Ferromagnetism and spin excitations in topological Hubbard models with a flatband
Xiao-Fei Su, Zhao-Long Gu, Zhao-Yang Dong, Shun-Li Yu, Jian-Xin Li
FFerromagnetism and spin excitations in topological Hubbard models with a flatband
Xiao-Fei Su,
1, 2
Zhao-Long Gu, Zhao-Yang Dong,
3, 1
Shun-Li Yu,
1, 4 and Jian-Xin Li
1, 4, ∗ National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China School of Physics and Electronic Information, Huaibei Normal University, Huaibei 235000, China Department of Applied Physics, Nanjing University of Science and Technology, Nanjing 210094, China. Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China (Dated: January 11, 2019)We study the spin-1 excitation spectra of the flatband ferromagnetic phases in interacting topo-logical insulators. As a paradigm, we consider a quarter filled square lattice Hubbard model whosefree part is the π flux state with topologically nontrivial and nearly-flat electron bands, which canrealize either the Chern or Z Hubbard model. By using the numerical exact diagonalization methodwith a projection onto the nearly-flat band, we obtain the ferromagnetic spin-1 excitation spectrafor both the Chern and Z Hubbard models, consisting of spin waves and Stoner continuum. Thespectra exhibit quite distinct dispersions for both cases, in particular the spin wave is gapless forthe Chern Hubbard model, while gapped for the Z Hubbard model. Remarkably, in both cases, thenonflatness of the free electron bands introduces dips in the lower boundary of the Stoner continuum.It significantly renormalizes the energies of the spin waves around these dips downward and leadsto roton-like spin excitations. We elaborate that it is the softening of the roton-like modes thatdestabilizes the ferromagnetic phase, and determine the parameter region where the ferromagneticphase is stable.
I. INTRODUCTION
Electronic bands with nonzero topological indices re-side on the center of a substantial amount of topologicalphenomena in condensed matter physics . It was pro-posed in a pioneering work by Haldane that a spinlessfermionic model on a honeycomb lattice exhibits integerquantum Hall effect without an external magnetic field.This model, serving as the first example of Chern insu-lator, breaks the time reversal symmetry with a complexnext-nearest-neighbor hopping and is characterized by anonzero Chern number . Later, the concept was general-ized to time reversal symmetric systems with spin-orbitalcoupling (SOC), such as the graphene and HgTe/CdTequantum wells . The SOC there generates complexhopping terms similar to that proposed by Haldane butwith opposite chiralities for electrons with up spins anddown spins, resulting in the quantum spin Hall insulatorcharacterized by a Z index.The lattice models with nontrivial band topology sharemuch similarity with the two-dimensional electron gas(2DEG) under a strong magnetic field with Landau lev-els, e.g. the existence of topologically protected gaplessedge states . Thus more novel phases other thanthe Chern insulator or Z insulator are expected whenCoulomb interactions are taken into account, as is simi-lar to the fractional quantum Hall effect in the 2DEGwith Landau levels. However, different from Landau lev-els, energy bands in lattice models usually have noneli-gible dispersions, which weakens the effect of Coulombinteractions. Therefore, in recent years, much effort hasbeen devoted to the design and search of tight-bindingmodels that host nearly-flat electron bands with non-trivial topology . Analogous exotic phases, such asthe fractional Chern insulator and fractional topologi-cal insulator were numerically verified to emerge in such nearly-flat topological bands when strong Coulomb inter-actions are turned on .Another involved intriguing phenomenon arising fromCoulomb repulsions in flat or nearly-flat bands is the itin-erant ferromagnetism . It was proved by Tasaki andMielke that the ground state of a flat electron band witha filling factor not more than but sufficiently close to 1 / . Afterwards,this ferromagnetism was shown to be stable against smallnonflatness of the electron bands if and only if the Hub-bard interaction exceeds a critical value . Spin wave ex-citations over this ferromagnetic ground state were alsostudied and itinerant topological magnons have beenreported quite recently .The interplay between flatband ferromagnetism andnontrivial band topology enriches the related physics. Infact, ferromagnetism is essential in the generation of sta-ble fractional Chern insulators in the proposals where thespin degrees of freedom of electrons are considered .Furthermore, ferromagnetism can also lead to possi-ble high-temperature quantum anomalous Hall effect(QAHE) when the nearly-flat