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Topological Hubbard model and its high-temperature quantum Hall effect Titus Neupert,1 Luiz Santos,2 Shinsei Ryu,3 Claudio Chamon,4 and Christopher Mudry1 1Condensed Matter Theory Group, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland 2Department of Physics, Harvard University, 17 Oxford Street, Cambridge, Massachusetts 02138, USA 3Department of Physics, University of Illinois at Urbana-Champaign, 1110 W. Green Street, Urbana, Illinois 61801-3080, USA 4Physics Department, Boston University, Boston, Massachusetts 02215, USA (Dated: January 26, 2012) The quintessential two-dimensional lattice model that describes the competition between the kinetic energy of electrons and their short-range repulsive interactions is the repulsive Hubbard model. We study a time-reversal symmetric variant of the repulsive Hubbard model defined on a planarlattice: Whereastheinteractionisunchanged,anyfullyoccupiedbandsupportsaquantized 2 spin Hall effect. We show that at 1/2 filling of this band, the ground state develops spontaneously 1 andsimultaneouslyIsingferromagneticlong-rangeorderand aquantizedchargeHalleffectwhenthe 0 2 interactionissufficientlystrong. Weponderonthepossiblepracticalapplications,beyondmetrology, that the quantized charge Hall effect might have if it could be realized at high temperatures and n without external magnetic fields in strongly correlated materials. a J 5 I. INTRODUCTION topologically nontrivial Bloch bands simultaneously dis- 2 playsIsingferromagneticlong-rangeorderand theIQHE ] High-temperature superconductivity1 and the quan- atsomecommensuratefillingfraction. Theenergyscales el tum Hall effect (QHE)2 have been two of the central thatcanbeattainedinlatticemodelsaretypicallyrather high,oftheorderofatomicmagnitudes,i.e.,electronvolt. - problems in condensed matter physics of the past three r If an interacting system with topological bands can be t decades. The former is related to electrons hopping on a s found so as to display the IQHE at high temperatures, two-dimensional (2D) lattice close to (but not at) half . t filling, while the latter focuses on fermions in doped it could be of practical use, as we shall explain after we a substantiate our claims. m semiconductor heterostructures or graphene in a high magnetic field. High-temperature superconductors are - d strongly interacting systems, with the potential energy n about an order of magnitude larger than the kinetic en- II. STUDY OF THE TOPOLOGICAL HUBBARD o ergy. In the QHE, the kinetic energy is quenched by the MODEL c external magnetic field. Moreover, interactions are im- [ portant only in understanding the fractional QHE but We consider spinful electrons hopping on a bipartite 2 not in understanding the integer QHE (IQHE). squarelatticeΛ=A∪B withsublatticesAandB,where v The possibility that the IQHE could arise in a lattice each sublattice has N := L ×L sites. The Hubbard 6 x y HamiltonianwithouttheLandaulevelsinducedbyauni- Hamiltonian with repulsive interactions (U > 0) can be 9 2 form magnetic field was suggested by Haldane in 19883. written 1 The essence is that, despite the absence of an uniform (cid:88) (cid:88) (cid:88) . magnetic field, the system still lacks time-reversal sym- H := c† H c +U n n . (2.1a) 0 k k k r,↑,α r,↓,α metry. More recently, it was shown that the fractional 1 k∈BZ r α=A,B 1 QHE could also emerge in flat topological bands when 1 they are partially filled4–9 (see also 10). These recent The component c† of the operator-valued spinor c† : developments point to a natural marriage between the k,σ,α k v createsanelectronwithmomentumk fromtheBrillouin QHE and strongly correlated lattice systems at a high i zone (BZ) of sublattice A and with spin σ =↑,↓, whose X filling fraction. Fourier transform c† = N−1/2(cid:80) e−ik·rc† is r In this Letter, we study a quintessential strongly cor- r,σ,α k∈BZ k,σ,α a exclusively supported on sublattice α=A,B. The 4×4 relatedlattice2Dsystembutwithatwist. Weconsidera Hermitian matrix H obeys the time-reversal symmetry time-reversalsymmetricfermionicHubbardmodelinthe k (TRS) limit of large on site repulsion U compared to the band- width W of the hopping dispersion, but with hopping H =σ H∗ σ , (2.1b) terms yielding topologically nontrivial Bloch bands in +k 2 −k 2 that they each support a quantized spin Hall conductiv- and, owing to a strong intrinsic spin-orbit coupling, the ity when fully occupied11. The time-reversal symmetric residual spin-rotation symmetry (RSRS) HubbardmodelwithasinglehalffillednestedBlochband has a charge insulating ground state that supports anti- (cid:32) (cid:33) ferromagneticlong-rangeorder1. Incontrast,theground H =σ H σ ≡ h(k↑) 0 , (2.1c) stateofourtime-reversalsymmetricHubbardmodelwith +k 3 +k 3 0 h(↓) k 2 where the Pauli matrices σ , σ , and σ act on the elec- exactly flat with eigenvalues ±1. The case κ = 0 cor- 1 2 3 tronic spin-1/2 degrees of freedom. Hence, the two 2×2 responds to a tight-binding model on the square lattice Hermitian matrices h(kσ) with σ =↑,↓ obey that involves only nearest-neighbor (|t1| = 1) and next- nearest-neighbor hopping (t ) together with a staggered 2 h(↑) =h(↓) , ∀k∈BZ, α,β =A,B, chemicalpotentialµs thatbreaksthesymmetrybetween +k,αβ −k,βα sublattices A and B4. For κ ∈ (0,1], longer-range hop- (2.1d) ping is introduced. However, we stress that the Hamil- because of the condition of TRS (2.1b). Finally, the op- tonian remains local for all κ∈[0,1] since all correlation eratorn =c† c measurestheelectrondensity r,σ,α r,σ,α r,σ,α functions decay exponentially due to the presence of the on site r in sublattice α and with spin σ. band gap4. The Hubbard Hamiltonian defined by Eq. (2.1a) thus The topological properties of the lower pair of bands has a global Z ×U(1) symmetry that arises because of 2 are characterized by their spin Chern number the TRS (2.1b) and the RSRS (2.1c). We are going to show that TRS is spontaneously broken while the con- C :=(cid:0)C −C (cid:1)/2, (2.3a) s ↑ ↓ tinuous RSRS is shared by the ground state, when this HubbardHamiltonianacquiressuitabletopologicalprop- where Cσ is to be computed from the orbitals of spin-σ erties. electrons according to It is the choice for the matrix elements h(kσ,α)β en- (cid:90) d2k (cid:16) (cid:17) tering the kinetic energy (2.1c) that endows the Hub- C := ∇ ∧ χ† ∇ χ . (2.3b) σ 2πi k k,σ,− k k,σ,− bard Hamiltonian (2.1a) with topological attributes. We k∈BZ choose Time-reversalsymmetryimpliesC =−C andtherefore ↑ ↓ (cid:104) (cid:16) (cid:17) entails a vanishing of the total (charge) Chern number h(k↑,)AB =h(k↑,)B∗A :=wk e−iπ/4 1+e+i(ky−kx) Cc := (C↑+C↓)/2 of the lower bands. The spin Chern +e+iπ/4(cid:0)e−ikx +e+iky(cid:1)(cid:105), number of the lower pair of bands is given by 1(cid:16) (cid:17) h(↑) =−h(↑) :=w (cid:2)2t (cos k −cos k )+4µ (cid:3), Cs = 2 sgnh((↑0,)π),AA−sgnh((↑π),0),AA . (2.3c) k,AA k,BB k 2 x y s (2.2a) Hence, the Bloch bands are topologically trivial when- ever |t /µ | < 1, while the model at half filling exhibits where 2 s the physics of a quantum spin Hall insulator whenever |t /µ | > 1. In an open geometry, the spin Hall conduc- w−1 :=κε +(1−κ), κ∈[0,1], (2.2b) 2 s k k tivity is quantized to the value σsH = eC /(2π) where e xy s denotes the electric charge of the electron. and WenowconsiderthesystemwitharepulsiveHubbard (cid:113) interaction U > 0 at 1/2 filling of the lower band (1/4 ε := 1+cosk cosk +[2t (cosk −cosk )+4µ ]2. k x y 2 x x s filling of the lower and upper bands), i.e., with (2.2c) In the noninteracting limit (U = 0), this model fea- N =L ×L =N (2.4) e x y tures four bands with two distinct two-fold degener- electrons. In all that follows, we assume that U is much ate dispersions ±w ε 4. This two-fold degeneracy is a k k smaller than the gap ∆ induced by a strong intrinsic consequence of the Kramers degeneracy implied by the 0 spin-orbit coupling between the two pairs of bands. If TRS(2.1b). Ifwedenotethecorrespondingeigenspinors so,wecanrestricttheN -bodyHilbertspacetotheFock χ =(χ ),whereλ=±,andchoosethenormal- e k,σ,λ k,σ,λ,α spacearisingfromthesingle-particleHilbertspacesofthe ization χ† χ =δ , ∀k, then the kinetic energy k,σ,λ k,σ,λ(cid:48) λ,λ(cid:48) lower pair of bands. is diagonalized by using the fermionic creation operators In the limit of flat bands κ = 1 and at the commen- surate filling fraction (2.4), the kinetic energy (2.2e) at (cid:88) d†k,σ,λ := χ∗k,σ,λ,αc†k,σ,α, (2.2d) fixed spin polarization S := |(cid:104)σ3(cid:105)| = 0, 2, ..., N in units α=A,B of (cid:126)/2 has a ground state degeneracy as (cid:18) (cid:19)2 N N = . (2.5) H := (cid:88) (cid:88) (cid:88)λd† w ε d . (2.2e) gs N−|S| 0 k,σ,λ k k k,σ,λ 2 k∈BZσ=↑,↓λ=± The repulsive Hubbard interaction lifts this degeneracy whenever any one of these states allows a site of Λ to be Hence, the Bloch states created by d† are generically k,σ,λ doubly occupied with a finite probability. The only two spread on both sublattices A and B. We shall consider states with full spin polarization S =N, only the case in which these bands are separated by an (cid:89) energy gap, i.e., |t2| (cid:54)= |µs|. The parameter κ controls |Ψσ(cid:105)= d†k,σ,−|0(cid:105), σ =↑,↓, (2.6a) the bandwidth of these bands. For κ=1, the bands are k∈BZ 3 are immune to the presence of the Hubbard repulsion. states with more than one spin flipped are higher in en- More formally, observe that Hamiltonian H −µN is a ergy than those with one spin flipped. e positive semidefinite operator for κ = 1, U > 0, and the Ruling out the possibility that states with many spin chemical potential µ=−1. Since flips have lower energies than states with few spin flips relative to the Ising ferromagnetic ground state is plau- (cid:104)Ψ |(H +N )|Ψ (cid:105)=0, σ =↑,↓, (2.6b) sible in the regime when the intrinsic spin-orbit coupling σ e σ generates the largest energy scale (∆ (cid:29)U). 