Nearly flat band with Chern number C=2 on the dice lattice
aa r X i v : . [ c ond - m a t . s t r- e l ] M a r Nearly flat band with Chern number C = 2 on the dice lattice Fa Wang and Ying Ran Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA (Dated: March 21, 2012)We point out the possibility of a nearly flat band with Chern number C = 2 on the dice lattice in asimple nearest-neighbor tightbinding model. This lattice can be naturally formed by three adjacent(111) layers of cubic lattice, which may be realized in certain thin films or artificial heterostructures,such as the SrTiO /SrIrO /SrTiO trilayer heterostructure grown along the (111) direction. Theflatness of two bands is protected by the bipartite nature of the lattice. Including the Rashba spin-orbit coupling on nearest-neighbor bonds separate the flat bands from the others but maintains theirflatness. Repulsive interaction will drive spontaneous ferromagnetism on the Kramer pair of the flatbands and split them into two nearly flat bands with Chern number C = ±
2. We thus propose thatthis may be a route to the quantum anomalous Hall effect and further conjecture that the partialfilling of the C = 2 band may realize exotic fractional quantum Hall effects. PACS numbers: 71.10.Fd,73.43.Cd,73.20.At
A few years after the experimental discovery of inte-ger quantum hall effect(IQHE) , Haldane wrote downa tight-binding model on the honeycomb lattice withIQHE , explicitly showing that the essence of IQHE is not the external magnetic field. However, it takes morethan two decades for people to show that the similarstatement is also true for fractional quantum Hall ef-fect(FQHE). Recently several groups have proposed torealize FQHE without Landau levels . The basic ideais to engineer a nearly flat band in two dimensions(2D)with nonzero Chern number. Electron interaction in thispartially filled band may realize fractional quantum Halleffect, as suggested by exact diagonalization studies .In these proposals nearly flat bands are obtained byfine-tuning ratios between nearest-neighbor(NN), next-nearest-neighbor(NNN), and even further neighbor tight-binding parameters. In this paper we point out a routeto get completely flat bands without this fine-tuning byemploying a bipartite lattice with unequal number of twosubsets of sites . As a concrete example we consider thedice lattice as shown in Fig. 1. It is bipartite with unequalnumber of two subsets of sites (the coordination-number-3 sites are twice as many as the coordination-number-6sites). This system is inversion symmetric with respectto the coordination-number-6 sites. We consider a single s -orbital with spin-1/2 degrees of freedom(DOF) on ev-ery site, and mainly focus on systems close to half-filling,i.e., one electron per site. The NN tight-binding model,including the Rashba-type spin-orbit coupling(SOC) con-sistent with lattice symmetry, will produce two com-pletely flat bands separated from the other bands. Be-cause the two flat bands are half-filled, ferromagnetism isa natural consequence of correlation , which gives riseto a Zeeman field on the mean-field level. We demon-strate the spontaneous ferromagnetism by a variationalwave function study of Hubbard interactions.As a nice feature of the current model system, evena small Zeeman field can split this Kramer pair of flatbands and produce two separated nearly flat bands with Chern number C = ±
2. Filling one of them will then pro-duce quantized anomalous Hall(QAH) effect with σ xy =2 e h . This Zeeman field could also be extrinsic, e.g. grow-ing the system on a ferromagnetic substrate. Note thatin a usual ferromagnetic system, a realistic Zeeman split-ting would not completely separate the two bands withopposite spin polarizations, and a ferromagnetic metal results. This is partially why the QAH insulator, whichneeds to be a ferromagnetic insulator, has not been re-alized experimentally so far. The main advantage of thepresently studied system is the existence of the half-filledflat bands, which natually support well-separated bandsby a realistic Zeeman splitting. Material realizations:
This model hamiltonian may ac-tually be relevant to some real systems. Heterostructuresof transition metal oxide(TMO) perovskites, whose crys-tal structures are cubic, are becoming available owing tothe recent development in the fields of oxide super-lattices and oxide electronics(for a review, see ). In par-ticular, layered structures of TMO heterstructures cannow be prepared with atomic precision, thus offering ahigh degree of control over important material properties,such as lattice constant, carrier concentration, spin-orbitcoupling, and correlation strength.TMO heterostructures grown along the (111) directionhave been synthesized experimentally (e.g., Ref. ).Recently it was pointed out that TMO (111) bilayer het-erostructures are promising candidates hosting varioustopological phases of matter . The dice lattice here canbe formed by three adjacent (111) layers of cubic lattice,each of which is a triangular lattice (Fig. 1). Althoughwe considered only simple s -orbital on every site here,the result should be valid if the active orbital is a onedimensional representation of the D d group. Some ex-amples are the p Z -orbital ( p x + p y + p z ), and the a orbital ( d yz + d zx + d xy ) of d -electrons under cubic andtrigonal crystal potential.A particularly relevant example is the transition metaloxide SrTiO /SrIrO /SrTiO trilayer heterostructure.Note that although the crystal structure of the bulkSrIrO is a monoclinic distortion of the hexagonalBaTiO structure , thin films of perovskite SrIrO havebeen synthesized on substrates , which are reported tobe metallic . This indicates that an itinerant electronicmodel could be a good starting point of describing theSrTiO /SrIrO /SrTiO trilayer heterostructure. Due tothe strong spin-orbit coupling on the Ir ion, togetherwith the octahedral crystal field, the active orbital is ahalf-filled effective J eff = 1 / . The explictlyform of these doublet in the presence of cubic symmetryis | J z = 1 / i = √ (+ i | xy, ↑i − | xz, ↓i + i | yz, ↓i ), and | J z = − / i is its time-reversal partner. These half-filled orbitals hop around the dice lattice, and contributeto states close to the fermi level. indicating the corre-lation in the bulk system is intermediate. In a (111)heterostructure, cubic symmetry is reduced to trigonalsymmetry. Nevertheless, to the leading order with re-spect to trigonal distortion, the nearest neighbor hop-pings between these J = 1 / s -orbitals, which form the model hamil-tonian considered in this paper.Therefore we think that our proposal is a promisingroute to realize QAH effect. In the same spirit of previousworks on the FQHE without Landau levels, we conjecturethat fractional filling of these bands might produce exoticfractional quantum hall (FQH) states. The nature ofthese FQH states remains unclear and we leave it as asubject of future research. But it is worth pointing outthat, in a nearly flat band with Chern number C = 2, thenatural candidate ground states for ν = 1 /m ( m is oddinteger) filling fractions are non-abelian states describedby SU ( m ) Chern-Simons effective theory . NN model without SOC.
The dice lattice and coordi-nation system are defined in Fig. 1. Label the three sub-lattices by V , , respectively. Consider a single s -orbitalwith spin-1/2 DOF on every site. As a warmup considerNN spin-independent hopping only, H = − X
FIG. 1: (Color online) Top: The dice lattice. Small up-ward triangles (bottom layer), downward triangles (top layer),and hexagons (middle layer) indicate the three sublattices.1 , , e and e indicate the two trans-lations of the dice lattice. Thick green arrows labelled as D ij indicate the Rashba SOC directions on those bonds ij , withcoordination-number-6 site j . Red dotted arrows with thelabels + x, + y, + z indicate of the projection of cubic latticeaxis. Captial X, Y, Z are axis for spin space in Rashba SOC. Z is the original (111) direction. Bottom: Perspective view ofthree adjacent (111) layers of cubic lattice. The middle layerhas different color for easy recognition. The top view of thistri-layer is the dice lattice. The middle band is completely flat as required bythe bipartiteness. However the top band touches theflat bands quadratically at the Brillouin zone corners ± K = ± ( k = 4 π/ , k = 2 π/ or certain other models with flat bands .The effective two-band hamiltonian at the band touchingpoint ± K is 3 t ǫ (cid:18) | δ k | δk ± δk ∓ | δ k | (cid:19) + O ( δ k ) , (3)where δ k = k ∓ K , δk + = e i π/ ( δk X + i δk Y ) and δk − =( δk + ) ∗ .The flat band has Bloch wavefunction ( γ ∗ k , − γ k ,
0) onthe three sublattices. It has local Wannier functions re-siding on the six neighbors of a coordination-number-6 site (sublattice-3) with opposite amplitudes betweensublattice-1 and sublattice-2.
