Topological states on the breathing kagome lattice
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Topological states on the breathing kagome lattice
Adrien Bolens
1, 2 and Naoto Nagaosa
3, 4 Institute of Physics, ´Ecole Polytechnique F´ed´erale de Lausanne (EPFL), 1015 Lausanne, Switzerland Department of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan RIKEN Center for Emergent Matter Sciences (CEMS), Wako, Saitama 351-0198, Japan Department of Applied Physics, University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan (Dated: April 29, 2019)We theoretically study the topological properties of the tight-binding model on the breathingkagome lattice with antisymmetric spin-orbit coupling (SOC) between nearest neighbors. We showthat the system hosts nontrivial topological phases even without second-nearest-neighbor hopping,and that the weakly dispersing band of the kagome lattice can become topological. The main resultsare presented in the form of phase diagrams, where the Z topological index is shown as a function ofSOC (intrinsically allowed and Rashba) and lattice trimerization. In addition, exact diagonalizationis compared with effective low-energy theories around the high-symmetry points. We find that theweakly dispersing band has a very robust topological property associated with it. Moreover, theRashba SOC can produce a topological phase rather than hinder it, in contrast to the honeycomblattice. Finally, we consider the case of a fully spin polarized (ferromagnetic) system, breakingtime-reversal symmetry. We find a phase diagram that includes systems with finite Chern numbers.In this case too, the weakly dispersing band is topologically robust to trimerization. I. INTRODUCTION
A series of theoretical predictions and experimentalobservations of topological band insulators (TBIs) hasstimulated the emergence of a wide collection of topolog-ical quantum materials and topological phenomena [1, 2].The topological phases are classified according to an in-variant in the bulk, as was originally shown for the non-interacting integer quantum Hall effect [3], whereas gap-less excitations appears at the border between topolog-ically distinct phases. Time-reversal invariant nonmag-netic insulators have been shown to reveal a topological Z classification which divides them into two categoriesdescribed by the Z topological invariant ν : trivial in-sulators ( ν = 0), adiabatically connected to a trivialstate, and “topological” insulators ( ν = 1) that are notconnected to a trivial state without a bulk gap closure[4, 5]. In the original Kane-Mele model, intrinsically al-lowed spin-orbit coupling (SOC) between second nearestneighbor was shown to be essential to achieve a topolog-ical nontrivial phase [4]. In addition, both the inclusionof Rashba SOC, which breaks the reflection symmetryacross the plane of the two-dimensional (2D) system, andof an inversion asymmetric on-site potential drive the in-sulator in the trivial phase when sufficiently strong.The topological properties of the tight-binding modelon the kagome lattice have also been investigated in thepast [6–12]. In particular, 2D organometallic topologi-cal insulators have been predicted for both the hexag-onal and kagome lattices [12–15]. Recently, coordina-tion nanosheets with atomic thickness have been inten-sively studied both experimentally and theoretically be-cause of their attractive physical and chemical properties[16]. The synthesis of such materials is an exciting wayof potentially engineering new materials with nontrivialtopological properties. In addition, on the kagome lat-tice, topological properties of Floquet-Bloch band struc- tures have been studied in Floquet systems [17], and themagnon bands of kagome