0 thetwostates(2.6a)belongtothegroundstatemanifold Equation (2.8) is a reasonable assumption when the of H +Ne for any U >0, t2, and µs. Blochstatesstemfromabandwithnonzero(spin)Chern WearegoingtoarguethatthispairofdegenerateIsing number, since the spinor χ maps out the entire sur- k,σ ferromagnets spans the ground state manifold for any face of the unit sphere as k takes values in the BZ. The U >0 and |t /µ |=(cid:54) 1. This is achieved by arguing that assumption underlying Eq. (2.8) can also be understood 2 s they are separated from excited states by a many-body by constructing the Wannier wavefunctions, centered at gap, a departure from the usual ferromagnetism in flat the lattice point z, of the lowest energy band with spin bands when full spin-1/2 SU(2) symmetry is not explic- σ and Chern number: C σ itlybroken12. First,particle-holeexcitationsof|Ψ (cid:105)that σ keep S = N fixed, cost an energy ∆0 > 0 and are thus ψ := 1 (cid:88) eik·(r−z)χ . (2.10a) gapped. Second,weaskwhetherexcitationsof|Ψ (cid:105)that z,r,σ,−,α N k,σ,−,α σ k∈BZ flip one spin (S = N −2) are gapped as well. Any such state can be written as Thegauge-invariantpartoftheirspreadfunctional13 sat- isfies |Φ (cid:105)= (cid:88) A(Q)d† d |Ψ (cid:105), σ =↑,↓, σ,Q k k+Q,σ¯,− k,σ,− σ (cid:104)ψ |r2|ψ (cid:105)−(cid:88) |(cid:104)ψ |r|ψ (cid:105)|2 k∈BZ 0,σ,− 0,σ,− 0,σ,− z,σ,− (2.7a) z where the center of mass momentum Q is a good quan- ≥|C |A /(2π), σ c tum number and thus (cid:104)Φσ,Q|Φσ,Q(cid:48)(cid:105) = δQ,Q(cid:48) if the nor- (2.10b) malization (cid:80) A(Q)∗A(Q) = 1 is imposed. One veri- k∈BZ k k where A denotes the area of the unit cell. This inequal- fies that (see Appendix) c ity relates the Chern number of the band and the “min- imum width” of the Wannier states (see Appendix). In (cid:104)Φ |(H +N )|Φ (cid:105) σ,Q e σ,Q particular, in the nontopological phase, one can imagine (cid:12) (cid:12)2 =U − NU (cid:88)(cid:12)(cid:12)(cid:12) (cid:88) A(kQ)χ−k−Q,σ,−,αχk,σ,−,α(cid:12)(cid:12)(cid:12) , agilviemnitsuinbwlahtticiche,thwehwilaevtehfuenncotnioznerioseCnhtierrenlynluomcabliezredinotnhae (cid:12) (cid:12) α k∈BZ topologicalphaseimpliesthattheWannierwavefunction (2.7b) has amplitudes on both sublattices. While Eq. (2.9) is strongly suggestive of the existence where the lowest energy state with one spin flipped is of a many-body gap ∆, it does not provide informa- characterized by the A(kQ) that minimizes Eq. (2.7b) tion about its size. To quantify ∆, we diagonalized the while satisfying the normalization condition. For exam- model(2.1)exactlynumericallyinthelimitofflatbands ple, if the single-particle orbitals are fully sublattice po- κ = 1 at the commensurate filling fraction (2.4). We larized, e.g., χ† ∝ (1,0) (topologically trivial), the varied the ratio |t /µ |, keeping t2 +µ2 = 1/2 constant k,σ,− 2 s 2 s choice A(Q) = N−1/2 minimizes Eq. (2.7b) with the to drive the system from the topological to the trivial k right-hand side equal to zero. Hence, the fully spin- phase. The results are shown in Fig. 1. First, they sup- polarizedstate|Ψ (cid:105)isagapless groundstateinthiscase. port the assumption that all states with more than one σ On the other hand, let us assume that spin flipped are higher in energy than the many-body one-spin-flipped gap provided |t /µ | > 1. Second, we 2 s χ† (cid:54)∝(1,0) and χ† (cid:54)∝(0,1) (2.8) find ∆ ≈ 0.3U as an extrapolation to the thermody- k,σ,− k,σ,− namic limit for µ = 0 deep in the topological phase, s holds almost everywhere in the BZ, i.e., up to a set while ∆ monotonically decreases toward a much smaller of measure zero. In the thermodynamic limit, where nonvanishing value for t2 = 0 in the topologically triv- the sum over k becomes an integral, this delivers from ial phase set by the unit of energy |t1| = 1. Finally, Eq. (2.7b) the strict inequality (see Appendix) it should be noted that neglecting the states from the upper band of the noninteracting Hamiltonian delivers (cid:10) (cid:11) Φ |(H +N )|Φ >0. (2.9) the correct excitation many-body gap ∆ not only in the σ,Q e σ,Q aforementionedlimitU (cid:28)∆ ,butalsoundertheweaker 0 Hence, assumption (2.8) is sufficient to show that the condition ∆<∆ , if the limit of flat bands is taken. 0 spin-polarized state |Ψ (cid:105) is a gapped ground state of the Deep in the topologically nontrivial regime |t /µ | (cid:29) σ 2 s Hamiltonianwithflatbandsinthethermodynamiclimit, 1, the states |Ψ (cid:105) and |Ψ (cid:105) are degenerate ground states σ σ¯ provided that one also assumes that the lowest energy related by TRS for any finite N. 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Plotted is the energy x y bands κ = 1 at the commensurate filling fraction (2.4). oftheloweststateindifferentsectorsoftotalspinS (inunits Markers show the energy of the lowest state in different sec- of (cid:126)/2) measured with respect to the ground√state energy in tors of total spin S (in units of (cid:126)/2) measured with respect the topological phase with µ = 0, t = 1/ 2. The ground s 2 to the ground state energy for L = 3, L = 4. Here, stateisgappedandfullyspin-polarizedforW/U <0.7,while x y g := (2/π)arctan|µ /t | so that g > 0.5 and g < 0.5 cor- it is unpolarized for W/U >0.7 (see Appendix). s 2 respond to the trivial and topological single-particle bands, respectively. Since there is only one state in the fully po- larized sector |S| = 12, the difference between the asterisks III. PRACTICAL APPLICATIONS and the squares is the many-body excitation gap ∆(g). The thick blue line shows the extrapolation of ∆(g) to the ther- Sowhatisitgoodfor, amaterialwithaQHEatroom modynamic limit. In the inset, exact diagonalization in the sectorwithonespinflippedawayfromthefullypolarizedsec- temperaturewithoutappliedexternalmagneticfieldsbe- √ tor is presented for µ =0, t =1/ 2 and L =L ranging sides metrology14,15? First, we recall that the quantiza- s 2 x y from 6 to 30. The straight lines are a guide to the eye and tion of the Hall resistance and the accompanying van- make evident an even-odd effect in L = L . Deep in the ishing of the longitudinal resistance is exact only at zero x y topologically nontrivial regime g (cid:28)0.5, we observe a sizable temperature. Thelongitudinalresistanceincreasesexpo- ∆(g (cid:28) 0.5). The topologically trivial regime g > 0.5 is also nentiallyfastwithincreasingtemperature15. However,if characterizedbyagap∆(g>0.5)inthesectorwithonespin aQHEwithgapsoftheorderofhundredsofmeVoreven flippedawayfromthefullypolarizedsector,howeverthisgap eV scales could arise in a strongly correlated lattice ma- is much smaller than ∆(g(cid:28)0.5). We refer the reader to the terial, exceptionally low resistivities could be attained. Appendix for a discussion of the regime ∆(g>0.5). The resistance of a Hall bar depends on its aspect ratio and the Hall angle δ = arctan(ρ /ρ )16, but for long xy xx fromtheirexcitationsbyagapthatsurvivesthethermo- systems (“wires”) near the quantized regime, the lon- dynamic limit N → ∞. Spontaneous breaking of TRS gitudinal resistance scales as Rxx = L/W ρxx, and the takes place in the thermodynamic limit N → ∞ by se- 2D resistivity ρ ∼ R e−∆/T, where ∆ is the exci- xx K lectingthegroundstatetobe|Ψ (cid:105), say. Itisthenmean- tation gap. For gaps of the order of 100meV to 1eV, ↑ ingful to discuss the quantized electromagnetic response onewouldobtainroomtemperature2Dresistivitiesfrom of |Ψ (cid:105), since TRS is spontaneously broken. The trans- ρ ∼ 103Ω to ρ ∼ 10−13Ω, respectively. Obviously ↑ xx xx verse charge response σH of |Ψ (cid:105) is proportional to the the exponential behavior is responsible for this gigan- xy ↑ tic range. Small as they are, these are not perfect con- many-body Chern number C . The latter takes into |Ψ↑(cid:105) ductors. For a benchmark, we consider the conductivity account the occupation of the Bloch states4. Since all of copper at room temperature per atomic layer. Us- Bloch states of the lower band with spin σ are occupied ing the value for the 3D resistivity of copper at 20◦C of in |Ψ↑(cid:105), while all Bloch states with spin ↓ are empty, ρ3D =1.68×10−8Ωm 17 and that the lattice parameter C|Ψ (cid:105) ≡ C↑. Hence, the ground state has the quantized foCruFCClatticeis3.61˚A,weobtainρ2D =93.3Ω. There- ↑ Cu Hall response fore, for gaps above ∆ ≈ .2eV, the Hall system starts to be better conducting than copper at room tempera- |σH |=|C |×e2/h=e2/h. (2.11) xy ↑ ture, and for ∆ ≈ .3eV it is already almost 3 orders of Remarkably, the selection by the repulsive Hubbard magnitude better conducting than copper. interaction of a ground state supporting simultaneously TheRSRS(2.1c)isnotexactinpractice. Forexample, Isingferromagnetismand theIQHEisrobusttoasizable a Rashba spin-orbit coupling violates this RSRS. How- bandwidth as is suggested by numerical exact diagonal- ever, our analysis of transport at room temperature still ization. As shown in Fig. (2), the fully spin-polarized applies provided the characteristic energy scale associ- state |Ψ (cid:105) remains the gapped ground state of the sys- ated to the breaking of the RSRS (2.1c) is much smaller σ tem up to a bandwidth W/U ≈0.7. than the largest energy scale ∆ induced by the intrinsic 0 5 spin-orbitcoupling. Materialsthatrealizea2DZ topo- in the N -many-body state 2 e logical