topological band is half-filled . In this paper, we study the ferromagnetism andspin excitations from the ferromagnetic ground state innearly-flat topological bands. As a paradigm, we con-sider a square lattice Hubbard model whose free part is a π flux model with topologically nontrivial and nearly-flatelectron bands. Depending on the nearest-neighbor hop-ping, the model Hamiltonian either explicitly breaks thetime-reversal symmetry but preserves the spin SU (2) ro-tation symmetry (Chern Hubbard model), or preservesthe time-reversal symmetry but explicitly breaks thespin SU (2) rotation symmetry ( Z Hubbard model).When the model is quarter filled (or correspondingly, thelower nearly-flat band is half filled), the ground state a r X i v : . [ c ond - m a t . s t r- e l ] J a n is spin fully polarized due to the ferromagnetism andexhibits QAHE because of the nonzero Chern numberof a single-spin band. Consequently, the charge excita-tions are gapped and the low energy physics is domi-nated by the one-spin-flip excitations. These spin-1 exci-tations have been studied by a generalized bosonizationscheme where the interacting fermionic model is mappedto a free bosonic model describing spin-wave excitationsat the harmonic approximation . The ferromagnetismwas shown to be stable against such spin wave excita-tions, which are gapless in the Chern Hubbard modeland gapped in the Z Hubbard model. However in thisbosonization scheme, the free part of the electron modelplays no role in the spin wave excitations other than con-tributing a global constant, suggesting that it should faildue to the competition between the kinetic energy andpotential energy of electrons when the nonflatness of theelectron bands is not negligible . In fact, in a strictlylocal periodic tight-binding model, an energy band witha nonzero Chern number cannot be exactly flat . There-fore it remains an open question on whether the groundstate is stable against the spin-1 excitations and how thenonflatness of the electron bands manifests itself in thespin-1 excitation spectra in such models.To elucidate these questions, we adopt the numericalexact diagonalization method with a projection onto thelower nearly-flat band to take close investigations on thespin-1 excitations of the models. A critical magnitudeof the Hubbard interaction is found for both the ChernHubbard model and Z Hubbard model, below whichthe ferromagnetic phase is unstable. Furthermore, thespin-1 excitation spectra are shown to consist of collec-tive modes (spin waves) and individual modes (Stonercontinuum). For the Chern Hubbard model, the spinwave is gapless while for the Z Hubbard model, thespin wave is gapped. Remarkably, for both cases, thenonflatness of the free electron bands introduces dips ofthe lower boundary of the Stoner continuum, and signif-icantly renormalizes the energies of the collective modesaround these dips downward, which leads to roton-likespin wave excitations. With the increase of this non-flatness, the energy of the induced roton-like modes goesdown and finally touches zero, which results in the desta-bilization of the ferromagnetic phase. Therefore, we elab-orate the mechanism of the instability of this flat-bandferromagnetism as the softening of the emergent roton-like modes with the increase of noflatness. We also makea comparison of our results in the flatband limit withthose obtained by the bosonization scheme . Quali-tative agreements are observed for the Chern Hubbardmodel, yet notable discrepancy appears for the Z Hub-bard model. We attribute the discrepancy to the multi-magnon processes, which are ignored in the bosonizationscheme but treated exactly in our method after the pro-jection.The rest of the paper is organized as follows. In Sec.II, we introduce the Chern Hubbard model and Z Hub-bard model studied in this paper, discuss the nontrivial (a)
XY MO k x k y (b) FIG. 1. (Color online) (a) Schematic representation of the π -flux model. Blue and red solid circles denote the A and B sub-lattices, respectively. The nearest-neighbor hopping ampli-tudes (solid black lines) are equal to t exp( iα σ π/
4) (see text)along the direction of the arrows, the next-nearest-neighborhopping amplitudes are equal to t (dashed green lines) and − t (dashed purple lines). The shaded area denotes the unitcell. (b) The first Brillouin zone. The next-nearest-neighbordistance is set to one, so Γ = (0 , X = ( π, Y = (0 , π ), M = ( π, π ), O = ( π , π ). band topology of their free parts, interpret the emergenceof QAHE resulting from the interplay between flatbandferromagnetism and nontrivial band topology, and formu-late the exact diagonalization method with a projectiononto the lower nearly-flat band on details. In Sec. III,we discuss the phase diagram and elaborate the spin-1excitation spectra of both models. Section IV provides asummary and discussion. II. MODEL AND METHODA. Introduction to model