NN model with Rashba SOC.
Rashba SOC induced byelectric fields can be included as H , SOC = H − X
00 0 0 0 0 γ k − γ k + − γ k − − γ ∗ k + − γ k + − γ ∗ k − , (5)where γ k ± = 1 + e i ( k ± π/ + e i ( k ± π/ , and the ba-sis is ( c k ↑ , c k ↓ , c k ↑ , c k ↓ , c k ↑ , c k ↓ ). It has three dou-bly degenerate bands with dispersions E = − ǫ/ − p ǫ / t | γ k | + λ ( | γ k , − | + | γ k , + | ), E = 0, E = − ǫ/ p ǫ / t | γ k | + λ ( | γ k , − | + | γ k , + | ). If λ ≪ ǫ , the effective four-band hamiltonian at the originalband touching point ± K is3 t ǫ (cid:18) | δ k | δk ± δk ∓ | δ k | (cid:19) ⊗ × + 3 √ λ t ǫ × ⊗ (cid:18) k ∓ k ± (cid:19) + 9 λ ǫ [ × ± τ z ⊗ σ z ] + O ( λ δ k ) + O ( δ k ) , (6)where Pauli matrix τ z acts on the sublattices-1 , i π ( e ,0) (− ,0) ( e ,0) π ( e ,0) π i π ( e ,0)(i,e ) λ π (1,i ) λ π (e ,i ) λ π −2i3 π (e ,e ) λ −2i3 π (−i,e ) λ λ (e ,e ) π π (− ,0) −2i3 π (e ,e ) λ π (1,i ) λ −2i3 (e ,i ) λ π π (e ,i ) λ λ (e ,e ) π π π −2i3 π (e ,e ) λ (i,e ) λ π −2i3 π (−i,e ) λ (e ,e ) λ −5i6 π π −2i3 (e ,i ) λ π (e ,e ) λ −5i6 π π −2i3 π (e ,e ) λ π FIG. 3: (Color online) Local Wannier function of one of theflat bands of NN model on dice lattice (black dotted lines)with Rashba SOC λ ( t = 1 for simplicity). The two compo-nent vector (blue) on each coordination-number-3 site indi-cates spin-up and spin-down amplitudes on that site. Am-plitudes on coordination-number-6 sites vanish, therefore theparameter ǫ has no effect. The small hexagon is the “guidingcenter” of this Wannier function. The mass term τ z ⊗ σ z has opposite sign between the twoband touching points ± K , similar to the Haldane model .The dispersions are illustrated in Fig. 2(b). There arestill two completely flat band dictated by the hamiltonianstructure.The flat bands have Bloch wavefunctions( t γ ∗ k Γ k , i λ γ ∗ k + Γ ∗ k , − t γ k Γ k , − i λ γ k − Γ ∗ k , ,
0) and itsKramer pair, where Γ k = γ k γ ∗ k + − γ ∗ k γ k − . Thereforethese flat bands also have local Wannier functions. Oneof the Wannier functions is illustrated in Fig. 3. Theother Wannier functions can be produced by translationand time-reversal. Note that the spin-up componentof the illustrated Wannier function acquires a phase2 π/ C = 2. Nearly flat band with Chern number C = 2 . The pre-vious double-degeneracy of the flat bands is protected bytime-reversal symmetry. Consider magnetic field effect, H , SOC+ B = H , SOC − g X i B i · S i (7)where S i = (1 / P α,β c † iα σ αβ c iβ is electron spin, theBohr magneton µ B is omitted, and g = 2 ( g = − J eff = 1 / ion ) is assumed hereafter.The field on sublattice-1 , B i = 0 . t along (111) direc-tion ( Z direction) in Fig. 2(c). As expected the Kramerpair of flat bands split into two nearly flat bands. Directcomputation of Chern numbers shows that they carryChern number C = ± C = ± λ for the mo-ment. Note that the sublattice-1,2 form a honeycomblattice by themselves. We could turn on another artificial iλ σ · (1 , ,