band insulator18,19 with a band gap ∆ are thus 0 candidates to realize a QHE at room temperature if (i) the band gap is larger than the correlation energy and (ii) the chemical potential can be tuned to half-filling of (iii)areasonablyflatvalenceband. HgTequantumwells |Φ (cid:105)= (cid:88) A(Q)d† d |Ψ (cid:105), σ =↑,↓, σ,Q k k+Q,σ¯,− k,σ,− σ with an inverted band structure realize a 2D Z topo- 2 k∈BZ logicalbandinsulatorwithasmallRashbacoupling20,21. (A1b) The design of a material with the functionalities (i)-(iii) which has one spin flipped as compared to the – up to has been proposed in Ref. 8 . Cold atoms trapped in an time-reversal symmetry and the global gauge phase fac- optical honeycomb lattice22,23 might offer an alternative tor – unique normalized N -many-body state with full e to realizing the topological Hubbard model discussed in spin polarization this Letter. We would like to close by mentioning that examples such as the topological Hubbard model discussed in this Letter, as well as lattice models displaying the fractional (cid:89) quantum Hall effect studied in Refs. 4–7, could serve as |Ψ (cid:105)= d† |0(cid:105), σ =↑,↓. (A1c) σ k,σ,− benchmarks for numerical methods of fermionic models k∈BZ in 2D such as dynamical mean-field theory and meth- ods based on tensor product states24. In contrast to the single-band repulsive Hubbard model, for which lit- tle is known exactly at fractional filling, the topological Here, Q denotes the center of mass momentum and the Hubbardmodel(2.1),becauseofthenonvanishingChern orthonormalization condition numbers of its bands, leads to much better understood (topological) ground states. It can thus serve as a yard- stick for the performance of these methods. This work was supported in part by DOE Grant No. DEFG02-06ER46316 and by the Swiss National Science (cid:104)Φ |Φ (cid:105)=δ (A1d) Foundation. σ,Q σ,Q(cid:48) Q,Q(cid:48) Appendix A: Intermediary steps enforces the normalization 1. Derivation of Eq. (2.7b) For the limit of flat bands κ = 1, we are going to compute the expectation value (cid:88) A(Q)∗A(Q) =1. (A1e) k k (cid:104)Φ |(H +N )|Φ (cid:105) (A1a) σ,Q e σ,Q k∈BZ We rewrite |Φ (cid:105) as σ,Q    |Φσ,Q(cid:105)= (cid:88) A(kQ) (cid:88) χ∗k+Q,σ¯,−,αc†k+Q,σ¯,α (cid:88) χk,σ,−,βck,σ,β|Ψσ(cid:105) k∈BZ α=A,B β=A,B (cid:34) (cid:35) = (cid:88) (cid:88) (cid:88) A(kQ)χ∗ χ e−i(k+Q)·re+ik·r(cid:48) c† c |Ψ (cid:105) (A2a) N k+Q,σ¯,−,α k,σ,−,α r,σ¯,α r(cid:48),σ,β σ α,β=A,Br,r(cid:48)∈A k∈BZ ≡ (cid:88) (cid:88) M(σ) c† c |Ψ (cid:105), r,α,r(cid:48),β r,σ¯,α r(cid:48),σ,β σ α,β=A,Br,r(cid:48)∈A where we have introduced the short-hand notation M(σ) := (cid:88) A(kQ)χ∗ χ e−i(k+Q)·re+ik·r(cid:48). (A2b) r,α,r(cid:48),β N k+Q,σ¯,−,α k,σ,−,β k∈BZ 6 in terms of which (cid:88) (cid:88) (cid:104)Φ |(H +N )|Φ (cid:105)=U (cid:104)Φ |n n |Φ (cid:105) ↑,Q e ↑,Q ↑,Q r˜,↑,δ r˜,↓,δ ↑,Q r˜∈Aδ=A,B =U (cid:88) (cid:88) (cid:12)(cid:12)M(↑) (cid:12)(cid:12)2(cid:104)Ψ |c† c n n c† c |Ψ (cid:105) (cid:12) r,α,r(cid:48),β(cid:12) ↑ r(cid:48),↑,β r,↓,α r˜,↑,δ r˜,↓,δ r,↓,α r(cid:48),↑,β ↑ r˜,r,r(cid:48)∈Aα,β,δ=A,B (A3) =U (cid:88) (cid:88) (cid:12)(cid:12)M(↑) (cid:12)(cid:12)2(1−δ δ )δ δ (cid:12) r,α,r(cid:48),β(cid:12) r,r(cid:48) α,β r˜,r δ,α r˜,r,r(cid:48)∈Aα,β,δ=A,B =U (cid:88) (cid:88) (cid:12)(cid:12)M(↑) (cid:12)(cid:12)2−U (cid:88) (cid:88) (cid:12)(cid:12)M(↑) (cid:12)(cid:12)2. (cid:12) r,α,r(cid:48),β(cid:12) (cid:12) r,α,r,α(cid:12) r,r(cid:48)∈Aα,β=A,B r∈Aα=A,B Using the definition (A2b), the first term becomes (cid:88) (cid:88) (cid:12)(cid:12)M(↑) (cid:12)(cid:12)2 = (cid:88) (cid:88) (cid:88) A(kQ)A(kQ(cid:48))∗χ∗ χ χ χ∗ e−i(k−k(cid:48))·(r−r(cid:48)) (cid:12) r,α,r(cid:48),β(cid:12) N2 k+Q,↓,−,β k,↑,−,α k(cid:48)+Q,↓,−,β k(cid:48),↑,−,α r,r(cid:48)∈Aα,β=A,B r,r(cid:48)∈Aα,β=A,Bk,k(cid:48)∈BZ (cid:88) (cid:88) A(Q)A(Q)∗ = k k(cid:48) χ∗ χ χ χ∗ N2δ N2 k+Q,↓,−,β k,↑,−,α k(cid:48)+Q,↓,−,β k(cid:48),↑,−,α k,k(cid:48) α,β=A,Bk,k(cid:48)∈BZ    = (cid:88) A(kQ)A(kQ)∗ (cid:88) χ∗k+Q,↓,−,βχk+Q,↓,−,β (cid:88) χk,↑,−,αχ∗k,↑,−,α k,∈BZ β=A,B α=A,B = (cid:88) A(Q)A(Q)∗ k k k,∈BZ =1, (A4) where both the normalization conditions on A(Q) in Eq. (A1e) and on χ have been used in the last two lines. k k,σ,−,α The second term in Eq. (A3) becomes (cid:88) (cid:88) (cid:12) (cid:12)2 (cid:88) (cid:88) (cid:88) A(Q)A(Q)∗ (cid:12)M(↑) (cid:12) = k k(cid:48) χ∗ χ χ χ∗ (cid:12) r,α,r,α(cid:12) N2 k+Q,↓,−,α k,↑,−,α k(cid:48)+Q,↓,−,α k(cid:48),↑,−,α r∈Aα=A,B r∈Aα=A,Bk,k(cid:48)∈BZ (cid:88) (cid:88) A(Q)A(Q)∗ = k k(cid:48) χ∗ χ χ χ∗ N k+Q,↓,−,α k,↑,−,α k(cid:48)+Q,↓,−,α k(cid:48),↑,−,α α=A,Bk,k(cid:48)∈BZ (A5) (cid:12) (cid:12)2 = 1 (cid:88) (cid:12)(cid:12) (cid:88) A(Q)χ∗ χ (cid:12)(cid:12) N (cid:12) k k+Q,↓,−,α k,↑,−,α(cid:12) (cid:12) (cid:12) α=A,B k∈BZ (cid:12) (cid:12)2 = 1 (cid:88) (cid:12)(cid:12) (cid:88) A(Q)χ χ (cid:12)(cid:12) , N (cid:12) k −k−Q,↑,−,α k,↑,−,α(cid:12) (cid:12) (cid:12) α=A,B k∈BZ where we used the identity χ = χ∗ which follows from the time-reversal symmetry of the Hamiltonian. k,σ,λ,α −k,σ¯,λ,α Putting everything together, we obtain Eq. (2.7b) of the Letter, (cid:12) (cid:12)2 (cid:104)Φ |(H +N )|Φ (cid:105)=U − U (cid:88) (cid:12)(cid:12) (cid:88) A(Q)χ χ (cid:12)(cid:12) . (A6) σ,Q e σ,Q N (cid:12) k −k−Q,σ,−,α k,σ,−,α(cid:12) (cid:12) (cid:12) α=A,B k∈BZ 2. Derivation of Eq. (2.9) fora,b,c,d∈[0,1]. Inparticular,theequalityonlyholds when either a = b = c = d = 0 or a = b = c = d = 1. Before starting with the derivation of the inequality To show this, one can rewrite the inequality in terms of Eq. (2.9), let us establish the following inequality (cid:112) (cid:112) (cid:112) (cid:112) 1≥abcd+ 1−a2 1−b2 1−c2 1−d2 (A7) 7 trigonometric functions with angles α,β,γ,δ ∈[0,π/2] strict inequality 1≥ sin α sin β sin γ sin δ +cos α cos β cos γ cos δ (cid:12) (cid:12)2 =1[cos(α+β)cos(γ+δ)+cos(α−β)cos(γ−δ)]. U − U (cid:88) (cid:12)(cid:12) (cid:88) A(Q)χ χ (cid:12)(cid:12) >0 2 N (cid:12) k −k−Q,σ,−,α k,σ,−,α(cid:12) (cid:12) (cid:12) (A8) α=A,B k∈BZ (A9) holds under the assumption that Eq. (2.8) is satisfied Hence the inequality holds and the equality is true only almost everywhere in the BZ, i.e., up to a set of measure if either α = β = γ = δ = 0 or α = β = γ = δ = π/2 as zero. announced. Wewillnowshowthatinthethermodynamiclimitthe We rewrite, for very large N, (cid:12) (cid:12)2 U (cid:88) (cid:12)(cid:12) (cid:88) A(Q)χ χ (cid:12)(cid:12) N (cid:12) k −k−Q,σ,−,α k,σ,−,α(cid:12) (cid:12) (cid:12) α=A,B k∈BZ →NU(cid:90) d2k (cid:90) d2k(cid:48) A(Q)A(Q)∗ (cid:88) χ χ χ∗ χ∗ (2π)2 (2π)2 k k(cid:48) −k−Q,σ,−,α k,σ,−,α −k(cid:48)−Q,σ,−,α k(cid:48),σ,−,α α=A,B ≤NU(cid:90) (d2π2k)2 (cid:90) (d22πk)(cid:48)2 (cid:12)(cid:12)(cid:12)A(kQ)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)A(kQ(cid:48))∗(cid:12)(cid:12)(cid:12) (cid:88) (cid:12)(cid:12)χ−k−Q,σ,−,α(cid:12)(cid:12)(cid:12)(cid:12)χk,σ,−,α(cid:12)(cid:12)(cid:12)(cid:12)χ∗−k(cid:48)−Q,σ,−,α(cid:12)(cid:12)(cid:12)(cid:12)χ∗k(cid:48),σ,−,α(cid:12)(cid:12) (A10) α=A,B (cid:90) d2k (cid:90) d2k(cid:48) (cid:12) (cid:12)(cid:12) (cid:12) <NU (cid:12)A(Q)(cid:12)(cid:12)A(Q)∗(cid:12) (2π)2 (2π)2 (cid:12) k (cid:12)(cid:12) k(cid:48) (cid:12) (cid:90) d2k (cid:12) (cid:12)2 ≤NU (cid:12)A(Q)(cid:12) (2π)2 (cid:12) k (cid:12) =U. Due to the normalization of the eigenspinors, the assumption (2.8), and the inequality (A7), (cid:88) (cid:12)(cid:12)χ−k−Q,σ,−,α(cid:12)(cid:12)(cid:12)(cid:12)χk,σ,−,α(cid:12)(cid:12)(cid:12)(cid:12)χ∗−k(cid:48)−Q,σ,−,α(cid:12)(cid:12)(cid:12)(cid:12)χ∗k(cid:48),σ,−,α(cid:12)(cid:12)<1 (A11) α=A,B holds for almost all k, k(cid:48). This allows the use of the strict inequality in the third-last inequality from the chain of inequalities (A10). The penultimate inequality from the chain of inequal- is non-vanishing under the assumption (2.8). ities (A10) is a consequence of H¨older’s inequality (cid:90) d2k (cid:12) (cid:12) (cid:18)(cid:90) d2k (cid:12) (cid:12)2(cid:19)1/2 (cid:12)A(Q)(cid:12)≤ (cid:12)A(Q)(cid:12) . (A12) 3. Proof of Eq. (2.10b) (2π)2 (cid:12) k (cid:12) (2π)2 (cid:12) k (cid:12) The last inequality in the chain of inequalities (A10) fol- We are now going to work exclusively with the single- lows from the representation particle eigenstates of the band Hamiltonian (2.2e). We start with the completeness relation (cid:90) d2k (cid:12) (cid:12)2 1 (cid:12)A(Q)(cid:12) = (A13) (2π)2 (cid:12) k (cid:12) N (cid:88) (cid:88) |r,α(cid:105)(cid:104)α,r|=11 (A16a) of the normalization (A1e) in the thermodynamic limit. r∈Aα=A,B In summary, we have shown that onthebipartitelatticeΛ=A∪B. ThenormalizedBloch (cid:10) (cid:11) Φ |(H +N )|Φ >0 (A14) states |ϕ (cid:105) are defined in terms of the single-particle σ,Q e σ,Q k,σ,λ eigenstates χ by their overlaps holds in the thermodynamic limit under the assump- k,σ,λ,α tion (2.8). In other words, the many-body gap e+ik·r (cid:10) (cid:11) (cid:104)α,r|ϕ (cid:105):= √ χ (A16b) ∆σ,Q := Φσ,Q|H|Φσ,Q −(cid:104)Ψσ|H|Ψσ(cid:105)>0 (A15) k,σ,λ N k,σ,λ,α 8 for any given (Ising) spin σ =↑,↓ and band λ = ±. The resolves the identity since normalization and phase factors are chosen so that the orthonormality condition (cid:88) ψ† ψ =δ δ (A17e) r,z,σ,λ r,z(cid:48),σ,λ(cid:48) z,z(cid:48) λ,λ(cid:48) (cid:104)ϕ |ϕ (cid:105)=δ (A16c) k,σ,λ k,σ,λ(cid:48) λ,λ(cid:48) r∈A with λ,λ(cid:48) = ± holds for any given wavenumber k ∈ BZ and any given spin σ =↑,↓. The overlap (A16b) is in- for any given spin σ =↑,↓. We want to estimate the variant under the translation k→k+Q for any Q that profile in space of the Wannier states (A17a). belongs to the reciprocal lattice of sublattice A owing to For that purpose, we