We consider a generalized π -flux Hubbard model onthe square lattice, whose Hamiltonian can be written as H = H + H U , where H is the spinfull genralization ofthe original spinless model proposed in Ref. 21, H = (cid:88) (cid:104) ij (cid:105) ,σ ( t ij,σ c † iσ c jσ + H.c.) + (cid:88) (cid:104)(cid:104) ij (cid:105)(cid:105) ,σ ( t ij c † iσ c jσ + H.c.) , (1)and H U is the Hubbard interaction H U = U (cid:88) i n i ↑ n i ↓ . (2)Here, c † iσ ( c iσ ) creates (annihilates) a spin σ electron atsite i , n iσ = c † iσ c iσ is the particle-number operator, (cid:104) ij (cid:105) denotes the nearest-neighbor (NN) bonds and (cid:104)(cid:104) ij (cid:105)(cid:105) thenext-nearest-neighbor (NNN) bonds. As shown in Fig.1(a), the spin-dependent NN hopping amplitude t ij,σ andthe spin-independent NNN hopping amplitude t ij aregiven by t ij,σ = t exp (cid:16) iδ ij α σ π/ (cid:17) (3)and t ij = t δ ij , (4)respectively. Here, δ ij = +1 if the NN electron hopping isalong the direction of the solid black arrow and δ ij = − δ ij = +1 if the NNNelectron hopping is along the dashed green lines and δ ij = − α σ breaks the time-reversal symmetry butpreserves the spin SU (2) rotation symmetry if α ↑ = α ↓ =+1, whereas it preserves the time-reversal symmetry butbreaks the spin SU (2) rotation symmetry if α ↑ = +1 and α ↓ = − π phase as it hops around a plaquette alongthe direction of the black arrows as indicated in Fig.1(a). Therefore, H describes free electrons hopping on asquare lattice in the presence of a fictitious staggered π -flux pattern . For the time-reversal-symmetry-breakingcase, α ↑ = α ↓ , the fluxes experienced by spin-up elec-trons and spin-down electrons are the same, while forthe time-reversal-symmetry-preserving case, α ↑ = − α ↓ ,they are opposite. B. Topology of free term H Gapped noninteracting fermionic systems can be topo-logically classified by their Hamiltonians in the momen-tum space in the presence of symmetries . After theFourier transformation, the free part H of our modelreads H = (cid:88) k σ ψ † k σ h k σ ψ k σ , (5)where ψ † k σ = ( c † A k σ , c † B k σ ) and h k σ = D k σ · τ . (6)Here τ = ( τ , τ , τ ) is a 2 × τ , τ , τ are the three Pauli matrices for the sublattice degrees offreedom. The components of D k σ are given by D , k = 2 √ t cos k x k y ,D , k = 2 √ t α σ sin k x k y ,D , k = 2 t (cos k x − cos k y ) . (7) H can be diagonalized with the transformation c A k σ = µ , k σ d k σ + µ , k σ f k σ ,c B k σ = µ ∗ , k σ f k σ − µ ∗ , k σ d k σ , (8) where µ , k σ = D , k − iα σ D , k (cid:112) D k ( D k + D , k ) ,µ , k σ = D k + D , k (cid:112) D k ( D k + D , k ) , (9)with D k = (cid:113) D , k + D , k + D , k . The diagonalized H is given by H = (cid:88) k σ ε d ( k ) c † k σ c k σ + (cid:88) k σ ε f ( k ) f † k σ f k σ , (10)where ε d ( k ) = − D k , ε f ( k ) = D k . It can be seen thatthere exists a gap between the d band and f band when t (cid:54) = 0 and t (cid:54) = 0.When there is a gap between the d band and f band, these bands can be shown to be topologically non-trivial by calculating their Chern numbers (for the time-reversal-symmetry-breaking case) or Z indices (forthe time-reversal-symmetry-preserving case). The Chernnumber for a single spin component of the d band or the f band can be expressed in terms of the coefficients D i, k , C d/fσ = ± π (cid:90) BZ d k ˆ D k σ · ( ∂ k x ˆ D k σ × ∂ k y ˆ D k σ ) = ± α σ , (11)with ˆ D k σ ≡ D k σ /D k .When the system breaks the time-reversal symmetry,i.e. α ↑ = α ↓ = 1, C d ↑ = C d ↓ = 1 and C f ↑ = C f ↓ = −
1, thetotal Chern number of the d band is C d = C d ↑ + C d ↓ = 2.Therefore, the ground state of H will be a noninter-acting Chern insulator and exhibits quantum anomalousHall effect(QAHE) when the lower d band is fully filled.When the system preserves the time-reversal symmetry,i.e. α ↑ = − α ↓ = 1, C d ↑ = − C d ↓ = 1 and C f ↑ = − C f ↓ = − d band is zero. However,the Z index, which is defined as ν = 12 ( C ↑ − C ↓ ) mod 2 , (12)of the d band is 1 and nontrivial. As a consequence, theground state of H will be a noninteracting Z insulatorand exhibits quantum spin Hall effect(QSHE) when thelower d band is fully filled. C. Emergence of QAHE in half-filled nearly-flattopological bands
