Tuning into the Kitaev spin liquid phase:A spin model on the honeycomb lattice with two types of Heisenberg exchange couplings

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by  Yue Yu

Symmetric spin-orbital-Hubbard model on honeycomb lattice: An origin of Kitaev
model and zig-zag antiferromagnetic order
Yue Yu and Shaojing Qin1
1State Key Laboratory of Theoretical physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, P.O. Box 2735, Beijing 100190, China
(Dated: February 28, 2012)
We present a single band Hubbard model on honeycomb lattice with link-symmetric spin-orbital
interaction. InthelargeU limit,thismodelismappedtoaconventionalantiferromangentic(AFM)
Heisenberg model combining with a ’zig-zag’ AFM Heisenberg one. The latter AFM order which
comes from the symmetric spin-orbital coupling was favored by a recent experiment for A IrO
2 3
(A=Na, Li)[6–9]. Competition between two different AFM orders leads to a spin liquid phase
2
characterized by Kitaev model that happens if the spin-orbital coupling has the same strength as
1
the hopping amplitude. This deepens our understanding of the origin of the Kitaev coupling: It
0
comesfromthebalancebetweenthehoppingandspin-orbitalcouplingbuttogetridofthehopping.
2
In a moderate U, this model has a charge-spin gapped phase with two different spinon excitations.

b Inaweakinteraction,theoriginalgaplessDiracfermionisgappedbythespin-orbitalcouplingwhile
e a new gapless Dirac fermion appears at the same Dirac points.
F
PACSnumbers: 71.20.Be,75.25.Dk,75.30.Et,75.10.Jm
6
2

Kitaev model on honeycomb lattice has attracted a alsoappearinthecompetitionbetweentwodifferentFM
]
l lot of interests because of its exact solubility, Majo- phases .
e
- rana fermion excitation, non-trivial topological orders The SO interaction which can lead to previously men-
r
t and amazing non-abelian anyon [1]. This non-abelian tioned physics is a time-reversal invariant SO interac-
s
. anyon is the key object to design a device of topological tionwiththenearestneighborsite,i.e.,link-dependence.
t
a quantum computer [2]. A possible way to realize Ki- The SO couplings with an orientation-dependence ex-
m taev model on optical lattice has been proposed [3]. Re- ist in many Mott insulators with orbital orders and
- cently,studiesofiridiumoxidesshowedthatthispeculiar fluctuations[18]. In this Letter, we include a particu-
d quantum spin liquid (SL) is possible to be realized[4, 5]. lar link-dependent SO coupling to the Hubbard model
n
o WhiletheexperimentsforA2IrO3 (A=Na,Li)supported and call it the symmetric SO-Hubbard (SSO-Hubbard)
c a ’zig-zag’ antiferromagnetic(AFM) phase [6–9], Kitaev model. The strong coupling Mott physics has been sum-
[ SLispossibletobeobservedinahighpressure[7]. These marized in the previous paragraph. For a moderate U, a

progresses drive a number of further researches [10–17] . gapped SL phase has been predicted even for the Hub-
2
v It is argued that the strong spin-orbtial(SO) inter- bard model without a SO coupling[19]. A slave boson
0 action plays an important role in deriving the Kitaev- theory [20, 21] also showed that there is such a SL state
1 Heisenberg (KH) model [4]. However, the exact soluble at the mean field level but may be broken down due to
6
1 Kitaev model is on the opposite side of the Neel order, a staggered U(1) gauge fluctuation for some symmetries
. i.e., one has to get rid of the hopping to obtain it. are broken [22]. This SL is also found in a Kane-Mele-
2
Hubbard model [23]. What we find is that, by using the
0 In this Letter, we will directly show how the SO in-
slave boson method [22], the SSO coupling will enlarge
2 teraction induces a Kitaev coupling in the context of a
1 the original spinon gap while a new branch of spinon ex-
single band repulsive Hubbard model. What we observe
: citation takes the gap of the original one. In the weak
v is as follows: For a large U, while the hopping between
coupling limit, we also find that the well-known Dirac
Xi the nearest neighbor sites contributes to the Neel order fermion on the honeycomb lattice is gapped by the SSO-
of spin S, the SO coupling offers a ’zig-zag’ AFM order
r coupling while a new gapless Dirac fermion whose veloc-
a with a variant spin S˜. The latter one is the most likely
ityissolelydeterminedbySSOcouplingisexactlylocates
spin structure in A IrO according to the recent experi-
2 3 at the original Dirac points.