consider the spread functional the periodicity χ =χ . (A16d) k,σ,λ,α k+Q,σ,λ,α R(2) :=(cid:104)ψ |r2|ψ (cid:105) σ,λ z=0,σ,λ z=0,σ,λ (A18) WannierstatesaredefinedintermsoftheBlochstates −|(cid:104)ψ |r|ψ (cid:105)|2. z=0,σ,λ z=0,σ,λ by the unitary transformation |ψ (cid:105):= √1 (cid:88) e−ik·z|ϕ (cid:105) (A17a) Here, we are assuming, for simplicity, that there is only z,σ,λ k,σ,λ N onebandλthatisoccupied. Ifmorebandsareoccupied, k∈BZ we have to carry out a summation over all the occupied for any given unit cell of sublattice A labeled by z and bands. Observe that by translational invariance R(2) is σ,λ any given spin σ =↑,↓. The orthonormality (A16c) of left unchanged under the global translation r,z → r+ the Bloch states thus carries over to the orthonormality R,z+RforanylatticevectorR. Hence,thechoicez =0 in Eq. (A18) can be done without loss of generality. (cid:104)ψ |ψ (cid:105)=δ (A17b) z,σ,λ z,σ,λ(cid:48) λ,λ(cid:48) We rewrite Eq. (A18), following Ref. [13], as of the Wannier states for any given unit cell labeled by z of sublattice A and any given spin σ =↑,↓. For any λ and α, the representation of the Wannier state on the R(2) :=R(2) +R˜(2), (A19a) bipartite lattice Λ=A∪B is the overlap σ,λ I|σ,λ σ,λ R(2) :=(cid:104)ψ |r2|ψ (cid:105) ψ :=(cid:104)r,α|ψ (cid:105) I|σ,λ z=0,σ,λ z=0,σ,λ r,z,σ,λ,α z,σ,λ (cid:88) = √1 (cid:88) e−ik·z(cid:104)r,α|ϕ (cid:105) − |(cid:104)ψz=0,σ,λ|r|ψz(cid:48),σ,λ(cid:105)|2, (A19b) k,σ,λ z(cid:48) N k∈BZ R˜(2) := (cid:88) |(cid:104)ψ |r|ψ (cid:105)|2. (A19c) 1 (cid:88) σ,λ z=0,σ,λ z(cid:48),σ,λ = e+ik·(r−z)χ (A17c) z(cid:48)(cid:54)=0 N k,σ,λ,α k∈BZ for any given unit cell of sublattice A labeled by z and BothR(2) andR˜(2) arenon-negativequantitiesthatcan any given spin σ =↑,↓. The set of Wannier spinors I|σ,λ σ,λ be expressed in terms of k-space summations as follows. (cid:0) (cid:1) ψ = ψ (A17d) First, r,z,σ,λ r,z,σ,λ,α R(2) = (cid:88) 1 (cid:88) (cid:104)(cid:16)∂ χ† (cid:17)(cid:16)∂ χ (cid:17)−(cid:16)∂ χ† χ (cid:17)(cid:16)χ† ∂ χ (cid:17)(cid:105) I|σ,λ N kj k,σ,λ kj k,σ,λ kj k,σ,λ k,σ,λ k,σ,λ kj k,σ,λ j=x,y k∈BZ (cid:88) 1 (cid:88) (cid:104) (cid:105) = (cid:104)∂ χ |∂ χ (cid:105)−(cid:104)∂ χ |χ (cid:105)(cid:104)χ |∂ χ (cid:105) N kj k,σ,λ kj k,σ,λ kj k,σ,λ k,σ,λ k,σ,λ kj k,σ,λ j=x,y k∈BZ (cid:88) 1 (cid:88) (cid:0) (cid:1) = N (cid:104)∂kjχk,σ,λ| 11−|χk,σ,λ(cid:105)(cid:104)χk,σ,λ| |∂kjχk,σ,λ(cid:105) (A20a) j=x,y k∈BZ (cid:88) 1 (cid:88) ≡ (cid:104)∂ χ |Q |∂ χ (cid:105) N kj k,σ,λ k,σ,λ kj k,σ,λ j=x,y k∈BZ 1 (cid:88) (cid:2) (cid:3) = tr g , N k,σ,λ k∈BZ 9 where and A¯ := 1 (cid:88) A (A21c) Q :=11−|χ (cid:105)(cid:104)χ |, (A20b) σ,λ|j N k,σ,λ|j k,σ,λ k,σ,λ k,σ,λ k denotetheBerryconnectionandtheaverageoftheBerry is the single-particle projector operator on all the Bloch connection, respectively. Under the gauge transforma- states orthogonal to |χ (cid:105), the 2×2 matrix [g ] k,σ,λ k,σ,λ tion has the components |χk,σ,λ(cid:105)→eiϕk|χk,σ,λ(cid:105), (A22) (cid:104) (cid:105) g :=Re (cid:104)∂ χ |Q |∂ χ (cid:105) , k,σ,λ|i,j ki k,σ,λ k,σ,λ kj k,σ,λ it can be shown that RI(|2σ),λ remains invariant while R˜σ(2,λ) (A20c) does not. labeled by the Euclidean indices i,j ∈ {x,y} of two- We will now establish a lower bound on the gauge in- dimensional space, and tr denotes the trace over i,j ∈ variantquantityR(2) . Tothisend,wenoticethat,since {x,y}. Second, I|σ,λ Q isaprojectionoperatorwitheigenvalues0,1,then, k,σ,λ for any single-particle state |Ψ (cid:105), it follows that k,σ,λ R˜(2) = (cid:88) 1 (cid:88) (cid:16)A −A¯ (cid:17)2, (A21a) σ,λ N k,σ,λ|j σ,λ|j 0≤(cid:104)Ψ |Q |Ψ (cid:105). (A23) j=x,y k k,σ,λ k,σ,λ k,σ,λ In particular, if we choose where |Ψ (cid:105):=|∂ χ (cid:105)±i|∂ χ (cid:105), (A24) ±,k,σ,λ kx k,σ,λ ky k,σ,λ A :=i(cid:104)χ |∂ χ (cid:105), (A21b) the inequality (A23) delivers k,σ,λ|j k,σ,λ k k,σ,λ j (cid:2) (cid:3) (cid:16) (cid:17) tr g ≥∓i (cid:104)∂ χ |∂ χ (cid:105)−(cid:104)∂ χ |∂ χ (cid:105) . (A25) k,σ,λ k k,σ,λ k k,σ,λ k k,σ,λ k k,σ,λ x y y x Summing (A25) over k and taking the thermodynamic limit N →∞, delivers R(2) = Ac (cid:90) d2ktr(cid:2)g (cid:3) I|σ,λ (2π)2 k,σ,λ BZ A (cid:90) d2k (cid:104) (cid:105) ≥ ± c (cid:104)∂ χ |∂ χ (cid:105)−(cid:104)∂ χ |∂ χ (cid:105) (A26) 2π 2πi kx k,σ,λ ky k,σ,λ ky k,σ,λ kx k,σ,λ BZ A = ± c C , 2π σ,λ where A is the area of the unit cell of sublattice A and Appendix B: Competing Phases in the Hubbard c C are the spin- and band-resolved Chern numbers. model σ,λ Since R(2) ≥0, it then follows I|σ,λ Any translationally invariant electronic single-particle Hamiltonianisdescribedinmomentumspacebythedis- A persions and Bloch eigenfunctions of its bands. Given R(2) ≥ c |C |, (A27) I|σ,λ 2π σ,λ thehoppingamplitudesofatight-bindingmodelinposi- tion space, the dispersions and Bloch eigenfunctions are uniquelydetermined. Onceinteractionsareincluded,the many-body ground state of the model depends crucially as a lower bound, proportional to the band Chern num- on the properties of the dispersions (bandwidths; shapes ber,onthegaugeinvariantpartofspreadoftheWannier of Fermi surface, e.g., nesting; ...) and the properties states(A17a). ItwasshowninRef.[13]that,inthetopo- of the Bloch states or Wannier states (Chern numbers, logicalphaseoftheHaldanemodel3,R(2) isfinitewhile I|σ,λ localization in position space, ...). The power of our ap- R˜(2) displays a logarithmic divergence which is related proach, the flattening of the band with the factor w σ,λ k to the non-zero Chern number of the bands. fromEq.(2.2),istodisentanglethepropertiesofthedis- 10 a) b) c) π 3 1.4 gical ky12 t1 olo 00 1.2 p 1 o t 2 –π3 1.0 –π3 2 1 00 1kx2 π3 A B π 3 0.8 0.3 ky2 0.25 vial 001 0.2 tri 1 0.6 0.15 0.1 2 –π3 0.05 –π3 2 1 00 1k 2 π3 0.4 x 10-3 FIG. 3. (Color online) Comparison of the topological (top; √ 0.2 t =1/ 2,µ =0)andnon-topological(bottom;t =0,µ = 2√ s 2 s 1/ 2) single-particle model. (a) The shaded area represents the Fermi see of the lower band at the commensurate filling 0.0 0.2 0.4 0.6 0.8 1.0 fractionN =N whenκ<1. (b)Theeigenspinorχ ,when g e k,σ interpretedasapointonthesurfaceoftheunitsphere,swipes out the full surface of this sphere (a small portion of this FIG. 4. (Color online) Numerical exact diagonalization re- spherenearonepole)ask takesvalueseverywhereintheBZ sults for flat bands κ = 1 for the system size L = L = 25 for the topological (non-topological) band structure. (c) The x y at the commensurate filling fraction N =N. Plotted is the spread of the Wannier states (A17c) in real space indicates e value of the many-body excitation gap ∆/U between the en- theirdelocalized(localized)characterforthetopological(non- ergy of the fully spin polarized state S = N and the lowest topological) band structure. energystateinthesectorwithonespinflippedS =N−2asa function of g :=(2/π)arctan|µ /t | and the nearest-neighbor s 2 hopping parameter t . The thick red line corresponds to the persionsfromthoseofthespinorstosomeextendandto 1 parameterchoicemadefortheplotinFig.1oftheLetter. As study their effects individually. discussed in the text, the spinors of the Bloch band become fully sublattice polarized in the limit t → 0 and the fer- 1 romagnetic ground state becomes degenerate in energy with 1. Possible phases states from other spin sectors (∆→0). Let us list the anticipated ground states for some lim- with t (cid:28) U, i.e., on the t-J model. An antiferro- iting properties of the spinors and the dispersions for magnetic ground state will be selected. An alter- N =N electrons when the two pairs of bands are sepa- e native way to lift the macroscopic degeneracy is to rated by a gap in the absence of interactions. add longer range interactions, in which case other 1. (Almost) flat band and sublattice-polarized spinors. ground states may be stabilized25. Note that the If the band is completely flat (κ = 1), the ground fullysublatticepolarizedspinorsdiscussedhereim- state is macroscopically degenerate in the absence ply that the non-interacting model is topologically of interactions. Moreover, when the Bloch states trivial. arefullysublatticepolarized, i.e., whenχ† =(1,0) k 2. Flat band and not sublattice polarized spinors. As say, then the kinetic energy becomes diagonal in was shown in the Letter, if the spinors are suffi- position space [ψ† = δ (1,0)]. Hence, any r,z,σ,− r,z ciently spread between the two sublattices, i.e., if state with no less and no more than one electron condition(2.8)ismet,thegroundstateinthelimit persiteonsublatticeAwillbeagroundstateofthe of flat bands is an Ising ferromagnet with full spin Hamiltonian, regardless of the spin-configuration polarization. of the electrons. Therefore, the ground state re- mains macroscopically degenerate, in spite of the 3. Strongly dispersing unnested band. For a strongly presence of the Hubbard interaction. This degen- dispersingbandwithoutanestedFermisurface,the eracycanbeliftedbyintroducingasmallbutfinite Fermi liquid ground state is expected to be stable bandwidth, while keeping the full sublattice polar- against sufficiently small repulsive Hubbard inter- ization of the spinors intact. Then, the low-energy actions. One generic instability at finite interac- degrees of freedom of the model map on the con- tion strength is Stoner ferromagnetism which, in ventional one-band Hubbard model at half filling contrast to the flat band ferromagnetism, does not