For a free fermionic system hosting an energy bandwith a nonzero Chern number or Z index, the distin-guished phenomenon resulting from this nontrivial bandtopology, such as QAHE or QSHE, only manifests itselfwhen the topological band is fully filled. At any frac-tional filling, the ground state of such a system will bea trivial metal. Intriguingly, when the Coulomb inter-actions between electrons is introduced, the physics ofthe nontrivial band topology becomes more involved, es-pecially when the band is nearly flat so that the effectsof the Coulomb interactions are highly enhanced. Com-bined with strong Coulomb interactions, nontrivial topo-logical phases can emerge from fractionally filled topolog-ical bands. In this article, we are interested in half-filledstrongly-correlated nearly-flat topological bands whereQAHE can arise . The essence for the occurrence ofthis nontrivial phase is the emergence of itinerant ferro-magnetism on nearly-flat bands , which fully polarizesall electron spins. Therefore, only one spin componentof the topological band will be fully filled exactly, whichleads to QAHE due to the nonzero Chern number of thatspin component of the band.The Chern Hubbard model and Z Hubbard modelon square lattice described above serve as the paradigm,where half filling of the lower electron band correspondsto quarter filling of the whole system because of the ex-istence of AB sublattices. When t /t takes values ina selected region, the d band and f band are quite flatin that the flatness ratio ∆ /W , which is defined as theratio of the gap ∆ between these two bands to the band- width W of the lower band, can be as large as 4 . .In the next subsection, we will introduce the exact di-agonalization method with a projection onto the lowernearly-flat electron band to study the spin-1 excitationsof the ferromagnetic phases in these two topological Hub-bard models. D. Exact diagonalization with projection
Exact diagonalization method with a projection ontothe low-energy Hilbert space has been widely ap-plied to systems that host flat or nearly-flat energybands . This approach applies when the en-ergy gap between the flat or nearly-flat band and otherbands is larger than the Coulomb interaction.For the model we study in this article, the relevantlow-energy subspace is the lower electron band, i.e. the d band. Let P denote the corresponding projector, thenthe Hamiltonian after the projection is P † HP = (cid:88) k σ ε d ( k ) d † k σ d k σ + UN (cid:88) a =1 , (cid:88) kk (cid:48) q ( µ ∗ a, k + q ↑ µ ∗ a, k (cid:48) − q ↓ µ a, k (cid:48) ↓ µ a, k ↑ ) d † k + q ↑ d † k (cid:48) − q ↓ d k (cid:48) ↓ d k ↑ . (13)Let | FM (cid:105) denotes the spin-up fully polarized state on the d band, | FM (cid:105) = (cid:89) k ∈ FBZ d † k ↑ | (cid:105) , (14)where FBZ denotes the first Brillouin zone [see Fig. 1(b)]and | (cid:105) is the fermion vaccum. Here, d † k ↑ creates a spin-up electron with momentum k . Then the basis of thespin-1 excitations with a center-of-mass momentum q over this reference state can be written as | k i (cid:105) q = d † k i − q ↓ d k i ↑ | FM (cid:105) , (15)which labels a spin-1 scattering channel with the index k i . The dimension of this Hilbert space scales linearlywith respect to the number of electron momentums , soa much larger system can be numerically accessed thanthe usual exact diagonalization without projection. It en-ables us to analyze the properties of the spin-1 excitationspectra in detail in the whole first Brillouin zone ratherthan some restricted discrete points solely. The matrixelement of the projected Hamiltonian on this spin-1 ex-citation basis can be easily obtained after some algebra, q (cid:104) k j | P † HP | k i (cid:105) q = ε d ( k i − q ) − ε d ( k i ) + UN (cid:88) a =1 , (cid:88) p (cid:54) = k i | µ a, p ↑ | | µ a, k i − q ↓ | δ k j , k i − UN (cid:88) a =1 , µ ∗ a, k i ↑ µ a, k j ↑ µ ∗ a, k j − q ↓ µ a, k i − q ↓ (cid:0) − δ k j , k i (cid:1) . (16)Here δ k j , k i is the Kronecker delta function. Then the fullspin-1 excitation spectra can be obtained by the diago-nalization of the matrix whose elements are defined by Eq. (16). It is noted that | FM (cid:105) is the true ground stateonly if the whole spin-1 excitation spectra have no neg-ative energies. Thus we can use this as the criterion to t / t U / t NFMFM (a) 0.4 0.5 0.6 0.7 0.8 t / t U / t NFMFM (b) U / W FIG. 2. (Color online) Phase diagrams of the quarter-filled (a)Chern Hubbard model and (b) Z Hubbard model. The col-ormap represents the ratio
U/W of the lower electron band,where U is the Hubbard interaction strength and W is thelower electron bandwidth. Red star in (a) marks the param-eter used in Figs. 3 and 4, blue stars in (a) mark the param-eters used in Fig. 5, green stars in (b) mark the parametersused in Fig. 6. determine the destabilization of the ferromagnetic phase.We also want to give some remarks on the flatbandlimit in the framework of this method. The flatband-limitHamiltonian shares the same single-particle eigenfunc-tions as well as the interaction terms with the originalone, yet it has exactly-flat single-particle energy bands .Thus the free part h flat k σ of the flatband-limit Hamiltoniancan be defined as h flat k σ = h k σ | ε d ( k ) | = ˆ D k σ · τ. (17)Apparently, the eigenfunctions of h flat k σ are the same with h k σ ’s but its energy bands are exactly flat with the eigen-values being ±