ments[6–9]. Thecompetitionbetweenthesetwodifferent
We consider the SSO-Hubbard model on honeycomb
AFM orders gives birth to a SL phase delegated by Ki-
lattice with links a = x,y,z described by the following
taev model with anyonic excitations. In fact, when the
Hamiltonian
SO coupling strength is the same as the hopping ampli-
tude, the resulting spin model is exactly Kitaev model. (cid:88)
H = H +H +H =−t (c† c +h.c)
Thisisthecrucialdifferenceofourresultfromthatinthe SSOH T SSO U iσ jσ
HK model: The Kitaev phase is in the midway between (cid:104)ij(cid:105)
(cid:88) (cid:88)
the hopping amplitude and the SO coupling but getting + iλ ν c† σa c +U n n (1)
ij iσ σσ(cid:48) jσ(cid:48) i↑ i↓
ridofthehopping. WenoticethattheKitaevphasemay
a;(cid:104)ij(cid:105)a i
2
where ν = −ν = 1. c is the electron annihilation involves in the nearest neighbor sites instead of the next
ij ji iσ
operator at site i. σa are Pauli matrices. U ≥ 0 is nearest neighbors.
Hubbard potential. (cid:104)ij(cid:105) labels the summation over the Strong coupling limit. We consider H and H as the
T SSO
nearest neighbor sties and (cid:104)ij(cid:105) means over all the a- perturbations. Besidesthewell-knownAFMcouplingas-
a
links. Different from that in Kane-Mele model [24], here sociated with H , the H contribution, to the second
T SSO
the spin orbital coupling is x,y,z-cycling symmetric and order for a given link a , is given by,
2 2 J˜ J˜
− (H ) (H ) =− (iλν c† σa c )(iλνac† σa c )=J˜SaSa−J˜SbSb−J˜ScSc+ n n − n (2)
U SSO a,ij SSO a,ji U ij iσ σσ(cid:48) jσ(cid:48) ji jτ ττ(cid:48) iτ(cid:48) i j i j i j 4 i j 2 i
whereJ˜= 4λ2,a(cid:54)=b(cid:54)=candS= 1c†(cid:126)σ c istheelec- One immediately sees that if t=λ, we arrives at Kitaev
U 2 σ σσ(cid:48) σ(cid:48)
tron spin operator. At the half filling, the SSO-Hubbard model with J =2J. It is coaction both of the hopping
K
model reduces to andtheSSOcouplinggivesbirthtotheKitaevcoupling.
H =J(cid:88)S ·S +J˜ (cid:88) (SaSa−SbSb−ScSc),(3) Noticethatthesystem,eitherforJ˜=0orJ =0,isinthe
JJ˜ i j i j i j i j conventional Landau symmetry breaking ordered phase.
(cid:104)ij(cid:105) a;(cid:104)ij(cid:105)a The Kitaev phase, however, is a SL phase without any
whereJ = 4t2 aswell-known. Thecrucialdifferencehere Landau-typeorderparameter. Thatistosay,theKitaev
U
from that in the KH model is both J and J˜are positive, physicsistheresultofcompetitionbetweentwodifferent
which means we have two different AFM orders. For AFM orders. There are quantum phase transitions from
λ=0,itisAFMHeisenbergmodel. Ift=0,itisanother the Neel phase to the Kitaev SL and then to the ’zig-
quantumliquidwhichdiffersasignfromthestripyAFM zag’ AFM phase. The ground state energy per site for
model and is of a ’zig-zag’ AFM order (See Fig. 1). It the Neel order is estimated as eN/J ≈ (1 − β/3)(cid:104)Si ·
seexepmersimtoebnteatlhreepmoorsttslcikoemlybisnpiinngowrdiethritnhAe2fiIrrsOt3pirninrceicpelnest uSpj(cid:105)pefrorboβun=daJr˜y/Jofatnhdat(cid:104)oSfit·heSjK(cid:105)it≈aev−S0L.37is[e4K]/wJh<∼ile(1th+e
calculation [6–8] or a direct measurement [9]. This ’zig- β)(cid:104)SiaSja(cid:105) for (cid:104)SiaSja(cid:105) = −0.13 [25]. This estimates the
zag’AFMmodelinfactcanbemappedtoanAFMmodel critical point βc <∼ 0.95. According Eq. (5), another
HJ˜ = J˜(cid:80)(cid:104)ij(cid:105)S˜i ·S˜j with S˜i defined as follows: Divide critical point is 1/βc ≈ 1.05. By exact diagonalization
thehoneycomblatticeintofoursublattices,keepS˜ =S calculations,upto24latticesites,weconfirmthecritical
in one sublattice and change in other three, e.g.,iS˜ =i pointisveryclosetotheaboveestimatebutareunableto
i determine that according to the ground state properties.