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Topological Hubbard model and its high-temperature quantum Hall effect Titus Neupert,1 Luiz Santos,2 Shinsei Ryu,3 Claudio Chamon,4 and Christopher Mudry1 1Condensed Matter Theory Group, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland 2Department of Physics, Harvard University, 17 Oxford Street, Cambridge, Massachusetts 02138, USA 3Department of Physics, University of Illinois at Urbana-Champaign, 1110 W. Green Street, Urbana, Illinois 61801-3080, USA 4Physics Department, Boston University, Boston, Massachusetts 02215, USA (Dated: January 26, 2012) The quintessential two-dimensional lattice model that describes the competition between the kinetic energy of electrons and their short-range repulsive interactions is the repulsive Hubbard model. We study a time-reversal symmetric variant of the repulsive Hubbard model defined on a planarlattice: Whereastheinteractionisunchanged,anyfullyoccupiedbandsupportsaquantized 2 spin Hall effect. We show that at 1/2 filling of this band, the ground state develops spontaneously 1 andsimultaneouslyIsingferromagneticlong-rangeorderand aquantizedchargeHalleffectwhenthe 0 2 interactionissufficientlystrong. Weponderonthepossiblepracticalapplications,beyondmetrology, that the quantized charge Hall effect might have if it could be realized at high temperatures and n without external magnetic fields in strongly correlated materials. a J 5 I. INTRODUCTION topologically nontrivial Bloch bands simultaneously dis- 2 playsIsingferromagneticlong-rangeorderand theIQHE ] High-temperature superconductivity1 and the quan- atsomecommensuratefillingfraction. Theenergyscales el tum Hall effect (QHE)2 have been two of the central thatcanbeattainedinlatticemodelsaretypicallyrather high,oftheorderofatomicmagnitudes,i.e.,electronvolt. - problems in condensed matter physics of the past three r If an interacting system with topological bands can be t decades. The former is related to electrons hopping on a s found so as to display the IQHE at high temperatures, two-dimensional (2D) lattice close to (but not at) half . t filling, while the latter focuses on fermions in doped it could be of practical use, as we shall explain after we a substantiate our claims. m semiconductor heterostructures or graphene in a high magnetic field. High-temperature superconductors are - d strongly interacting systems, with the potential energy n about an order of magnitude larger than the kinetic en- II. STUDY OF THE TOPOLOGICAL HUBBARD o ergy. In the QHE, the kinetic energy is quenched by the MODEL c external magnetic field. Moreover, interactions are im- [ portant only in understanding the fractional QHE but We consider spinful electrons hopping on a bipartite 2 not in understanding the integer QHE (IQHE). squarelatticeΛ=A∪B withsublatticesAandB,where v The possibility that the IQHE could arise in a lattice each sublattice has N := L ×L sites. The Hubbard 6 x y HamiltonianwithouttheLandaulevelsinducedbyauni- Hamiltonian with repulsive interactions (U > 0) can be 9 2 form magnetic field was suggested by Haldane in 19883. written 1 The essence is that, despite the absence of an uniform (cid:88) (cid:88) (cid:88) . magnetic field, the system still lacks time-reversal sym- H := c† H c +U n n . (2.1a) 0 k k k r,↑,α r,↓,α metry. More recently, it was shown that the fractional 1 k∈BZ r α=A,B 1 QHE could also emerge in flat topological bands when 1 they are partially filled4–9 (see also 10). These recent The component c† of the operator-valued spinor c† : developments point to a natural marriage between the k,σ,α k v createsanelectronwithmomentumk fromtheBrillouin QHE and strongly correlated lattice systems at a high i zone (BZ) of sublattice A and with spin σ =↑,↓, whose X filling fraction. Fourier transform c† = N−1/2(cid:80) e−ik·rc† is r In this Letter, we study a quintessential strongly cor- r,σ,α k∈BZ k,σ,α a exclusively supported on sublattice α=A,B. The 4×4 relatedlattice2Dsystembutwithatwist. Weconsidera Hermitian matrix H obeys the time-reversal symmetry time-reversalsymmetricfermionicHubbardmodelinthe k (TRS) limit of large on site repulsion U compared to the band- width W of the hopping dispersion, but with hopping H =σ H∗ σ , (2.1b) terms yielding topologically nontrivial Bloch bands in +k 2 −k 2 that they each support a quantized spin Hall conductiv- and, owing to a strong intrinsic spin-orbit coupling, the ity when fully occupied11. The time-reversal symmetric residual spin-rotation symmetry (RSRS) HubbardmodelwithasinglehalffillednestedBlochband has a charge insulating ground state that supports anti- (cid:32) (cid:33) ferromagneticlong-rangeorder1. Incontrast,theground H =σ H σ ≡ h(k↑) 0 , (2.1c) stateofourtime-reversalsymmetricHubbardmodelwith +k 3 +k 3 0 h(↓) k 2 where the Pauli matrices σ , σ , and σ act on the elec- exactly flat with eigenvalues ±1. The case κ = 0 cor- 1 2 3 tronic spin-1/2 degrees of freedom. Hence, the two 2×2 responds to a tight-binding model on the square lattice Hermitian matrices h(kσ) with σ =↑,↓ obey that involves only nearest-neighbor (|t1| = 1) and next- nearest-neighbor hopping (t ) together with a staggered 2 h(↑) =h(↓) , ∀k∈BZ, α,β =A,B, chemicalpotentialµs thatbreaksthesymmetrybetween +k,αβ −k,βα sublattices A and B4. For κ ∈ (0,1], longer-range hop- (2.1d) ping is introduced. However, we stress that the Hamil- because of the condition of TRS (2.1b). Finally, the op- tonian remains local for all κ∈[0,1] since all correlation eratorn =c† c measurestheelectrondensity r,σ,α r,σ,α r,σ,α functions decay exponentially due to the presence of the on site r in sublattice α and with spin σ. band gap4. The Hubbard Hamiltonian defined by Eq. (2.1a) thus The topological properties of the lower pair of bands has a global Z ×U(1) symmetry that arises because of 2 are characterized by their spin Chern number the TRS (2.1b) and the RSRS (2.1c). We are going to show that TRS is spontaneously broken while the con- C :=(cid:0)C −C (cid:1)/2, (2.3a) s ↑ ↓ tinuous RSRS is shared by the ground state, when this HubbardHamiltonianacquiressuitabletopologicalprop- where Cσ is to be computed from the orbitals of spin-σ erties. electrons according to It is the choice for the matrix elements h(kσ,α)β en- (cid:90) d2k (cid:16) (cid:17) tering the kinetic energy (2.1c) that endows the Hub- C := ∇ ∧ χ† ∇ χ . (2.3b) σ 2πi k k,σ,− k k,σ,− bard Hamiltonian (2.1a) with topological attributes. We k∈BZ choose Time-reversalsymmetryimpliesC =−C andtherefore ↑ ↓ (cid:104) (cid:16) (cid:17) entails a vanishing of the total (charge) Chern number h(k↑,)AB =h(k↑,)B∗A :=wk e−iπ/4 1+e+i(ky−kx) Cc := (C↑+C↓)/2 of the lower bands. The spin Chern +e+iπ/4(cid:0)e−ikx +e+iky(cid:1)(cid:105), number of the lower pair of bands is given by 1(cid:16) (cid:17) h(↑) =−h(↑) :=w (cid:2)2t (cos k −cos k )+4µ (cid:3), Cs = 2 sgnh((↑0,)π),AA−sgnh((↑π),0),AA . (2.3c) k,AA k,BB k 2 x y s (2.2a) Hence, the Bloch bands are topologically trivial when- ever |t /µ | < 1, while the model at half filling exhibits where 2 s the physics of a quantum spin Hall insulator whenever |t /µ | > 1. In an open geometry, the spin Hall conduc- w−1 :=κε +(1−κ), κ∈[0,1], (2.2b) 2 s k k tivity is quantized to the value σsH = eC /(2π) where e xy s denotes the electric charge of the electron. and WenowconsiderthesystemwitharepulsiveHubbard (cid:113) interaction U > 0 at 1/2 filling of the lower band (1/4 ε := 1+cosk cosk +[2t (cosk −cosk )+4µ ]2. k x y 2 x x s filling of the lower and upper bands), i.e., with (2.2c) In the noninteracting limit (U = 0), this model fea- N =L ×L =N (2.4) e x y tures four bands with two distinct two-fold degener- electrons. In all that follows, we assume that U is much ate dispersions ±w ε 4. This two-fold degeneracy is a k k smaller than the gap ∆ induced by a strong intrinsic consequence of the Kramers degeneracy implied by the 0 spin-orbit coupling between the two pairs of bands. If TRS(2.1b). Ifwedenotethecorrespondingeigenspinors so,wecanrestricttheN -bodyHilbertspacetotheFock χ =(χ ),whereλ=±,andchoosethenormal- e k,σ,λ k,σ,λ,α spacearisingfromthesingle-particleHilbertspacesofthe ization χ† χ =δ , ∀k, then the kinetic energy k,σ,λ k,σ,λ(cid:48) λ,λ(cid:48) lower pair of bands. is diagonalized by using the fermionic creation operators In the limit of flat bands κ = 1 and at the commen- surate filling fraction (2.4), the kinetic energy (2.2e) at (cid:88) d†k,σ,λ := χ∗k,σ,λ,αc†k,σ,α, (2.2d) fixed spin polarization S := |(cid:104)σ3(cid:105)| = 0, 2, ..., N in units α=A,B of (cid:126)/2 has a ground state degeneracy as (cid:18) (cid:19)2 N N = . (2.5) H := (cid:88) (cid:88) (cid:88)λd† w ε d . (2.2e) gs N−|S| 0 k,σ,λ k k k,σ,λ 2 k∈BZσ=↑,↓λ=± The repulsive Hubbard interaction lifts this degeneracy whenever any one of these states allows a site of Λ to be Hence, the Bloch states created by d† are generically k,σ,λ doubly occupied with a finite probability. The only two spread on both sublattices A and B. We shall consider states with full spin polarization S =N, only the case in which these bands are separated by an (cid:89) energy gap, i.e., |t2| (cid:54)= |µs|. The parameter κ controls |Ψσ(cid:105)= d†k,σ,−|0(cid:105), σ =↑,↓, (2.6a) the bandwidth of these bands. For κ=1, the bands are k∈BZ 3 are immune to the presence of the Hubbard repulsion. states with more than one spin flipped are higher in en- More formally, observe that Hamiltonian H −µN is a ergy than those with one spin flipped. e positive semidefinite operator for κ = 1, U > 0, and the Ruling out the possibility that states with many spin chemical potential µ=−1. Since flips have lower energies than states with few spin flips relative to the Ising ferromagnetic ground state is plau- (cid:104)Ψ |(H +N )|Ψ (cid:105)=0, σ =↑,↓, (2.6b) sible in the regime when the intrinsic spin-orbit coupling σ e σ generates the largest energy scale (∆ (cid:29)U). 