1. To approach this limit, long-range hop-ping terms in the real space must be included . To re-veal the physics related to the nonflatness of the d band,we also calculated the spin-1 excitation spectra in theflatband limit for comparison. This can be done by sim-ply throwing away the [ ε d ( k i − q ) − ε d ( k i )] δ k j , k i term inEq. (16), because of the same single-particle eigenfunc-tions shared by the flatband-limit Hamiltonian and theoriginal one. III. NUMERICAL RESULTS
Before the detailed discussion on the spin-1 excitationspectra of the quarter-filled Chern Hubbard model and Z Hubbard model, we present their phase diagrams first,which are shown in Fig. 2. Here, the colormap repre-sents the ratio of the Hubbard interaction strength U tothe lower electron bandwidth W . FM denotes the ferro-magnetic phase and NFM denotes the non-ferromagneticphase. As discussed in Sec. II D, the phase boundary isdetermined by the onset parameter at which the spin-1excitation spectrum starts to acquire zero energy at a fi-nite momentum. The phase diagrams of these two modelsappear quite similar. It is obvious that a critical Hubbard X M q00.20.40.6 E / U q q (a) k x / k y / (b ) k x / ) k x / ) k x / ) FIG. 3. (Color online) (a) Spin-1 excitation spectra of theChern Hubbard model in the FM phase. Green solid linesdenote the spin waves and the grey region denotes the Stonercontinuum. Blue solid lines denote the upper and lowerboundaries of the Stoner continuum determined by the band-splitting picture shown in Fig. 4(a). Black arrows markthe local minima of the Stoner continuum. (b -b ) Spectralweights for the lowest four eigen levels of the spin-1 excita-tion spectra with q = (0 , t = 1 . t = 0 .
490 and U = 2 .
3. Green dashed lines in (a) representthe corresponding spectra in the flatband limit. interaction strength is needed to maintain the ferromag-netically ordered ground state when the electron bandhas finite nonflatness. Furthermore, the phase bound-aries always lie near the contour line with
U/W = 2 . A. Chern Hubbard model
In this subsection, we focus on the quarter-filled ChernHubbard model. In Fig. 3(a), we present the spin-1 ex-citation spectra in the FM phase along a high symme-try path in the first Brillouin zone with the parametersmarked by the red star in Fig. 2(a). It is shown that thereare two kinds of obviously different excitations: one con-tains two low-lying modes labeled by the green solid linesexhibiting well defined band structures, and the otherforms a high-energy continuum as labeled by the shadedarea. We find that, for these two kinds of modes, the pat-terns of the contributions from each spin-1 particle-holescattering channel of electrons, which are embodied inthe eigenvectors Ψ q ( k i ) of Eq. (16), are quite different.In Fig. 3(b), the spectral weights of each scattering chan-nel, i.e. | Ψ q ( k i ) | as a function of k i , for the four lowestlevels with the center-of-mass momentum q = (0 ,
0) (Γpoint) are shown. It is clear that for the modes in thegreen solid lines [Fig. 3(b ) and Fig. 3(b )], the spectralweights come from a quite broad range of scattering chan-nels; indeed, the spectral weight for the lowest mode atΓ point is even homogeneous for all scattering channels.However, for those in the shaded area [Fig. 3(b ) andFig. 3(b )], the spectral weights almost come from a sin-gle scattering channel. Therefore, the modes in the greensolid lines are collective spin-1 excitations and are iden-tified as the spin waves, while the modes in the shadedarea are individual spin-1 excitations and are identifiedas the Stoner continuum. The spin waves consist of anacoustic branch and an optical branch due to the AB sublattice of the model. The acoustic branch is gaplessat the Γ point and disperses quadratically away from theΓ point, which is the character of a ferromagnetic exci-tation with a Goldstone mode as the spin-fully polarizedground state spontaneously breaks the spin SU (2) rota-tion symmetry. Additionally, the acoustic and opticalbranches are degenerate along the X - M path.In the flatband limit, the spin-1 excitation spectra asshown in Fig. 3(a) as the green dashed lines show severalsimilarities with the dispersive case, including the consti-tution of the spin waves and Stoner continuum exhibitingas a flat line at E = 0 . U , the gapless acoustic spin waveand its quadratical dispersion near the Γ point, and