magnets with SOC have beenshown to be topological and lead to a finite magnon ther-mal Hall conductivity [18–20].On the kagome lattice, in contrast to the honey-comb one, there is no inversion symmetry centered onthe middle of the nearest-neighbor bonds, and anti-symmetric SOC is allowed between nearest neighbors[21, 22]. Kagome systems with such intrinsic SOC be-tween nearest neighbors were shown to host nontrivialphases [9, 11, 12]. Moreover, the energy spectrum of thetight-binding model on the kagome lattice has an extradispersion-less band which can also become topologicalthanks to SOC [9]. Topological flat bands have beenstudied in the context of the fractional quantum Hall ef-fect and topological flat-band lattice models [11, 23–25],and are an important feature of some TBIs.In real materials, the mirror symmetry about the planecontaining the 2D system is often not protected, resultingin Rashba SOC between nearest neighbors, in addition tothe intrinsic SOC. For instance, the breaking of the mir-ror symmetry can be caused by the interaction with asubstrate, by buckling of the 2D material, or simply byan external electric field [14, 26]. In addition, inversionsymmetry-breaking lattice trimerization (i.e., the breath-ing kagome lattice) can also be important for materialson the kagome lattice [27, 28].In this paper, we thus study systems with both typesof SOC and with the lattice trimerization. We show thatthe inclusion of Rashba SOC drives the system into atopological phase by itself, in contrast to graphene whereit hinders the nontrivial phase. In addition, we show thatthe flat band is intrinsically topological, in a very robustway. We note that the trimerization term has alreadybeen considered in Ref. [9], but only perturbatively, andnot in combination with Rashba SOC.The present paper is structured as follows. First, in FIG. 1. (a) Breathing kagome lattice with the hopping termsof Eqs. (5) and (6) indicated and the definition of the latticevectors a , a , and a . White, black, and gray dots representssites in the A, B, and C sublattices, respectively. (b) SOCvectors ˆd , ˆd , and ˆd defined in Eq. (3) for pure RashbaSOC ( α = 0). (c) Same SOC vectors for pure intrinsic SOC( α = π/ Sec. II, we introduce the model considered. We thencalculate the Z topological index in Sec. III and drawthe full phase diagram considering nearest-neighbor SOC(intrinsic and Rashba) and lattice trimerization.Subsequently, in Sec. IV, we consider the same modelbut for a system with ferromagnetically polarized spinsand derive an effective three-band tight-binding model.We calculate the Chern numbers of the three bands anddraw the phase diagram accordingly.Finally, we conclude in Sec. V. II. MODEL
Let us consider the tight-binding model on the kagomelattice as depicted in Fig. 1. The spin-independentHamiltonian is written as H = − X R [ t c † R B c R A + t ′ c † ( R − a ) B c R A + t c † R C c R B + t ′ c † ( R − a ) C c R B + t c † R A c R C + t ′ c † ( R − a ) A c R C ] + H . c ., (1)where c † R α = ( c † R α ↑ , c † R α ↓ ) with α ∈ { A, B, C } for thethree sublattices. Here R labels the position of theunit cell composed of three sites forming an upward triangle, and a = ( − / , −√ / a = (1 , a = ( − / , √ /