0 thetwostates(2.6a)belongtothegroundstatemanifold Equation (2.8) is a reasonable assumption when the of H +Ne for any U >0, t2, and µs. Blochstatesstemfromabandwithnonzero(spin)Chern WearegoingtoarguethatthispairofdegenerateIsing number, since the spinor χ maps out the entire sur- k,σ ferromagnets spans the ground state manifold for any face of the unit sphere as k takes values in the BZ. The U >0 and |t /µ |=(cid:54) 1. This is achieved by arguing that assumption underlying Eq. (2.8) can also be understood 2 s they are separated from excited states by a many-body by constructing the Wannier wavefunctions, centered at gap, a departure from the usual ferromagnetism in flat the lattice point z, of the lowest energy band with spin bands when full spin-1/2 SU(2) symmetry is not explic- σ and Chern number: C σ itlybroken12. First,particle-holeexcitationsof|Ψ (cid:105)that σ keep S = N fixed, cost an energy ∆0 > 0 and are thus ψ := 1 (cid:88) eik·(r−z)χ . (2.10a) gapped. Second,weaskwhetherexcitationsof|Ψ (cid:105)that z,r,σ,−,α N k,σ,−,α σ k∈BZ flip one spin (S = N −2) are gapped as well. Any such state can be written as Thegauge-invariantpartoftheirspreadfunctional13 sat- isfies |Φ (cid:105)= (cid:88) A(Q)d† d |Ψ (cid:105), σ =↑,↓, σ,Q k k+Q,σ¯,− k,σ,− σ (cid:104)ψ |r2|ψ (cid:105)−(cid:88) |(cid:104)ψ |r|ψ (cid:105)|2 k∈BZ 0,σ,− 0,σ,− 0,σ,− z,σ,− (2.7a) z where the center of mass momentum Q is a good quan- ≥|C |A /(2π), σ c tum number and thus (cid:104)Φσ,Q|Φσ,Q(cid:48)(cid:105) = δQ,Q(cid:48) if the nor- (2.10b) malization (cid:80) A(Q)∗A(Q) = 1 is imposed. One veri- k∈BZ k k where A denotes the area of the unit cell. This inequal- fies that (see Appendix) c ity relates the Chern number of the band and the “min- imum width” of the Wannier states (see Appendix). In (cid:104)Φ |(H +N )|Φ (cid:105) σ,Q e σ,Q particular, in the nontopological phase, one can imagine (cid:12) (cid:12)2 =U − NU (cid:88)(cid:12)(cid:12)(cid:12) (cid:88) A(kQ)χ−k−Q,σ,−,αχk,σ,−,α(cid:12)(cid:12)(cid:12) , agilviemnitsuinbwlahtticiche,thwehwilaevtehfuenncotnioznerioseCnhtierrenlynluomcabliezredinotnhae (cid:12) (cid:12) α k∈BZ topologicalphaseimpliesthattheWannierwavefunction (2.7b) has amplitudes on both sublattices. While Eq. (2.9) is strongly suggestive of the existence where the lowest energy state with one spin flipped is of a many-body gap ∆, it does not provide informa- characterized by the A(kQ) that minimizes Eq. (2.7b) tion about its size. To quantify ∆, we diagonalized the while satisfying the normalization condition. For exam- model(2.1)exactlynumericallyinthelimitofflatbands ple, if the single-particle orbitals are fully sublattice po- κ = 1 at the commensurate filling fraction (2.4). We larized, e.g., χ† ∝ (1,0) (topologically trivial), the varied the ratio |t /µ |, keeping t2 +µ2 = 1/2 constant k,σ,− 2 s 2 s choice A(Q) = N−1/2 minimizes Eq. (2.7b) with the to drive the system from the topological to the trivial k right-hand side equal to zero. Hence, the fully spin- phase. The results are shown in Fig. 1. First, they sup- polarizedstate|Ψ (cid:105)isagapless groundstateinthiscase. port the assumption that all states with more than one σ On the other hand, let us assume that spin flipped are higher in energy than the many-body one-spin-flipped gap provided |t /µ | > 1. Second, we 2 s χ† (cid:54)∝(1,0) and χ† (cid:54)∝(0,1) (2.8) find ∆ ≈ 0.3U as an extrapolation to the thermody- k,σ,− k,σ,− namic limit for µ = 0 deep in the topological phase, s holds almost everywhere in the BZ, i.e., up to a set while ∆ monotonically decreases toward a much smaller of measure zero. In the thermodynamic limit, where nonvanishing value for t2 = 0 in the topologically triv- the sum over k becomes an integral, this delivers from ial phase set by the unit of energy |t1| = 1. Finally, Eq. (2.7b) the strict inequality (see Appendix) it should be noted that neglecting the states from the upper band of the noninteracting Hamiltonian delivers (cid:10) (cid:11) Φ |(H +N )|Φ >0. (2.9) the correct excitation many-body gap ∆ not only in the σ,Q e σ,Q aforementionedlimitU (cid:28)∆ ,butalsoundertheweaker 0 Hence, assumption (2.8) is sufficient to show that the condition ∆<∆ , if the limit of flat bands is taken. 0 spin-polarized state |Ψ (cid:105) is a gapped ground state of the Deep in the topologically nontrivial regime |t /µ | (cid:29) σ 2 s Hamiltonianwithflatbandsinthethermodynamiclimit, 1, the states |Ψ (cid:105) and |Ψ (cid:105) are degenerate ground states σ σ¯ provided that one also assumes that the lowest energy related by TRS for any finite N. They are separated 4  E–UE00000gs....4268 • • • • • • • • • UΔ• 0000000000..........•333333333311111111113445563456•505050U0S•0.010•2500.1.0•0011008•.01560••0.0.024•200.02•2L5x×10•LyLx•1Ly E–UEg00000s....4268 0• • • • • •0.•5• • • • •1.0• • •W•/U• • • •• S1186420•20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 g FIG. 2. (Color online) Numerical exact diagonalization re- sultsatthecommensuratefillingfraction(2.4)asafunctionof FIG. 1. Numerical exact diagonalization results for flat the bandwidth W for L =3, L =4. Plotted is the energy x y bands κ = 1 at the commensurate filling fraction (2.4). oftheloweststateindifferentsectorsoftotalspinS (inunits Markers show the energy of the lowest state in different sec- of (cid:126)/2) measured with respect to the ground√state energy in tors of total spin S (in units of (cid:126)/2) measured with respect the topological phase with µ = 0, t = 1/ 2. The ground s 2 to the ground state energy for L = 3, L = 4. Here, stateisgappedandfullyspin-polarizedforW/U <0.7,while x y g := (2/π)arctan|µ /t | so that g > 0.5 and g < 0.5 cor- it is unpolarized for W/U >0.7 (see Appendix). s 2 respond to the trivial and topological single-particle bands, respectively. Since there is only one state in the fully po- larized sector |S| = 12, the difference between the asterisks III. PRACTICAL APPLICATIONS and the squares is the many-body excitation gap ∆(g). The thick blue line shows the extrapolation of ∆(g) to the ther- Sowhatisitgoodfor, amaterialwithaQHEatroom modynamic limit. In the inset, exact diagonalization in the sectorwithonespinflippedawayfromthefullypolarizedsec- temperaturewithoutappliedexternalmagneticfieldsbe- √ tor is presented for µ =0, t =1/ 2 and L =L ranging sides metrology14,15? First, we recall that the quantiza- s 2 x y from 6 to 30. The straight lines are a guide to the eye and tion of the Hall resistance and the accompanying van- make evident an even-odd effect in L = L . Deep in the ishing of the longitudinal resistance is exact only at zero x y topologically nontrivial regime g (cid:28)0.5, we observe a sizable temperature. Thelongitudinalresistanceincreasesexpo- ∆(g (cid:28) 0.5). The topologically trivial regime g > 0.5 is also nentiallyfastwithincreasingtemperature15. However,if characterizedbyagap∆(g>0.5)inthesectorwithonespin aQHEwithgapsoftheorderofhundredsofmeVoreven flippedawayfromthefullypolarizedsector,howeverthisgap eV scales could arise in a strongly correlated lattice ma- is much smaller than ∆(g(cid:28)0.5). We refer the reader to the terial, exceptionally low resistivities could be attained. Appendix for a discussion of the regime ∆(g>0.5). The resistance of a Hall bar depends on its aspect ratio and the Hall angle δ = arctan(ρ /ρ )16, but for long xy xx fromtheirexcitationsbyagapthatsurvivesthethermo- systems (“wires”) near the quantized regime, the lon- dynamic limit N → ∞. Spontaneous breaking of TRS gitudinal resistance scales as Rxx = L/W ρxx, and the takes place in the thermodynamic limit N → ∞ by se- 2D resistivity ρ ∼ R e−∆/T, where ∆ is the exci- xx K lectingthegroundstatetobe|Ψ (cid:105), say. Itisthenmean- tation gap. For gaps of the order of 100meV to 1eV, ↑ ingful to discuss the quantized electromagnetic response onewouldobtainroomtemperature2Dresistivitiesfrom of |Ψ (cid:105), since TRS is spontaneously broken. The trans- ρ ∼ 103Ω to ρ ∼ 10−13Ω, respectively. Obviously ↑ xx xx verse charge response σH of |Ψ (cid:105) is proportional to the the exponential behavior is responsible for this gigan- xy ↑ tic range. Small as they are, these are not perfect con- many-body Chern number C . The latter takes into |Ψ↑(cid:105) ductors. For a benchmark, we consider the conductivity account the occupation of the Bloch states4. Since all of copper at room temperature per atomic layer. Us- Bloch states of the lower band with spin σ are occupied ing the value for the 3D resistivity of copper at 20◦C of in |Ψ↑(cid:105), while all Bloch states with spin ↓ are empty, ρ3D =1.68×10−8Ωm 17 and that the lattice parameter C|Ψ (cid:105) ≡ C↑. Hence, the ground state has the quantized foCruFCClatticeis3.61˚A,weobtainρ2D =93.3Ω. There- ↑ Cu Hall response fore, for gaps above ∆ ≈ .2eV, the Hall system starts to be better conducting than copper at room tempera- |σH |=|C |×e2/h=e2/h. (2.11) xy ↑ ture, and for ∆ ≈ .3eV it is already almost 3 orders of Remarkably, the selection by the repulsive Hubbard magnitude better conducting than copper. interaction of a ground state supporting simultaneously TheRSRS(2.1c)isnotexactinpractice. Forexample, Isingferromagnetismand theIQHEisrobusttoasizable a Rashba spin-orbit coupling violates this RSRS. How- bandwidth as is suggested by numerical exact diagonal- ever, our analysis of transport at room temperature still ization. As shown in Fig. (2), the fully spin-polarized applies provided the characteristic energy scale associ- state |Ψ (cid:105) remains the gapped ground state of the sys- ated to the breaking of the RSRS (2.1c) is much smaller σ tem up to a bandwidth W/U ≈0.7. than the largest energy scale ∆ induced by the intrinsic 0 5 spin-orbitcoupling. Materialsthatrealizea2DZ topo- in the N -many-body state 2 e logical band insulator18,19 with a band gap ∆ are thus 0 candidates to realize a QHE at room temperature if (i) the band gap is larger than the correlation energy and (ii) the chemical potential can be tuned to half-filling of (iii)areasonablyflatvalenceband. HgTequantumwells |Φ (cid:105)= (cid:88) A(Q)d† d |Ψ (cid:105), σ =↑,↓, σ,Q k k+Q,σ¯,− k,σ,− σ with an inverted band structure realize a 2D Z topo- 2 k∈BZ logicalbandinsulatorwithasmallRashbacoupling20,21. (A1b) The design of a material with the functionalities (i)-(iii) which has one spin flipped as compared to the – up to has been proposed in Ref. 8 . Cold atoms trapped in an time-reversal symmetry and the global gauge phase fac- optical honeycomb lattice22,23 might offer an alternative tor – unique normalized N -many-body state with full e to realizing the topological Hubbard model discussed in spin polarization this Letter. We would like to close by mentioning that examples such as the topological Hubbard model discussed in this Letter, as well as lattice models displaying the fractional (cid:89) quantum Hall effect studied in Refs. 4–7, could serve as |Ψ (cid:105)= d† |0(cid:105), σ =↑,↓. (A1c) σ k,σ,− benchmarks for numerical methods of fermionic models k∈BZ in 2D such as dynamical mean-field theory and meth- ods based on tensor product states24. In contrast to the single-band repulsive Hubbard model, for which lit- tle is known exactly at fractional filling, the topological Here, Q denotes the center of mass momentum and the Hubbardmodel(2.1),becauseofthenonvanishingChern orthonormalization condition numbers of its bands, leads to much better understood (topological) ground states. It can thus serve as a yard- stick for the performance of these methods. This work was supported in part by DOE Grant No. DEFG02-06ER46316 and by the Swiss National Science (cid:104)Φ |Φ (cid:105)=δ (A1d) Foundation. σ,Q σ,Q(cid:48) Q,Q(cid:48) Appendix A: Intermediary steps enforces the normalization 1. Derivation of Eq. (2.7b) For the limit of flat bands κ = 1, we are going to compute the expectation value (cid:88) A(Q)∗A(Q) =1. (A1e) k k (cid:104)Φ |(H +N )|Φ (cid:105) (A1a) σ,Q e σ,Q k∈BZ We rewrite |Φ (cid:105) as σ,Q    |Φσ,Q(cid:105)= (cid:88) A(kQ) (cid:88) χ∗k+Q,σ¯,−,αc†k+Q,σ¯,α (cid:88) χk,σ,−,βck,σ,β|Ψσ(cid:105) k∈BZ α=A,B β=A,B (cid:34) (cid:35) = (cid:88) (cid:88) (cid:88) A(kQ)χ∗ χ e−i(k+Q)·re+ik·r(cid:48) c† c |Ψ (cid:105) (A2a) N k+Q,σ¯,−,α k,σ,−,α r,σ¯,α r(cid:48),σ,β σ α,β=A,Br,r(cid:48)∈A k∈BZ ≡ (cid:88) (cid:88) M(σ) c† c |Ψ (cid:105), r,α,r(cid:48),β r,σ¯,α r(cid:48),σ,β σ α,β=A,Br,r(cid:48)∈A where we have introduced the short-hand notation M(σ) := (cid:88) A(kQ)χ∗ χ e−i(k+Q)·re+ik·r(cid:48). (A2b) r,α,r(cid:48),β N k+Q,σ¯,−,α k,σ,−,β k∈BZ 6 in terms of which (cid:88) (cid:88) (cid:104)Φ |(H +N )|Φ (cid:105)=U (cid:104)Φ |n n |Φ (cid:105) ↑,Q e ↑,Q ↑,Q r˜,↑,δ r˜,↓,δ ↑,Q r˜∈Aδ=A,B =U (cid:88) (cid:88) (cid:12)(cid:12)M(↑) (cid:12)(cid:12)2(cid:104)Ψ |c† c n n c† c |Ψ (cid:105) (cid:12) r,α,r(cid:48),β(cid:12) ↑ r(cid:48),↑,β r,↓,α r˜,↑,δ r˜,↓,δ r,↓,α r(cid:48),↑,β ↑ r˜,r,r(cid:48)∈Aα,β,δ=A,B (A3) =U (cid:88) (cid:88) (cid:12)(cid:12)M(↑) (cid:12)(cid:12)2(1−δ δ )δ δ (cid:12) r,α,r(cid:48),β(cid:12) r,r(cid:48) α,β r˜,r δ,α r˜,r,r(cid:48)∈Aα,β,δ=A,B =U (cid:88) (cid:88) (cid:12)(cid:12)M(↑) (cid:12)(cid:12)2−U (cid:88) (cid:88) (cid:12)(cid:12)M(↑) (cid:12)(cid:12)2. (cid:12) r,α,r(cid:48),β(cid:12) (cid:12) r,α,r,α(cid:12) r,r(cid:48)∈Aα,β=A,B r∈Aα=A,B Using the definition (A2b), the first term becomes (cid:88) (cid:88) (cid:12)(cid:12)M(↑) (cid:12)(cid:12)2 = (cid:88) (cid:88) (cid:88) A(kQ)A(kQ(cid:48))∗χ∗ χ χ χ∗ e−i(k−k(cid:48))·(r−r(cid:48)) (cid:12) r,α,r(cid:48),β(cid:12) N2 k+Q,↓,−,β k,↑,−,α k(cid:48)+Q,↓,−,β k(cid:48),↑,−,α r,r(cid:48)∈Aα,β=A,B r,r(cid:48)∈Aα,β=A,Bk,k(cid:48)∈BZ (cid:88) (cid:88) A(Q)A(Q)∗ = k k(cid:48) χ∗ χ χ χ∗ N2δ N2 k+Q,↓,−,β k,↑,−,α k(cid:48)+Q,↓,−,β k(cid:48),↑,−,α k,k(cid:48) α,β=A,Bk,k(cid:48)∈BZ    = (cid:88) A(kQ)A(kQ)∗ (cid:88) χ∗k+Q,↓,−,βχk+Q,↓,−,β (cid:88) χk,↑,−,αχ∗k,↑,−,α k,∈BZ β=A,B α=A,B = (cid:88) A(Q)A(Q)∗ k k k,∈BZ =1, (A4) where both the normalization conditions on A(Q) in Eq. (A1e) and on χ have been used in the last two lines. k k,σ,−,α The second term in Eq. (A3) becomes (cid:88) (cid:88) (cid:12) (cid:12)2 (cid:88) (cid:88) (cid:88) A(Q)A(Q)∗ (cid:12)M(↑) (cid:12) = k k(cid:48) χ∗ χ χ χ∗ (cid:12) r,α,r,α(cid:12) N2 k+Q,↓,−,α k,↑,−,α k(cid:48)+Q,↓,−,α k(cid:48),↑,−,α r∈Aα=A,B r∈Aα=A,Bk,k(cid:48)∈BZ (cid:88) (cid:88) A(Q)A(Q)∗ = k k(cid:48) χ∗ χ χ χ∗ N k+Q,↓,−,α k,↑,−,α k(cid:48)+Q,↓,−,α k(cid:48),↑,−,α α=A,Bk,k(cid:48)∈BZ (A5) (cid:12) (cid:12)2 = 1 (cid:88) (cid:12)(cid:12) (cid:88) A(Q)χ∗ χ (cid:12)(cid:12) N (cid:12) k k+Q,↓,−,α k,↑,−,α(cid:12) (cid:12) (cid:12) α=A,B k∈BZ (cid:12) (cid:12)2 = 1 (cid:88) (cid:12)(cid:12) (cid:88) A(Q)χ χ (cid:12)(cid:12) , N (cid:12) k −k−Q,↑,−,α k,↑,−,α(cid:12) (cid:12) (cid:12) α=A,B k∈BZ where we used the identity χ = χ∗ which follows from the time-reversal symmetry of the Hamiltonian. k,σ,λ,α −k,σ¯,λ,α Putting everything together, we obtain Eq. (2.7b) of the Letter, (cid:12) (cid:12)2 (cid:104)Φ |(H +N )|Φ (cid:105)=U − U (cid:88) (cid:12)(cid:12) (cid:88) A(Q)χ χ (cid:12)(cid:12) . (A6) σ,Q e σ,Q N (cid:12) k −k−Q,σ,−,α k,σ,−,α(cid:12) (cid:12) (cid:12) α=A,B k∈BZ 2. Derivation of Eq. (2.9) fora,b,c,d∈[0,1]. Inparticular,theequalityonlyholds when either a = b = c = d = 0 or a = b = c = d = 1. Before starting with the derivation of the inequality To show this, one can rewrite the inequality in terms of Eq. (2.9), let us establish the following inequality (cid:112) (cid:112) (cid:112) (cid:112) 1≥abcd+ 1−a2 1−b2 1−c2 1−d2 (A7) 7 trigonometric functions with angles α,β,γ,δ ∈[0,π/2] strict inequality 1≥ sin α sin β sin γ sin δ +cos α cos β cos γ cos δ (cid:12) (cid:12)2 =1[cos(α+β)cos(γ+δ)+cos(α−β)cos(γ−δ)]. U − U (cid:88) (cid:12)(cid:12) (cid:88) A(Q)χ χ (cid:12)(cid:12) >0 2 N (cid:12) k −k−Q,σ,−,α k,σ,−,α(cid:12) (cid:12) (cid:12) (A8) α=A,B k∈BZ (A9) holds under the assumption that Eq. (2.8) is satisfied Hence the inequality holds and the equality is true only almost everywhere in the BZ, i.e., up to a set of measure if either α = β = γ = δ = 0 or α = β = γ = δ = π/2 as zero. announced. Wewillnowshowthatinthethermodynamiclimitthe We rewrite, for very large N, (cid:12) (cid:12)2 U (cid:88) (cid:12)(cid:12) (cid:88) A(Q)χ χ (cid:12)(cid:12) N (cid:12) k −k−Q,σ,−,α k,σ,−,α(cid:12) (cid:12) (cid:12) α=A,B k∈BZ →NU(cid:90) d2k (cid:90) d2k(cid:48) A(Q)A(Q)∗ (cid:88) χ χ χ∗ χ∗ (2π)2 (2π)2 k k(cid:48) −k−Q,σ,−,α k,σ,−,α −k(cid:48)−Q,σ,−,α k(cid:48),σ,−,α α=A,B ≤NU(cid:90) (d2π2k)2 (cid:90) (d22πk)(cid:48)2 (cid:12)(cid:12)(cid:12)A(kQ)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)A(kQ(cid:48))∗(cid:12)(cid:12)(cid:12) (cid:88) (cid:12)(cid:12)χ−k−Q,σ,−,α(cid:12)(cid:12)(cid:12)(cid:12)χk,σ,−,α(cid:12)(cid:12)(cid:12)(cid:12)χ∗−k(cid:48)−Q,σ,−,α(cid:12)(cid:12)(cid:12)(cid:12)χ∗k(cid:48),σ,−,α(cid:12)(cid:12) (A10) α=A,B (cid:90) d2k (cid:90) d2k(cid:48) (cid:12) (cid:12)(cid:12) (cid:12) <NU (cid:12)A(Q)(cid:12)(cid:12)A(Q)∗(cid:12) (2π)2 (2π)2 (cid:12) k (cid:12)(cid:12) k(cid:48) (cid:12) (cid:90) d2k (cid:12) (cid:12)2 ≤NU (cid:12)A(Q)(cid:12) (2π)2 (cid:12) k (cid:12) =U. Due to the normalization of the eigenspinors, the assumption (2.8), and the inequality (A7), (cid:88) (cid:12)(cid:12)χ−k−Q,σ,−,α(cid:12)(cid:12)(cid:12)(cid:12)χk,σ,−,α(cid:12)(cid:12)(cid:12)(cid:12)χ∗−k(cid:48)−Q,σ,−,α(cid:12)(cid:12)(cid:12)(cid:12)χ∗k(cid:48),σ,−,α(cid:12)(cid:12)<1 (A11) α=A,B holds for almost all k, k(cid:48). This allows the use of the strict inequality in the third-last inequality from the chain of inequalities (A10). The penultimate inequality from the chain of inequal- is non-vanishing under the assumption (2.8). ities (A10) is a consequence of H¨older’s inequality (cid:90) d2k (cid:12) (cid:12) (cid:18)(cid:90) d2k (cid:12) (cid:12)2(cid:19)1/2 (cid:12)A(Q)(cid:12)≤ (cid:12)A(Q)(cid:12) . (A12) 3. Proof of Eq. (2.10b) (2π)2 (cid:12) k (cid:12) (2π)2 (cid:12) k (cid:12) The last inequality in the chain of inequalities (A10) fol- We are now going to work exclusively with the single- lows from the representation particle eigenstates of the band Hamiltonian (2.2e). We start with the completeness relation (cid:90) d2k (cid:12) (cid:12)2 1 (cid:12)A(Q)(cid:12) = (A13) (2π)2 (cid:12) k (cid:12) N (cid:88) (cid:88) |r,α(cid:105)(cid:104)α,r|=11 (A16a) of the normalization (A1e) in the thermodynamic limit. r∈Aα=A,B In summary, we have shown that onthebipartitelatticeΛ=A∪B. ThenormalizedBloch (cid:10) (cid:11) Φ |(H +N )|Φ >0 (A14) states |ϕ (cid:105) are defined in terms of the single-particle σ,Q e σ,Q k,σ,λ eigenstates χ by their overlaps holds in the thermodynamic limit under the assump- k,σ,λ,α tion (2.8). In other words, the many-body gap e+ik·r (cid:10) (cid:11) (cid:104)α,r|ϕ (cid:105):= √ χ (A16b) ∆σ,Q := Φσ,Q|H|Φσ,Q −(cid:104)Ψσ|H|Ψσ(cid:105)>0 (A15) k,σ,λ N k,σ,λ,α 8 for any given (Ising) spin σ =↑,↓ and band λ = ±. The resolves the identity since normalization and phase factors are chosen so that the orthonormality condition (cid:88) ψ† ψ =δ δ (A17e) r,z,σ,λ r,z(cid:48),σ,λ(cid:48) z,z(cid:48) λ,λ(cid:48) (cid:104)ϕ |ϕ (cid:105)=δ (A16c) k,σ,λ k,σ,λ(cid:48) λ,λ(cid:48) r∈A with λ,λ(cid:48) = ± holds for any given wavenumber k ∈ BZ and any given spin σ =↑,↓. The overlap (A16b) is in- for any given