thedegeneracy between the acoustic and optical branchesalong the X - M path. Comparing results in the flatbandlimit with those obtained by the generalized bosoniza-tion method at the harmonic approximation [see Fig. 4in Ref. ], one will find that two results are qualitativelyquite consistent. In the scheme adopted in Ref. , theignorance of the interactions between magnons suggeststhat only the single-magnon excitations are captured. So,this consistence indicates that the interactions betweenmagnons shows a negligible effect in the flatband limit.However, when the free band becomes dispersive, consid-erable distinct features appears. Now, the spin-wave en-ergies around the X = (0 , π ) and O = ( π/ , π/
2) pointsare strongly suppressed, as a result two local minima areformed at X and O which can be ascribed to the emer-gence of the roton-like spin-wave excitations. Further-more, the continuum exhibiting as a straight line in theflatband limit extends to a large grey region shown inFig. 3(a) with dispersive boundaries, and the local min-ima of its lower boundary coincide with those in the spin-wave dispersion, as indicated by the black arrows in Fig.3(a). This observation shows that it is the extending ofthe Stoner continuum arising from the nonflatness of the d band that pushes down the spin-wave excitations. Inparticular, it suggests that the couplings between the in- X M k0.00.51.0 ( EE m i n ) / U (a) MXY Oq q q q (b) k x / k y / (c ) -0.06 0.06 k x / (c ) -0.06 0.06 k x / (c ) -0.06 0.06 k x / (c ) -0.06 0.06 FIG. 4. (Color online) (a) Illustration of the splitting of thelower electron bands for up spins (blue solid line) and downspins (blue dashed line) in the presence of ferromagnetism ofthe Chern Hubbard model. (b) Positions of the local max-ima (red solid circles) of the spin-up bands and local minima(purple solid circles) of the spin-down bands in the first Bril-louin zone. Green arrows denote the corresponding scatteringchannels of the minima of the Stoner continuum marked inFig. 3(a). (c -c ) Spectral weights subtracted by those inthe flatband limit for the lowest four eigen levels of the spin-1excitation spectra with q = ( π, dividual excitations and spin waves would be responsiblefor the emergence of the roton-like spin waves.The formation of the dispersive boundary of the Stonercontinuum can be understood by a simple band-splittingpicture as shown in Fig. 4(a). In the presence of ferro-magnetism, the spin-down and spin-up electron bandsare split in energy which is roughly proportional to U (cid:104) m (cid:105) , with U the Hubbard interaction and (cid:104) m (cid:105) theaverage magnetic moment per site. For a quarter-filled electron model with two inequivalent sublattices, (cid:104) m (cid:105) = 1 /
2. Consequently, an individual spin-1 exci-tation, i.e., a mode in the Stoner continuum, corre-sponds to the excitation of an electron from the fully-filled spin-up band to the empty spin-down band withan energy proportional to the difference in the initialand final states plus the band splitting U/
2. Indeed,the δ function-like distribution of the spectral weightsof a mode in the Stoner continuum [as has be seen inFig. 3(b ) and Fig. 3(b )] implies that its excitationenergy can be approximated by the corresponding diag-onal term in Eq. (16). Numerically, we also find that theterm UN (cid:80) a =1 , (cid:80) p (cid:54) = k i | µ a, p ↑ | | µ a, k i − q ↓ | ∼ U . There-fore, the excitation energies of the Stoner continuum areroughly ε d ↓ ( k i − q )+ U − ε d ↑ ( k i ), which is consistent withthe above analysis. The lower and upper boundaries ofthe Stoner continuum determined in this way are plottedas the blue solid lines in Fig. 3(a), which fits extremelywell with the true boundaries of the shaded area, thusverifies the validity of this simple argument.With this picture, the position of a local boundarybottom of the Stoner continuum is determined by thetransferred momentum of the particle-hole scattering be-tween a spin-up maximum and a spin-down minimum.Thus, in Fig. 4(b), we mark the local maxima of thespin-up band with red solid circles and the local minimaof the spin-down band with purple ones. One can seethat the transferred momenta coincide completely withthe positions of the local bottoms in Fig. 3(a) which areindicated here by green arrows.To understand the renormalization of the spin wavesobserved above, in Fig. 4(c), we plot the spectral weightsof the four lowest levels subtracted by those in the flat-band limit at momentum q = ( π,
0) ( X point). Thedifferences concentrate around the scattering channelslabeled by k i = ( ± π,
