproposalsofShitadeetal.and intermediate to strong interactions 3 < U < 50. In ad-
Jackeli and Khaliullin. dition, we show the result in the strong coupling limit
Zigzag antiferromagnetic phase. In general, Hubbard (U ) obtained from exact diagonalization (ED) for
→ ∞
models in d = 2 are not exactly soluble and one is re- afinitesystemwith24latticesiteswhichagreeswithpre-
4
vious studies such as Ref.[21]. As a central VCA result, 0.5 0.5
U=0.0 U=1.7
we find that within wide ranges of α, the topological in-
sulator phase is stable up to U 1 followed by a second 0 0
∼
order phase transition without gap closing into a zigzag
AFM regime. 0.5 0.5
Mappingoutthedetailedinterplayofdifferentphases, −0.5 0 Ukπ=x1.5 2π −0.5 0 Ukπ=x1.8 2π
we first observe a TI, NI and two semi-metallic points
0 0
at U = 0, see Fig.2. Up to U 1 the phase diagram
∼
remains largely unchanged except that the semi-metallic
pointbetweentheTIandNIphasessplitsintotwosemi- −00..55 0 π 2π −00..55 0 π 2π
metallic lines. For U 1 the TI phase, evolving over Uk=x1.6 Uk=x1.9
a wide range 0 < α <(cid:63)0.8, undergoes a phase transi-
0 0
tion into the zigzag AFM phase. Note that the inter-
action driven phase transition from the TI phase into a
0.5 0.5
collinearmagnetisingeneralexpectedfornon-frustrated − −
0 π 2π 0 π 2π
lattices[37, 43]. For U 1.6 to 1.8 a first-order phase kx kx
≈
transitionfromzigzagAFMtoaN´eelorderedAFMtakes
FIG. 3: Spectral function A(k ,ω) for the surface states at
x
place. While the N´eel phase extends up to α 0.8 for α = 0.4, C = λ = 1, t(cid:48) = 1.5t and different interaction
≈ 1 1
small interactions, with increasing U it slowly shrinks strength. For U < 1.5 magnetic order is absent and helical
down to α 0.4 for U = 50 which coincides with the edgestatesindicatetheTIphase. ForU >1.6wefindafinite
≈
phase transition in the ED phase diagram. For large α zigzag order parameter and edge states acquire a small gap
we detect a non-magnetic insulator (NMI) phase which sincetheprotectingtime-reversalsymmetryisspontaneously
broken.
persists up to U 10 and down to very small U at
≈
α 0.85. The origin of this phase is unclear: candidate
≈
states are quantum paramagnets including spin liquids
and valence bond solids but also ordered states which follows: (i) As mentioned before, symmetric off-diagonal
are incommensurate with respect to the cluster size such ΓexchangeisgeneratedinthespinHamiltonian(7). (ii)
as spiral magnets or large-unit cell magnets. At U 5 As a result of these couplings, the ED phase diagram
a stripy AFM becomes stable for large α and starts≈to at U shows an additional small 120◦ N´eel regime
→ ∞
spread over an extended α-range for increasing U. The between the stripy AFM phase and the Kitaev spin liq-
reader should notice that VCA is unable to identify the uid[21]. (iii) Changes in the non-interacting phase di-
Kitaevspinliquidphase,whichisclearlyresolvedbyED agram and in the interacting VCA phase diagram are
forα>0.8. Duetothesmallclustersizeandthesuppres- rather small indicating that up to minor shifts of the
sionoflong-rangeentanglement(whichisaninescapable phase boundaries, our previous discussion of the ideal-
consequenceofquantumclusterapproaches)thisisbyno izedscenariot(cid:48)1 =t1isstillvalidinthemorerealisticcase
means surprising. For large U and α 1, however, the t(cid:48)1 = 1.5t1. Similar arguments also hold for the overall
≈
VCA condensation energies of possible magnetic orders kineticenergyscaleC [seeEq.(9)]whichisanadditional
become almost indistinguishable. Such a nearly degen- free parameter. While so far we have set C =1, we also
erate ground state scenario might indicate the proximity checkedthedependenceoftheinteractingphasediagram
to a spin-liquid phase. on this parameter. Varying C in the range 0.8 < C < 2
On a conceptually more simple level, the presence of a partially moves the phase boundaries but does not gen-
zigzag AFM at intermediate interaction strength can be erate new phases or destroy existing ones. Thus we con-
furthersubstantiatedbyaHartree-Fock(HF)mean-field cludethatinthisrangeC onlyaffectsthephasediagram
analysis. Recently, the HF approach has been applied to quantitatively but not qualitatively.
a three-band Hubbard model which is similar to Eq.(1) We finally focus on the regime near the transition be-
butdiffers in details aboutthe interaction termsand the tween the TI and zigzag AFM phase (for t(cid:48) = 1.5t ).