(Sx,−Sy,−Sz) for the x-link direction sublattice (and
i i i Since the gapless excitation in the Kitaev phase is very
cycling for the y- and z-link’s. See Fig. 1(a) [4].)
different from that in the AFM phases: The former is
Majorana fermion while the latter is bosonic spin wave.
z Therefore, we can estimate the critical points by looking
y
x for the level crossing of the first excitation state energy
levels of two different phases in a finite size (See Fig. 2
(a)). (Wealsousethiscriteriatocheckthecriticalpoints
in the KH model which are the same as the results in
Ref. [4].) The 24-site data gives β ≈0.93 for the phase
c
(a)StripyAFM (b)Zig-zagAFM transition from the Neel order to the Kitaev phase.
InKitaevmodel,ifaperturbedexternalmagneticfield
FIG. 1: (Color online) (a) The stripy AFM order. Four
is applied, the SL becomes gapped and the phase we are
different colored spots label four sublattices. (b) The zig-zag
discussedturnstobetopologicallynon-trivialwithanon-
AFM order. The black and white spots divide the lattice in
vanishing Chern number and non-abelian anyons. We
to a bipartite one.
expect these non-trivial properties exist in the whole SL
phaseandagaplesschiralexcitationemergesintheedge
Rewrite Eq. (3) as
of the system. Both of the AFM states are also gapped.
H =(J −J˜)(cid:88)S ·S +2J˜ (cid:88) SaSa (4) At the critical points, the gaps collapse.
JJ˜ i j i j
(cid:104)ij(cid:105) a;(cid:104)ij(cid:105)a Whenthesystemdeviatesfromthehalffilling,onecan
=(J˜−J)(cid:88)S˜ ·S˜ +2J (cid:88) S˜aS˜a. (5) study the doped t-λ-J-J˜model.
i j i j
Original Kitaev model allows a phase transition from
(cid:104)ij(cid:105) a;(cid:104)ij(cid:105)a
3
Excitations for N = 8, 16, 24 SSO repulsive Hubbard model. Instead, we find that it
1.2
1 ~~JJ ~~ 01 NN = = 1 86 happenswhentwodifferentFMorders[26]compete,i.e.,
N = 24 if J <0 and J˜<0 in Eq. (4), one has
0.8
Ei - Eg 0.6 H(cid:48) =(|J˜|−|J|)(cid:88)S ·S −2|J˜| (cid:88) SaSa. (6)
0.4 JJ˜ i j i j
0.2 (cid:104)ij(cid:105) a;(cid:104)ij(cid:105)a
0 0 0.1 0.2 0.3 0.4~J 0(.J5J=~1 ) 0.6 0.7 0.8 0.9 1 (OrexpressedbyS˜ withJ˜↔J asEq. (5).) WhenJ˜=0,
Kitaev onehastheFMorderwhileitisthestripyAFMorder[4]
Zig-zag forJ =0. Again,KitaevmodelcomesoutatJ =J˜. The
phasetransitionfromtheKitaevphasetothestripyAFM
FM AFM J phase happens at J = 2J˜/3 according to the result by
Ref. [4]. Thus, the window of the Kitaev phase is larger
than the AFM case: 2/3 ≤ J˜/J ≤ 3/2. This captures
Kitaev Stripy (b) thepartoftheKHmodel[4]fromthestripyAFMtothe
Kitaev phases. Furthermore, the phase transition from
the AFM Neel order to the stripy AFM order [4] is the
FIG. 2: (Color online) (a) The level crossing of the energy result of the competition between J > 0 and J˜< 0. A
levelsofthefirstexcitedstatesforlatticesitesN =8,16and model with J < 0 and J˜ > 0 corresponds to a phase
24. (b) Schematic phase diagram in J-J˜plane. transition from the FM to the zig-zag AFM order. A
schematic phase diagram is depicted in Fig. 2(b).