0) and k i = (0 , ± π ). However,the results for the collective modes [Fig. 4(c ) and Fig.4(c )] are positive while those for the individual modes atthe Stoner continuum boundary bottom [Fig. 4(c ) andFig. 4(c )] are negative. Therefore a noticeable amountof spectral weights transfer from the latter to the former,which indicates a strong coupling between the spin wavesand the nearby Stoner continuum. This coupling renor-malizes the energies of the spin waves and leads to theoccurrence of a roton-like local minimum for the opticalbranch.The emerged roton-like spin waves are essential for thedestabilization of the ferromagnetic phase. In Fig. 5(a )and 5(b ), we plot the spin-1 excitation spectra of themodel near the FM/NFM phase boundary, with two setsof parameters marked by the blue stars in Fig. 2(a). Thecorresponding illustrations of the band-splitting pictureare shown in Fig. 5(a ) and 5(b ). Compared with Fig.3(a), it can be seen that the increase of the nonflatnessleads the boundary bottom marked by q to move towardlow energies further. In this process, a roton-like localminimum in the acoustic band appears nearby, whichcan be seen more clearly in Fig. 5(a ) and 5(b ). Withthe increase of the nonflatness of the electron band, theenergy of this newly formed roton-like mode goes downand touches zero, which leads to the destabilization ofthe ground state. From Fig. 5(a ) and 5(b ), we cansee that the ferromagnetic ground state is quite robustagainst the spin-1 flips until the top of the spin-up bandis approaching near to the bottom of the spin-down band.Considering that the energy splitting of these two bandsis U/
2, the destabilization point would be quite close to U/ ∼ W . B. Z Hubbard model
In this subsection, we focus on the quarter-filled Z Hubbard model. Its spin-1 excitation spectra along ahigh symmetry path in the first Brillouin zone with theparameters marked by the green stars in Fig. 2(b) areplotted in Fig. 6(a ) ( t = 0 . , U = 2 .
0) and Fig. 6(b )( t = 0 . , U = 2 . E / U q (a ) q X E / U (a ) X M ( EE m i n ) / U (a ) X M q00.10.20.30.40.5 E / U q (b ) q X E / U (b ) X M k0.00.51.0 ( EE m i n ) / U (b ) FIG. 5. (Color online) (a ) and (b ): Spin-1 excitation spec-tra of the Chern Hubbard model near the FM/NFM phaseboundary. Black arrow marks the minimum of the Stonercontinuum. (a ) and (b ): Low-energy parts of the spin-1 excitation spectra shown in (a ) and (b ) along the Γ- X path with an amplified resolution. (a ) and (b ): Illustrationof the splitting of the corresponding lower electron bands. t = 0 .
437 for (a) and t = 0 .
430 for (b). Other parametersare fixed at t = 1 . U = 2 . the Chern insulator discussed in Sec. III A, the low-lyinggreen solid lines are identified as the spin waves (collec-tive modes) and the high-energy shaded areas the Stonercontinuum (individual modes). Overall, the whole spec-tra are different from those in the Chern insulator shownin Fig. 3(a). In particular, the collective modes here aregapped because the Z Hubbard model explicitly breaksthe spin SU (2) rotation symmetry and no spontaneouscontinuous symmetry breaking occurs in the ferromag-netic phase. Besides, the acoustic and optical bands aredegenerate along the Γ- X path, instead of the X - M pathreported in the subsection III A.The corresponding spectra in the flatband limit arealso shown in Fig. 6(a ) and Fig. 6(b ) as the greendashed lines. Noticeable differences can be found whenwe compare our result in the flatband limit with that ob-tained by the generalized bosonization method [see Fig. 5in Ref. ]. On the one hand, the band bottom of the spinwaves in our results lies at the M points while theirs liesat the Γ point. On the other hand, the acoustic and op-tical branches are degenerate along the Γ- X path, whichis same as the results for a dispersive band as discussedin the above. This is in sharp contrast to that obtainedin Ref. , which are degenerate along the X - M path.These discrepancies are attributed to the effects of multi- E / U q (a ) X M ( EE m i n ) / U (a ) MXY Oq q (a ) X M q00.20.40.6 E / U q (b ) X M k0.00.51.0 ( EE m i n ) / U (b ) MXY Oq q (b ) FIG. 6. (Color online) (a ) and (b ): Spin-1 excitation spec-tra of the Z Hubbard model. Green solid lines denote thespin waves and grey regions denote the Stoner continuum.Green dashed lines represent the corresponding spectra in theflatband limit. Black arrows mark the minima of the Stonercontinuum. (a ) and (b ): Illustration of the splitting of thelower electron bands of the Z Hubbard model. (a ) and (b ):Positions of the corresponding local maxima (red solid circles)of the spin-up bands and local minima (purple solid circles) ofthe spin-down bands in the first Brillouin zone. Green arrowsdenote the corresponding scattering channels of the minimaof the Stoner continuum marked in (a ) and (b ). t = 0 . t = 0 .