1) spin-orbit coupling between the second-neighbors on these two sublattices only, with the samesigns as the Kane-Mele model . In this λ -only model,spin rotation along (111) direction is conserved so thatwe could consider each spin-poloarized subsystem sepa-rately. Energy gaps at K and − K , the quadratic band-touching points, in Fig. 2(a) are opened by λ . How-ever, it is well-known that the spin-orbit energy gap ata quadratic band-touching point transfers Chern numberone between the two bands. Therefore, two spin-orbitgaps at K and − K transfer Chern number two insteadof one as in the Kane-Mele case. Including a Zeeman field B along (111) split the C = ± λ -only model [see Fig. 2(c)]. Fix-ing a Zeeman field B , it turns out that one can adiabat-ically connect the λ -only model with the λ -only modelby interpolation while keeping all the six bands isolatedfrom one another. This adiabatic evolution perserves theChern numbers of each bands. We thus prove the Chernnumbers in Fig. 2(c). Spontaneous ferromagnetism.
The flat band is half-filled if the entire system is at half-filling. Add onsiteHubbard interactions in the hamiltonian, H int = H , SOC + X i U n i ↑ n i ↓ . (8)If SOC λ = 0, by Lieb’s theorem the ground state is fer-romagnetic with total spin S = (1 / N + N ) − N ] =(1 / N cell ( N , , is the number of sites on sublattice-1 , , N cell ). With Rashba λ there is no known proof of ferro-magnetism. We use a variational (mean field) treatmentof this problem.The ferromagnetic “mean field” hamiltonian is just thefree fermion hamiltonian with magnetic field H , SOC+ B .By inversion symmetry we assume field on sublattices-1 , B = B , but may be different fromthat on sublattice-3, B . The variational wavefunctionis the free fermion wavefunction by half-filling this meanfield hamiltonian. We then evaluate the energy expecta-tion value of the Hubbard model H int and try to min-imize it with respect to the variational parameters B and B . From preliminary numerical results, the systemis unstable to spontaneous ferromagnetism for infinites-mal repulsive U , consistent with the Stoner criterion ,however the energy gain is very insensitive to the fielddirections. For ǫ = 0 . t , λ = 0 . t , U = t , and the fielddirections along (111) ( Z direction), the field strength is B = 0 . t on sublattices-1 , B = − . t onsublattice-3. The mean field band structure is very sim- -0.6-0.4-0.2 0 0.2 0.4 0.6 Γ K M Γ E /t -0.6-0.4-0.2 0 0.2 0.4 0.6 -1 -0.5 0 0.5 1 E /t k / π FIG. 4: (Color online) Left: The two nearly flat bands (red)with Chern numbers C = ±
2. Parameters are ǫ = 0 . t , λ =0 . t , B = 0 . t , and B = − . t . Right: Dispersionof a cylinder with 32 unit cell open boundary condition along e and periodic boundary condition along e , showing theedge states between the nearly flat bands. ilar to Fig. 2(c) where a uniform B = 0 . t is used. Thetwo nearly flat mean field bands are drawn in Fig. 4. Thethree occupied mean field bands have total Chern num-ber C = 2 and exhibit anomalous quantum Hall effect.The edge state on a cylindrical geometry is also shownin Fig. 4. Conclusion.