2) [see Fig. 1(a)]. The trimerization ofthe lattice, corresponding to the breathing kagome lat-tice, is obtained by considering different hopping ampli-tudes on the two distinct sets of triangles: t = t + δ for upward triangles and t ′ = t − δ for downward trian-gles. Here δ is trimerization parameter, and (1) can bedecomposed as H = H ( t ) + H trim ( δ ).The SOC coupling is introduced as a spin-dependenthopping between nearest neighbors. Furthermore, we as-sume that SOC hopping amplitudes are similarly affectedby the trimerization of the lattice. The general nearest-neighbor SOC Hamiltonian is H SO = i X R [ λ c † R B ( ˆd · σ ) c R A + λ ′ c † ( R − a ) B ( ˆd · σ ) c R A + λ c † R C ( ˆd · σ ) c R B + λ ′ c † ( R − a ) C ( ˆd · σ ) c R B + λ c † R A ( ˆd · σ ) c R C + λ ′ c † ( R − a ) A ( ˆd · σ ) c R C ]+H . c ., (2)with λ = λ + δ λ and λ ′ = λ − δ λ so that λ/λ ′ = t/t ′ (i.e., δ λ = λ · δ/t ). Here, σ = ( σ x , σ y , σ z ) are the Paulimatrices and the ˆd i ’s are unit vectors which depend onthe type of SOC considered. The intrinsic SOC (i.e.,it respects all symmetries of the lattices including themirror symmetry across the 2D plane) corresponds to ˆd = ˆd = ˆd = ˆz [see Fig. 1(c)] with amplitudes denotedby λ = λ i = λ , i + δ λ, i and λ ′ = λ ′ i = λ , i − δ λ, i . In a sys-tem breaking the mirror symmetry, we also have RashbaSOC with amplitudes denoted by λ = λ R = λ , R + δ λ, R and λ ′ = λ ′ R = λ , R − δ λ, R , which corresponds to in-plane ˆd i ’s perpendicular to their respective bonds, as depictedin Fig. 1(b). We consider the general situation with bothtypes of SOC, so that ˆd = cos α ( √ , − ,
0) + sin α (0 , , , ˆd = cos α (0 , ,
0) + sin α (0 , , , ˆd = cos α ( − √ , − ,
0) + sin α (0 , , , . (3)Here α is the angle between the 2D plane and the ˆd i ’ssuch that α = 0 corresponds to pure Rashba SOC and α = π/ α = λ , i /λ , R .The full Hamiltonian H = H + H trim + H SO , i + H SO , R is H = − X R [ c † R B ˆ t c R A + c † ( R − a ) B ˆ t ′ c R A + c † R C ˆ t c R B + c † ( R − a ) C ˆ t ′ c R B + c † R A ˆ t c R C + c † ( R − a ) A ˆ t ′ c R C ] + H . c ., (4)where each hopping is accompanied by a spin rotation Γ K + Γ Γ−4−202 ε / t (a) (i)(ii)(iii) k x k y Γ K + Γ (b) Γ K + Γ Γ ε / t (c) FIG. 2. (a) Band structure with pure Rashba SOC ( α = 0)with (i) δ = 0 , λ , R = 0 . t , (ii) δ = 0 . t , λ , R = 0, and(iii) δ = λ , R = 0. (b) Brillouin zone with high-symmetrypoints. (c) Band structure with δ = 0 . t and λ , R = 0 . t which corresponds to a band-touching at the K ± points. defined byˆ t i = q t + λ + λ exp (cid:16) iφ ˆd i · σ (cid:17) , (5)ˆ t ′ i = q t ′ + λ ′ + λ ′ exp (cid:16) iφ ˆd i · σ (cid:17) , (6) φ = − arctan q λ , i + λ , R t . (7)After a Fourier transform c † R α = √ N P k c † k α e − i k · R ,where N is the number of unit cells, we obtain the BlochHamiltonian H ( k ). In the ( c † k A , c † k B , c † k C ) basis, the 6 × H ( k ) = − t † + ˆ t ′† e − i k · a ˆ t + ˆ t ′ e i k · a ˆ t + ˆ t ′ e i k · a t † + ˆ t ′† e − i k · a ˆ t † + ˆ t ′† e − i k · a ˆ t + ˆ t ′ e i k · a . (8)Without SOC nor trimerization, the Hamiltonian (8) hasthree doubly degenerate bands: two pairs of dispersingbands similar to graphene with two Dirac points at K ± =( ± π/ ,
0) and an additional pair of flat bands as shownin Fig. 2(a)(iii).A gap opens at the Dirac points when introducing ei-ther SOC or trimerization. However at the quadraticband touching point, a gap does not open from trimer-ization alone but only opens when either type of SOC isintroduced. In this case, the top bands are not perfectlyflat anymore, but are only weakly dispersing as long asSOC is small ( λ ≪ t ), as can be seen in Fig. 2(a)(i)and 2(c).We also note that the transformation φ → φ ± π isequivalent to t i → − t i and t ′ i → − t ′ i so that the topolog-ical properties at 1/3 filling with angle φ are the same asat 2/3 filling with angle φ ± π . III. Z TOPOLOGICAL INVARIANTA. Results
First we briefly discuss the topology of the systemwithout trimerization, δ = 0. In this case, the system isinvariant under inversion symmetry and the Z topologi-cal invariant ν is easily obtained from the parity eigenval-ues ξ m (Γ − ) of the 2 m th occupied bands with m = 1 , T -invariant)points Γ = Γ , Γ = (2 π, , Γ = ( π, π/ √ , Γ =( π, − π/ √