Weak interaction limit. When U = 0, the free-electron
the non-abelian phase to three equal gapped SL phase gas on honeycomb lattice has a gapless Dirac fermonic
withabeliananyon. ThiscanalsoberealizedinthisSSO- lowenergyspectrumintwovalleysneartheDiracpoints.
Hubbard model by introducing link-dependent hopping Now we discuss the effect of the SSO coupling to the
amplitudes tx,ty,tz and SO couplings λx,λy,λz. In this Dirac semi-metal. The honeycomb lattice is bipartite
way, we can have a whole parameter Kitaev phase as to A and B sublattices (See Fig. 1(b)). Using the z-
well as the fruitful phase diagram from the XYZ model link as the elemental cell, the electron annihilation op-
to Kitaev model and that in Kitaev model. erator may be labeled by the sublattice suffices, i.e.,
In the KH model studied in Ref. [4], there is a phase cσi → (cAσ,i,cBσ,i). Under the Fourier transformation
transition from the stripy AFM phase to the Kitaev cασ = √21N (cid:80)kcασkeik·ri,theHamiltonianwiththeSSO
phase. This can not be explained by the single band coupling is given by
  
0 −tf(k)+iλ 0 λg (k) c
− A↑k
HT +HSSO =(cid:88)(c†A↑k,c†B↑k,c†A↓k,c†B↓k)−tf∗(k0)−iλ λg+0(k) λg+0(k)∗ −tf(k0)−iλccBA↓↑kk  (7)
k λg (k)∗ 0 −tf∗(k)+iλ 0 c
− B↓k
where f(k) = 1+eik·n1 +eik·n2 and g√±(k) = i(eik·n1 ± The t-dependent branch (E+) of fermnionic excitation is
ieik·n2)√with the vectors n1 = (1/2, 3/2) and n2 = gapped while a new gapless Dirac fermion mode (E−)
(−1/2, 3/2). The spectra of quasi-electron excitations with a speed λ appears at the same Dirac points. This
then can be easily calculated: Dirac fermion differs from the free one as the latter does
not mix the spins and the former mixes the spins.
E2 =(6λ2+2t2|f|2)/2 (8)
±
(cid:112)
± (6λ2+2t2|f|2)2−4|λ2g g −t2f2−λ2|2/2 Moderate interaction. In a moderate U, it is predicted
+ −
that there is a featureless gapped SL phase [19]. A
When λ = 0, we recove E = ±t|f(k)| ∝ q near Dirac
slave boson study showed that there is also a charge-
points K = (±4π/3,0) ( q = k−K ). Now, because
± ± spin gapped phase in a moderate U region [22] although
|f|2 ∝q2 and |g (k)g (k)−1|∝q near K , we have
+ − ± themagnitudeofthisregionisnumericallydifferentthat
√
E ≈±λ|g g −1|∼±λq, E =± 6λ+O(q2). (9) fromthequantumMonteCarlostudyinRef. [19]. More-
− + − +
4
over, the SL state is broken down due to translation The eigen energies then are easily obtained by
symmetryandlatticerotationsymmetrybreakingfroma
staggered U(1) fluctuation. As we have seen before new
physical properties of the system arise from the SSO in- E2 =ξ2/4+[6λ˜2+2t˜2|f|2 (12)
0
teraction. We now use the salve boson technique to see (cid:113)
±(1/2) (6λ˜2+2t˜2|f|2)2−4|λ˜2g g −t˜2f2−λ˜2|2]
what is the effect of this SSO coupling to the system in + −
this interaction region.