In this paper we discuss a modelwith spin-orbit coupling on the dice-lattice and thecorrelation physics in it. A transition metal oxideSrTiO3/SrIrO3/SrTiO3 trilayer heterostructure grownalong the (111) direction, where this model may be re-alized, is proposed. In this system, two degenerate flatbands at half-filling are found. Stoner’s instability nat-urally leads to ferromagnetism and split the two bands,which gives rise two nearly flat bands with Chern number ±
2. This indicate a promising route to realize QAHE. Wefurther speculate that further doping into the nearly flatChern bands could lead to FQHE without an externalmagnetic field. We hope these results could encourageexperimental syntheses and characterization of the ma-terial proposed here, as well as future theoretical inves-tigations on the nature of the possible FQH states.FW thanks the Institute for Advanced Study at Ts-inghua University for hospitality where part of this workwas finished. YR is supported by the startup fund atBoston College. K. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. , 494 (1980). F. D. M. Haldane, Phys. Rev. Lett. , 2015 (1988). Evelyn Tang, Jia-Wei Mei, and Xiao-Gang Wen, Phys. Rev. Lett. , 236802 (2011). Kai Sun, Zhengcheng Gu, Hosho Katsura, and S. DasSarma, Phys. Rev. Lett. 106, 236803 (2011). Titus Neupert, Luiz Santos, Claudio Chamon, and
Christopher Mudry, Phys. Rev. Lett. , 236804 (2011). D.N. Sheng, Z.C. Gu, K. Sun, and L. Sheng, Nature Com-mun. , 389 (2011). X. Hu, M. Kargarian, G. A. Fiete, Phys. Rev. B , 155116(2011). N. Regnault, B. A. Bernevig, arXiv:1105.4867 (unpub-lished). E. H. Lieb, Phys. Rev. Lett. , 1201 (1989). A. Mielke, J. Phys. A , 3311 (1991). H. Tasaki, Phys. Rev. Lett. , 1608 (1992). E. C. Stoner, Phil. Mag. , 1018 (1933). M. Izumi, Y. Ogimoto, Y. Konishi, T. Manako,M. Kawasaki, and Y. Tokura. Materials Science and Engi-neering B, , 53 (2001). A. Ohtomo, D. A. Muller, J. L. Grazul, and H. Y. Hwang.Nature, , 378 (2002). A. Ohtomo, and H. Y. Hwang. Nature, , 423 (2004). J. Mannhart, and D. G. Schlom. Science, , 1607 (2010). S. Chakraverty, A. Ohtomo, and M. Kawasaki. AppliedPhysics Letters, , 243107 (2010). Benjamin Gray, Ho Nyung Lee, Jian Liu, J. Chakhalian,and J. W. Freeland. Applied Physics Letters, , 013105(2010). Di Xiao, Wenguang Zhu, Ying Ran, Naoto Nagaosa, andSatoshi Okamoto, arXiv:1106.4296 (unpublished). J. M. Longo, J. A. Kafalas, and R. J. Arnott, J. Solid StateChem. , 174 (1971) S. J. Moon, H. Jin, K.W. Kim, W. S. Choi, Y. S. Lee,J. Yu, G. Cao, A. Sumi, H. Funakubo, C. Bernhard, andT.W. Noh, Phys. Rev. Lett. , 226402 (2008). Y.K. Kim, A. Sumi, K. Takahashi, S. Yokoyama, S. Ito, T.Watanabe, K. Akiyama, S. Kaneko, K. Saito, and H. Fu-nakubo, Jpn. J. Appl. Phys. , L36 (2005); A. Sumi, Y.K.Kim, N. Oshima, K. Akiyama, K. Saito, H. Funakubo,Thin Solid Films , 182 (2005). Yuan-Ming Lu, and Ying Ran, arXiv:1109.0226 (unpub-lished). E. McCann, and V. I. Falko, Phys. Rev. Lett. , 086805(2006). D. L. Bergman, C. Wu, and L. Balents, Phys. Rev. B ,125104 (2008). B. Bleaney, and M. C. M. O’Brien, Proc. Phys. Soc. B ,1216 (1956). C. L. Kane, and E. J. Mele, Phys. Rev. Lett.95