3) [29]. For the kagome lattice, the parity oper-ator at the T -invariant points, which is spin-independent,is concisely written as a 3 × P ( k ) = diag(1 , e i a · k , e − i a · k ) , (9)where we chose the inversion center on an A site sothat P (Γ ) = (1 , , P (Γ ) = (1 , − , − P (Γ ) =(1 , − , P (Γ ) = (1 , , − Z index for the m th Kramers pair is given by ( − ν m = Q i ξ m (Γ i ) and,by explicitly calculating the wave functions of the BlochHamiltonian (8), we find that ν m = 1 for both the bot-tom and top pairs of bands independently of φ . Hence,the overall Z index is always ν = 1 for both 1/3 and 2/3filling (as long as the different Kramers pairs of bands arenot touching). Most interestingly, this is not only truefor the intrinsic SOC, but also for the Rashba SOC.Motivated by this preliminary result, we show the fullphase diagram in Fig. 3 as a function of both φ and theinversion symmetry-breaking δ for several values of α .The phase diagrams are plotted for α = 0 (pure Rashba), π/ π/
3, and π/ p ( k ) = Pf[ h u i ( k ) |T | u j ( k ) i ] , (10)which is the method originally described by Kane andMele [4]. Here T is the time-reversal operator, | u i ( k ) i are the band wave functions, and i ranges over the filledbands. Due to the D symmetry of the breathing kagomelattice, only zeros along the high-symmetry [Γ − Γ ] linesegment (along which k y = 0) are relevant. We observedthat the band-touchings happen at either k = K ± or k = Γ. All gap-closing-gap-opening transitions at the K ± Dirac points (which always happen simultaneouslyfor both points) result in a change of the topological in-dex. When φ = 0 or π , the pair of flat bands touches an-other dispersive pair of bands “quadratically” at k = Γ,but the topological index is not affected. Finally, in theunphysical case where t = − t ′ (or δ → ±∞ ), all sixbands are degenerate at the Γ point and ν changes in anontrivial way [30]. B. Discussion
In the following, we derive effective Hamiltonians intwo different limits to give some analytical insight to the −101 φ / π (a) α = 02/3 filling: ν = 1 ν = 0 (b) α = π/6 (c) α = π/3 (d) α = π/2-2 0 2δ/t −101 φ / π (e) α = 01/3 filling: -2 0 2δ/t (f) α = π/6 -2 0 2δ/t (g) α = π/3 -2 0 2δ/t (h) α = π/2 FIG. 3. Phase diagrams as a function of δ/t and φ for various SOC vector directions corresponding to α = 0 (pure Rashba), π/ π/
3, and π/ ν = 0 and 1 regions are indicated in white and gray, respectively. Both 2/3 filling (top)and 1/3 filling (bottom) cases are shown, which are related by a φ → φ ± π transformation. phase diagram: (1) the λ , δ ≪ t limit (small SOC andtrimerization) relevant at 1 / t ′ ≪ t limit (close to full trimerization δ/t = 1), relevant at2 /
1. Small SOC and trimerization
The gap-closing-gap-opening behavior near the Diracpoints, relevant at 1 / k = K ± . The effective Hamiltonian is obtainedafter a projection onto the subspace of the two pairs ofbands forming the Dirac cones. For the trimerization andintrinsic SOC, H τsσ ( k ) = v F ( k x σ x τ z + k y σ y )+ ( √ λ , i τ z s z − δ ) σ z + const ., (11)with v F = √ t / τ i , s i , and σ i ( i ∈ { x, y, z } ) arePauli matrices which refer to the K ± valleys, spins, andbands (in a given basis), respectively. The constant termin Eq. (11) includes all terms independent of the σ i ’s. Asimilar result has already been obtained in Ref. [9]. Wesee from Eq. (11) that the band at 1 / | λ , i | > √ | δ | .For the λ , R Rashba term, we obtain no contributioncoming out of this projection, which apparently contra-dicts what we observe numerically in Fig. 3. However,in order to obtain Eq. (11), the projection was done onthe eigenstates at the K ± points calculated without SOC and with δ = 0. Hence, it