1 1
signs of the parameters[44, 45]. In these works, a zigzag Assumingthattheexperimentallyobservedzigzagphase
AFM phase was likewise identified at intermediate inter- is indeed not deep in the Mott phase but much closer
action strength. to the weak-coupling regime[16, 27] it is interesting to
Discussion. So far we considered t = t(cid:48) which led to investigate whether remnants of the TI phase can still
1 1
the KH model in the strong coupling limit. Given the be detected in the zigzag phase. We compute the single-
microscopic situation of Na IrO , the parameter choice particle spectral function A(k ,ω) on a ribbon geometry
2 3 x
t <t(cid:48) certainlybetteraccountsfortheorbitalstructure which allows to track the edge states even in the pres-
1 1
and orientation of the t states. To make the model ence of interactions. In Fig.3 surface spectral functions
2g
morerealistic, weconsidert(cid:48) =1.5t intheremainderof for several values of U at α = 0.4 are shown. In the TI
1 1
the paper. The changes of the results are summarized as phase at U < 1.5 we clearly observe helical edge states
5
traversing the bulk gap while the magnetization remains 017205 (2009).
zero. At U = 1.5 the second order phase transition oc- [9] A. Kitaev, Ann. of Phys. (N.Y.) 321, 2 (2006).
curs marking the smooth onset of magnetization. Once [10] J.Chaloupka,G.Jackeli,andG.Khaliullin,PhysicalRe-
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magnetic order sets in, we immediately observe a small
[11] S. K. Choi et al., Phys. Rev. Lett. 108, 127204 (2012).
gap in the edge states at k =π, which is a consequence
x [12] F. Ye et al., Phys. Rev. B 85, 180403 (2012).
of the lost topological protection. Even around U =1.9,
[13] X. Liu et al., Phys. Rev. B 83, 220403(R) (2011).
i.e., close to the transition from the zigzag to the N´eel
[14] I.KimchiandY.-Z.You,Phys.Rev.B84,180407(2011).
AFM, we still observe the gapped edge states, see Fig.3. [15] Y. Singh et al., Phys. Rev. Lett. 108, 127203 (2012).
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2 3
[18] K. Foyevtsova et al., Phys. Rev. B 88, 035107 (2013).
persion have been revealed[28, 46]. It remains as an ex-
[19] J.Reuther,R.Thomale,andS.Rachel,Phys.Rev.B90,
citingtasktoverifywhetherthesemetallicsurfacestates
100405 (2014).
are the topologically trivial remnants of the TI phase as
[20] V. M. Katukuri et al., New J. Phys. 16, 013056 (2014).
suggested by our VCA analysis. [21] J.G.Rau,E.K.-H.Lee,andH.-Y.Kee,PhysicalReview
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2 3
Rev. B 90, 155126 (2014).
Shitadeetal.[7],theKHmodeladvocatedbyJackeliand
[24] I. Kimchi, R. Coldea, and A. Vishwanath, Phys. Rev. B
Khaliullin[8],andtheinelasticneutronscatteringresults
91, 245134 (2015).
of Choi et al.[11] indicating a zigzag antiferromagnetic
[25] I. Rousochatzakis et al., Phys. Rev. X 5, 041035 (2015).
ground state. Our three-band Hubbard model hosts for [26] S.M.Winter,Y.Li,H.O.Jeschke,andR.Valent´ı,Phys.
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[29] J.Reuther,R.Thomale,andS.Rachel,Phys.Rev.B86,
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155127 (2012).
which might have been observed in recent ARPES mea-
[30] S.R.Hassanetal.,Phys.Rev.Lett.110,037201(2013).
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[31] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802
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We acknowledge financial support by the DFG through Condens. Matter Phys. 4, 137 (2013).
SFB 1143 (ML, SR) and SFB 1170 (RT), by the ERC [33] L. de’ Medici, J. Mravlje, and A. Georges, Phys. Rev.
Lett. 107, 256401 (2011).
through the starting grant TOPOLECTRICS (ERC-
[34] D. S´en´echal, D. Perez, and D. Plouffe, Phys. Rev. B 66,
StG-Thomale-336012, RT), and by the Freie Universit¨at
075129 (2002).
BerlinwithintheExcellenceInitiativeoftheGermanRe-
[35] T. Maier, M. Jarrell, T. Pruschke, and M. H. Hettler,
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mationServicesandHighPerformanceComputing(ZIH) [36] M. Potthoff, Eur. Phys. J. B 36, 335 (2003).
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