We briefly review the slave boson theory [20, 21].
where t˜= t∆ and λ˜ = λ∆ . Notice that in Eq. (12),
There are four different states in a single site: |0(cid:105),| ↑ b b
except the first term ξ2/4, the rest is of the same form
(cid:105),|↓(cid:105),|↑↓(cid:105)withthecompleteness|0(cid:105)(cid:104)0|+|↑(cid:105)(cid:104)↑|+|↓(cid:105)(cid:104)↓ 0
as Eq. (8), that of the Dirac fermions at U = 0. There-
|+|↑↓(cid:105)(cid:104)↑↓|=1. Theelectronoperatorcanbeexpressed
fore, there are two branches of spinon excitations whose
asc =|0(cid:105)(cid:104)σ|+σ|−σ(cid:105)(cid:104)↑↓|. Athalf-filling,onecanthing
σ
energy minima are at the Dirac points and the gaps are
each site has averagely an electron charge. If subtract-
(cid:113)
ing this average charge, the state |0(cid:105) can be thought as |ξ |/2 and ξ2/4+6λ˜2 , respectively. As the same as
0 0
carrying a positive charge while | ↑↓(cid:105) carries a negative that in Ref. [22], our mean field states also suffer a stag-
charge and |σ(cid:105) carries spin degree with charge neutral. gered U(1) global gauge fluctuation as f →eiαf and
iA iA
In the operator formalism, one has c = h†f +σf† d. f →e−iαf and the SL state is possibly broken down
σ σ −σ iB iB
Thisistheslavebosontechniquewithaspin-chargesep- in a semiclassical limit. Besides the next nearest neigh-
aration if h† ∼|0(cid:105),d† ∼|↑↓(cid:105) , called holon and doublon, borhoppingmaydestroythisU(1)gaugesymmetry[22],
are bosonic operators carrying charge ± while f† ∼ |σ(cid:105) the Kane-Mele SO coupling [24] also concerns the next
σ
is called charge neutral fermonic spinon. The complete- nearest neighbor sites and breaks this U(1) to Z gauge
2
ness condition becomes a constraint on these charge and symmetrysothatthesystemisturnedtobeaSL.More-
(cid:80)
spin degrees: s =n +n + n −1=0 for every over, while the Kane-Mele SO coupling gives a mass to
i hi di σ fσi
site. Within the slave boson theory, then, the effective the new Dirac fermion at the weak coupling and the sys-
Hamiltonian is given by tem is expected to be a topological insulator with the
time reversal symmetry[24], the strongly-corrected be-
(cid:88) (cid:88) (cid:88)
Heff = U d†idi− [(tχ0f,ij −λχaf,ij)χb,ij haviors with this SO coupling are more intriguing [27].
i a a−links Finally, if the system is doped, those SL phases may be
+ (t∆0† −λ∆a† )∆ +h.c.]+i(cid:88)ξ s , (10) superconductorsandthecompetitionbetweenthed-wave
f,ij f,ij b,ji i i
superconductivity and a p-wave superconductivity is ex-
i
pected as U varys.
where χ0 = (cid:80) f† f , χa = (cid:80) σ(cid:48)ν f† σa f ,
f,ij σ iσ jσ f,ij σσ(cid:48) ij iσ σσ(cid:48) jσ In conclusions, we have studied the physical behaviors
χb,ij =h†ihj −d†idj. And ∆0f,ij =(cid:80)σσfi,−σfjσ, ∆af,ij = oftheSSO-Hubbardmodelinthestrong,weakandmod-
(cid:80)σσ(cid:48)σ(cid:48)νijfiσσσaσ(cid:48)fj,−σ(cid:48) and ∆b,ij = hidj +dihj. For a erate repulsive interactions. We showed that a SL phase
non-zero ∆f then ∆b, the pair nature of the boson wave can be a result of a competition between two Landau-
functionandtheholon-doublonsymmetryrequireχb =0 type ordered states: the Kitaev phase appears when two
and then tχ0f −λχaf =0. different AFM couplings become comparable, i.e., both
NowwefollowthesametrickusedinRef. [22]tosolve the hopping amplitude and the SO coupling play equal
the mean field theory. The boson problem in our model roles. Meanwhile, the hopping does not play any role in
is not much different from that in Ref. [22] if t∆0f in the limit of U → 0. A phase with both charge and spin
Ref. [22], the mean field value of ∆0 , is replaced by gapped emerges in the midway between the weak and
f,ij
t∆0−λ∆a ineverylink. Thechargeexcitationspectrum strong couplings. We do not determine the critical U for
f f
(cid:113) these different phases. Since our mean field equation for
is E± =±U + (U−2ξ0)2−|∆ |2 where ξ =(cid:104)iξ(cid:105) and
b 2 2 fk 0 the slave boson’s is formally the same as that given in
∆fk =(cid:80)a(t∆0f −λ∆af)eik·na for nx,y =n1,2 and nz =0 Ref. [22], the result is the same. The critical interaction
[22]. The mean field effective Hamiltonian for spinon is
into the AFM orders is also very close that when λ=0.