spin σ =↑,↓. We want to estimate the variant under the translation k→k+Q for any Q that profile in space of the Wannier states (A17a). belongs to the reciprocal lattice of sublattice A owing to For that purpose, we consider the spread functional the periodicity χ =χ . (A16d) k,σ,λ,α k+Q,σ,λ,α R(2) :=(cid:104)ψ |r2|ψ (cid:105) σ,λ z=0,σ,λ z=0,σ,λ (A18) WannierstatesaredefinedintermsoftheBlochstates −|(cid:104)ψ |r|ψ (cid:105)|2. z=0,σ,λ z=0,σ,λ by the unitary transformation |ψ (cid:105):= √1 (cid:88) e−ik·z|ϕ (cid:105) (A17a) Here, we are assuming, for simplicity, that there is only z,σ,λ k,σ,λ N onebandλthatisoccupied. Ifmorebandsareoccupied, k∈BZ we have to carry out a summation over all the occupied for any given unit cell of sublattice A labeled by z and bands. Observe that by translational invariance R(2) is σ,λ any given spin σ =↑,↓. The orthonormality (A16c) of left unchanged under the global translation r,z → r+ the Bloch states thus carries over to the orthonormality R,z+RforanylatticevectorR. Hence,thechoicez =0 in Eq. (A18) can be done without loss of generality. (cid:104)ψ |ψ (cid:105)=δ (A17b) z,σ,λ z,σ,λ(cid:48) λ,λ(cid:48) We rewrite Eq. (A18), following Ref. [13], as of the Wannier states for any given unit cell labeled by z of sublattice A and any given spin σ =↑,↓. For any λ and α, the representation of the Wannier state on the R(2) :=R(2) +R˜(2), (A19a) bipartite lattice Λ=A∪B is the overlap σ,λ I|σ,λ σ,λ R(2) :=(cid:104)ψ |r2|ψ (cid:105) ψ :=(cid:104)r,α|ψ (cid:105) I|σ,λ z=0,σ,λ z=0,σ,λ r,z,σ,λ,α z,σ,λ (cid:88) = √1 (cid:88) e−ik·z(cid:104)r,α|ϕ (cid:105) − |(cid:104)ψz=0,σ,λ|r|ψz(cid:48),σ,λ(cid:105)|2, (A19b) k,σ,λ z(cid:48) N k∈BZ R˜(2) := (cid:88) |(cid:104)ψ |r|ψ (cid:105)|2. (A19c) 1 (cid:88) σ,λ z=0,σ,λ z(cid:48),σ,λ = e+ik·(r−z)χ (A17c) z(cid:48)(cid:54)=0 N k,σ,λ,α k∈BZ for any given unit cell of sublattice A labeled by z and BothR(2) andR˜(2) arenon-negativequantitiesthatcan any given spin σ =↑,↓. The set of Wannier spinors I|σ,λ σ,λ be expressed in terms of k-space summations as follows. (cid:0) (cid:1) ψ = ψ (A17d) First, r,z,σ,λ r,z,σ,λ,α R(2) = (cid:88) 1 (cid:88) (cid:104)(cid:16)∂ χ† (cid:17)(cid:16)∂ χ (cid:17)−(cid:16)∂ χ† χ (cid:17)(cid:16)χ† ∂ χ (cid:17)(cid:105) I|σ,λ N kj k,σ,λ kj k,σ,λ kj k,σ,λ k,σ,λ k,σ,λ kj k,σ,λ j=x,y k∈BZ (cid:88) 1 (cid:88) (cid:104) (cid:105) = (cid:104)∂ χ |∂ χ (cid:105)−(cid:104)∂ χ |χ (cid:105)(cid:104)χ |∂ χ (cid:105) N kj k,σ,λ kj k,σ,λ kj k,σ,λ k,σ,λ k,σ,λ kj k,σ,λ j=x,y k∈BZ (cid:88) 1 (cid:88) (cid:0) (cid:1) = N (cid:104)∂kjχk,σ,λ| 11−|χk,σ,λ(cid:105)(cid:104)χk,σ,λ| |∂kjχk,σ,λ(cid:105) (A20a) j=x,y k∈BZ (cid:88) 1 (cid:88) ≡ (cid:104)∂ χ |Q |∂ χ (cid:105) N kj k,σ,λ k,σ,λ kj k,σ,λ j=x,y k∈BZ 1 (cid:88) (cid:2) (cid:3) = tr g , N k,σ,λ k∈BZ 9 where and A¯ := 1 (cid:88) A (A21c) Q :=11−|χ (cid:105)(cid:104)χ |, (A20b) σ,λ|j N k,σ,λ|j k,σ,λ k,σ,λ k,σ,λ k denotetheBerryconnectionandtheaverageoftheBerry is the single-particle projector operator on all the Bloch connection, respectively. Under the gauge transforma- states orthogonal to |χ (cid:105), the 2×2 matrix [g ] k,σ,λ k,σ,λ tion has the components |χk,σ,λ(cid:105)→eiϕk|χk,σ,λ(cid:105), (A22) (cid:104) (cid:105) g :=Re (cid:104)∂ χ |Q |∂ χ (cid:105) , k,σ,λ|i,j ki k,σ,λ k,σ,λ kj k,σ,λ it can be shown that RI(|2σ),λ remains invariant while R˜σ(2,λ) (A20c) does not. labeled by the Euclidean indices i,j ∈ {x,y} of two- We will now establish a lower bound on the gauge in- dimensional space, and tr denotes the trace over i,j ∈ variantquantityR(2) . Tothisend,wenoticethat,since {x,y}. Second, I|σ,λ Q isaprojectionoperatorwitheigenvalues0,1,then, k,σ,λ for any single-particle state |Ψ (cid:105), it follows that k,σ,λ R˜(2) = (cid:88) 1 (cid:88) (cid:16)A −A¯ (cid:17)2, (A21a) σ,λ N k,σ,λ|j σ,λ|j 0≤(cid:104)Ψ |Q |Ψ (cid:105). (A23) j=x,y k k,σ,λ k,σ,λ k,σ,λ In particular, if we choose where |Ψ (cid:105):=|∂ χ (cid:105)±i|∂ χ (cid:105), (A24) ±,k,σ,λ kx k,σ,λ ky k,σ,λ A :=i(cid:104)χ |∂ χ (cid:105), (A21b) the inequality (A23) delivers k,σ,λ|j k,σ,λ k k,σ,λ j (cid:2) (cid:3) (cid:16) (cid:17) tr g ≥∓i (cid:104)∂ χ |∂ χ (cid:105)−(cid:104)∂ χ |∂ χ (cid:105) . (A25) k,σ,λ k k,σ,λ k k,σ,λ k k,σ,λ k k,σ,λ x y y x Summing (A25) over k and taking the thermodynamic limit N →∞, delivers R(2) = Ac (cid:90) d2ktr(cid:2)g (cid:3) I|σ,λ (2π)2 k,σ,λ BZ A (cid:90) d2k (cid:104) (cid:105) ≥ ± c (cid:104)∂ χ |∂ χ (cid:105)−(cid:104)∂ χ |∂ χ (cid:105) (A26) 2π 2πi kx k,σ,λ ky k,σ,λ ky k,σ,λ kx k,σ,λ BZ A = ± c C , 2π σ,λ where A is the area of the unit cell of sublattice A and Appendix B: Competing Phases in the Hubbard c C are the spin- and band-resolved Chern numbers. model σ,λ Since R(2) ≥0, it then follows I|σ,λ Any translationally invariant electronic single-particle Hamiltonianisdescribedinmomentumspacebythedis- A persions and Bloch eigenfunctions of its bands. Given R(2) ≥ c |C |, (A27) I|σ,λ 2π σ,λ thehoppingamplitudesofatight-bindingmodelinposi- tion space, the dispersions and Bloch eigenfunctions are uniquelydetermined. Onceinteractionsareincluded,the many-body ground state of the model depends crucially as a lower bound, proportional to the band Chern num- on the properties of the dispersions (bandwidths; shapes ber,onthegaugeinvariantpartofspreadoftheWannier of Fermi surface, e.g., nesting; ...) and the properties states(A17a). ItwasshowninRef.[13]that,inthetopo- of the Bloch states or Wannier states (Chern numbers, logicalphaseoftheHaldanemodel3,R(2) isfinitewhile I|σ,λ localization in position space, ...). The power of our ap- R˜(2) displays a logarithmic divergence which is related proach, the flattening of the band with the factor w σ,λ k to the non-zero Chern number of the bands. fromEq.(2.2),istodisentanglethepropertiesofthedis- 10 a) b) c) π 3 1.4 gical ky12 t1 olo 00 1.2 p 1 o t 2 –π3 1.0 –π3 2 1 00 1kx2 π3 A B π 3 0.8 0.3 ky2 0.25 vial 001 0.2 tri 1 0.6 0.15 0.1 2 –π3 0.05 –π3 2 1 00 1k 2 π3 0.4 x 10-3 FIG. 3. (Color online) Comparison of the topological (top; √ 0.2 t =1/ 2,µ =0)andnon-topological(bottom;t =0,µ = 2√ s 2 s 1/ 2) single-particle model. (a) The shaded area represents the Fermi see of the lower band at the commensurate filling 0.0 0.2 0.4 0.6 0.8 1.0 fractionN =N whenκ<1. (b)Theeigenspinorχ ,when g e k,σ interpretedasapointonthesurfaceoftheunitsphere,swipes out the full surface of this sphere (a small portion of this FIG. 4. (Color online) Numerical exact diagonalization re- spherenearonepole)ask takesvalueseverywhereintheBZ sults for flat bands κ = 1 for the system size L = L = 25 for the topological (non-topological) band structure. (c) The x y at the commensurate filling fraction N =N. Plotted is the spread of the Wannier states (A17c) in real space indicates e value of the many-body excitation gap ∆/U between the en- theirdelocalized(localized)characterforthetopological(non- ergy of the fully spin polarized state S = N and the lowest topological) band structure. energystateinthesectorwithonespinflippedS =N−2asa function of g :=(2/π)arctan|µ /t | and the nearest-neighbor s 2 hopping parameter t . The thick red line corresponds to the persionsfromthoseofthespinorstosomeextendandto 1 parameterchoicemadefortheplotinFig.1oftheLetter. As study their effects individually. discussed in the text, the spinors of the Bloch band become fully sublattice polarized in the limit t → 0 and the fer- 1 romagnetic ground state becomes degenerate in energy with 1. Possible phases states from other spin sectors (∆→0). Let us list the anticipated ground states for some lim- with t (cid:28) U, i.e., on the t-J model. An antiferro- iting properties of the spinors and the dispersions for magnetic ground state will be selected. An alter- N =N electrons when the two pairs of bands are sepa- e native way to lift the macroscopic degeneracy is to rated by a gap in the absence of interactions. add longer range interactions, in which case other 1. (Almost) flat band and sublattice-polarized spinors. ground states may be stabilized25. Note that the If the band is completely flat (κ = 1), the ground fullysublatticepolarizedspinorsdiscussedhereim- state is macroscopically degenerate in the absence ply that the non-interacting model is topologically of interactions. Moreover, when the Bloch states trivial. arefullysublatticepolarized, i.e., whenχ† =(1,0) k 2. Flat band and not sublattice polarized spinors. As say, then the kinetic energy becomes diagonal in was shown in the Letter, if the spinors are suffi- position space [ψ† = δ (1,0)]. Hence, any r,z,σ,− r,z ciently spread between the two sublattices, i.e., if state with no less and no more than one electron condition(2.8)ismet,thegroundstateinthelimit persiteonsublatticeAwillbeagroundstateofthe of flat bands is an Ising ferromagnet with full spin Hamiltonian, regardless of the spin-configuration polarization. of the electrons. Therefore, the ground state re- mains macroscopically degenerate, in spite of the 3. Strongly dispersing unnested band. For a strongly presence of the Hubbard interaction. This degen- dispersingbandwithoutanestedFermisurface,the eracycanbeliftedbyintroducingasmallbutfinite Fermi liquid ground state is expected to be stable bandwidth, while keeping the full sublattice polar- against sufficiently small repulsive Hubbard inter- ization of the spinors intact. Then, the low-energy actions. One generic instability at finite interac- degrees of freedom of the model map on the con- tion strength is Stoner ferromagnetism which, in ventional one-band Hubbard model at half filling contrast to the flat band ferromagnetism, does not

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