741 for (b). Other parameters are fixed at t = 1 . U = 2 . magnon processes which are ignored in their scheme. Theabove observation suggests that the interactions betweenmagnons have an essential effect on the spin excitationsin the Z Hubbard model, though they have a negligibleeffect in the Chern Hubbard model as discussed above.Similar to the Chern Hubbard model, the introductionof the dispersion to the flatband leads to the appear-ance of three local minima (roton-like modes) along theΓ − X − M − Γ direction for both acoustic and opticalbands, respectively. The momentum positions of theseminima coincide the corresponding boundary bottoms ofthe Stoner continuum, suggesting that the renormaliza-tion of the spin-wave bands is due to the coupling be-tween the spin waves and Stoner continuum. In the band-splitting picture as proposed above, the position of a localboundary bottom of the Stoner continuum is determinedby the transfer momentum of the particle-hole scatter-ing between a spin-up band maximum and a spin-downband minimum. Therefore, we present in Fig. 6(a ) and6(b ) the dispersions for the lower electron bands in the Z Hubbard model showing the splitting of the spin-upand spin-down bands. The corresponding local maxima(red solid circles) of the spin-up bands and local minima(purple solid circles) of the spin-down bands in the firstBrillouin zone are shown in Fig. 6(a ) and 6(b ). Onecan see that the momentum positions of the boundarybottoms as marked by the black arrows in Fig. 6(a )and Fig. 6(b ) (here only q is plotted for illustration)match the transferred momenta between the spin-up andspin-down bands, which gives a strong support of thepicture based on the band splitting.From Fig. 6(a ) to Fig. 6(b ), the ratio of the Hub-bard interaction to the lower electron bandwidth U/W decreases. In this process, the roton-like mode with mo-mentum q approaches to zero, indicating the destabi-lization of the ferromagnetic ground state. So, the pa-rameters used to get the result in Fig. 6(b ) marks theboundary between a ferromagnetism phase and a nonfer-romagnetic phase as shown in the phase diagram Fig.2(b). From the band splittings shown in Fig. 6(a )and Fig. 6(b ), one can see that the relative energiesat the Γ and M points increases while that at the X ( Y )points decreases in this process. Consequently, althoughthe momentum positions of the forementioned bound-ary bottoms remain the same, the corresponding scat-tering channels are in fact different, which are shiftedfrom (0 , → ( π , π ) and ( − π, − π ) → ( − π , − π ) to( − π, → ( − π , π ) and (0 , − π ) → ( π , − π ), as shownin Fig. 6(a ) and Fig. 6(b ). In fact, we find that theenergy of the spin-up band at O almost overlaps withthe bottom of the spin-down band at X as shown in Fig.6(b ). This result demonstrates that the ferromagneticground state is also quite robust against spin flips un-til the indirect gap between the spin-down and spin-upbands vanishes, which means the phase boundary of theFM/NFM is also quite close to U/W = 2 from the dis-cussion in Sec. III A.
IV. SUMMARY AND DISCUSSION
In summary, we have considered the flatband ferro-magnetic phases in a quarter-filled Chern Hubbard modeland a quarter-filled Z Hubbard model in this paper. Byusing the numerical exact diagonalization method witha projection onto the lower nearly-flat electron band, wedetermine the critical Hubbard interaction strength be-low which the ferromagnetic phase is unstable, and elabo-rate the ferromagnetic spin-1 excitation spectra of thesemodels. Both spectra consist of collective modes (spinwaves) and individual modes (Stoner continuum). Forthe Chern Hubbard model, the spin wave is gapless whilefor the Z Hubbard model, the spin wave is gapped. Re-markably, in both cases, the nonflatness of the free elec-tron bands introduces dips in the lower boundary of theStoner continuum. As a result, it renormalizes signifi-cantly the energies of the collective modes around thesedips through the couplings between spin waves and in-dividual modes in the Stoner continuum, and leads toroton-like spin wave excitations. We find that the desta-bilization of the ferromagnetic phase arises from the soft-ening of the roton-like mode, whose energy goes downgradually with the increase of this nonflatness.We would like to remark that the downward renor-malization of the energies of the collective modes af-fected by the boundary bottoms of the Stoner contin-uum shares similarity with that observed in the two-dimensional antiferromagents, where the spin waves areinterpreted as bound states of confined spinons and thesuppression of energies of spin waves around the contin-uum bottom is attributed to their couplings to the nearbydeconfined spinon continuum . Intriguing flatbandphysics arises not only in models with repulsive interac-tions, but also in those with attractive ones , wherethe Bardeen-Cooper-Schrieffer superconducting state is the exact ground state in the flatband limit . Wehope our methods can be generalized to these systemsbeyond the flatband limit. In addition, although corre-lated nearly-flat topological bands are hard to realize inreal materials, they are possible to be designed in cold-atom systems. In fact great progress has been made veryrecently . We expect our results will stimulate re-lated investigations in future cold-atom experiments.
ACKNOWLEDGMENTS
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