given by
We have seen that the Kitaev phase appears due to the
(cid:88) competitionbetweentwodifferentAFMorders. Thisim-
H = −(ξ /2) (f† f +f† f ) (11)
f 0 Aσ,k Aσ,k Bσ,k Bσ,k plies the Kitaev phase must not appear before these two
k AFMphasesbutoncethesetwoorderedonesexist,there
(cid:88)
+ ∆ [f† f† (−tf(k)+iλ) should be this SL as a result of the competition.
b A↑,k B↓,−k
k
In conclusions, we have studied the SSO-Hubbard
(cid:88)
+ f† f† (tf(k)+iλ)]+ [f† f† λg (k)
A↓,k B↑,−k A↑,k B↑,−k − model for the repulsive Hubbard U. This work was sup-
k portedbyNNSFofChinaandthe973programofMOST
− f† f† λg (k)]∆ +h.c. of China.
A↓,k B↓,−k + b
5
Rev. B 83, 245104 (2011).
[13] J.KimichiandY-Z.You,Phys.Rev.B84,180407(2011).
[14] W-L. You and Y-L. Dong, Phys. Rev. B 84, 174426
[1] A. Kitaev, Ann. Phys. 321, 2(2006). (2011).
[2] A. Kitaev, Ann. Phys. 303, 2 (2003). [15] S. Bhattacharjee, S-S. Lee, and Y. B. Kim, arXiv:
[3] L-M. Duan, E. Demler, and M. D. Lukin, Phys. Rev. 1108.1806 (2011).
Lett. 91, 090402 (2003). [16] Y-Z. You, I. Kimchi, and A. Vishwanath, Arxiv
[4] J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev. 1109.4155 (2011) .
Lett. 105, 027204 (2010). [17] T. Hyart, A. R. Wright, G. Khaliullin, and B. Rosenow,
[5] G. Jackeli, and G. Khaliullin, Phys. Rev. Lett. 102 Arxiv 1109.6681 (2011).
017205 (2009). [18] G.Khaliullin,Prog.Theor.Phys.Suppl.16,155(2005).
[6] X. Liu, T. Berlijn, W.-G. Yin, W. Ku, A. Tsvelik, Y-J. [19] Z. Y. Meng, T. C. Lang, S. Wessel, F. F. Assaad and A.
Kim, H. Gretarsson, Y. Singh, P. Gegenwart, and J. P. Muramats, Nature 464, 847 (2010).
Hill1, Phys. Rev. B 83, 220403(R) (2011). [20] Z.ZouandP.W.Anderson,Phys.Rev.B37,627(1988).
[7] Y. Singh, S. Manni, J. Reuther, T. Berlijn, R. Thomale, [21] P. A. Lee, N. Nagaosa, and X-G. Wen, Rev. Mod. Phys.
W. Ku, S. Trebst, and P. Gegenwart, arXiv:1106.0429v2 78, 17 (2006).
(2011). [22] A. Vaezi and X-G. Wen, arXiv: 1010. 5744 (2010).
[8] S. K. Choi et al, arXiv: 1202.1268, to appear in Phys. [23] S-L. Yu, X. C. Xie, and J-X. Li, Phys. Rev. Lett. 107,
Rev. Lett. 010401 (2011).
[9] F. Ye, S. X. Chi, H. B. Cao, B. C. Chakoumakos, J. A. [24] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802
Fernandez-Baca,R.Custelcean,T.F.Qi,O.B.Korneta, (2005); 95, 226801 (2005).
and G. Cao, arXiv:1202.3995. [25] G. Baskaran, S. Mandal, and R. Shankar, Phys. Rev.
[10] F. Trousselet, G. Khaliullin, and P. Horsch, Phys. Rev. Lett. 98, 247201 (2007).
B 84, 054409 (2011). [26] WedonotinvestigatetheoriginofthesetwoFMphases
[11] J.Reuther,T.Thomale,andS.Trebst,Phys.Rev.B84 in this work.
100406 (2011). [27] L. Liang, S. J. Qin, and Y. Yu, in preparation.
[12] H.-C. Jiang, Z.-C. Gu, X.-L. Qi, and S. Trebst, Phys.