corresponds to a first orderperturbation theory in H trim , H SO , i , and H SO , R . Thefirst contribution from the Rashba SOC comes at secondorder (in λ , R and δ ) and we find H SO , R → √ δ λ, R ( σ x s y τ z − σ y s x ) + λ , R t τ z s z σ z , (12)where we can also write δ λ, R = λ , R δ/t . Hence, RashbaSOC drives the system in topological phase by itself when λ , R > | δ t | + O ( δ ). Both the linear and quadraticbehaviors can be seen around the ( δ/t = 0 , φ = 0) pointin Fig. 3(h) and Fig. 3(e), respectively.The Hamiltonians (11) and (12) have a limitation:They do not tell anything about the topology of the topbands. In particular, the gap opened at the quadraticband touching point Γ is stable under the inclusion ofthe trimerization δ , even when a gap closes at a Diracpoint, as shown in Fig. 2(c), which reflects the fact that δ by itself does not open a gap at Γ.The full phase diagram reveals that the pair of weaklydispersing bands is intrinsically topological. As soon asthe quadratic band touching at Γ is lifted, e.g., by aninfinitesimally small SOC contribution, the now well-defined Z index of the top bands is nontrivial untila band-closing-band-opening happens at K ± for largetrimerization or SOC: δ or λ ∼ t . Thus, the topologicalphase at 2 / δ . As opposed to the 1 / δ (as −403 ε / t k x −403 ε / t FIG. 4. Energy dispersion of a system periodic in the ˆx direc-tion and with edges in the ˆy direction, with α = 0 (RashbaSOC), φ = 0 . π , and (top): δ = 0 .
0, (bottom): δ = 0 . long as − < δ/t < ˆy direction for α = 0 (Rashba SOC), φ = 0 . π , and both δ = 0 (top)and δ = 0 . t (bottom). We clearly see the change inthe edge states at 1 / δ = 0 to the trivial phase at δ = 0 . t , while at 2 /
2. Fully trimerized limit
We now derive a second effective Hamiltonian for largetrimerization δ , valid near the fully trimerized limit δ = t , or t ′ = 0. We also assume small SOC, λ ≪ t .The fully trimerized system consists of independent up-ward triangles. The now localized system has no energydispersion, and the Hamiltonian is simply H ( k ) = − t ⊗ I , (13)where I is the identity in spin space, so that there aretwo pairs of bands with energy E = t and one pair withenergy E = − t . The effective theory at 2 / E = t states. Thedispersion comes from the small hopping t ′ between thenearly isolated triangles. Expanding H in Eq. (1) with finite t ′ around K ± we obtain H → v ′ F ( − k x σ z + k y σ x ) τ z + 3 t ′ σ y τ z , (14)where v ′ F = √ t ′ /
2, and the σ i Pauli matrices now referto the two E = t bands [we have chosen two real eigen-vectors of Eq. (13)]. The intrinsic SOC on each trimergives a gap term, H SO , i → √ λ i σ y s z , (15)and the contribution from Rashba SOC only comes atsecond order in λ R and t ′ , H SO , R → √ λ ′ R ( σ x s x + σ z s y ) τ z − λ t σ y s z , (16)where we can also write λ ′ R = λ R t ′ /t .When the intertriangle hopping t ′ = 0, the completelylocalized system is obviously topologically trivial. Thiscorresponds to the δ/t = 1 lines in Fig. 3. Then, as t ′ increases, the previously E = t degenerate bands splitin a topological manner. For small SOC and t ′ ( λ, t ′ ≪ t ), the phase boundary is given by Eqs. (14)-(16): Thesystem is topological if | λ i | < √ | t ′ | / λ R < | t t ′ | for pure Rashba SOC. The linearand quadratic behavior can be seen around the ( δ/t =1 , φ = 0) point in Fig. 3(d) and Fig. 3(a), respectively.It is interesting to note that, effectively, the trimerizedsystem at 2 / H andthe on-site two-band Hamiltonian obtained after project-ing out the third pair of bands.The effective model with, say, intrinsic SOC can bewritten on a triangular lattice (made of the original unitcells at positions R ) as H eff = X R " X i =1 f † R + a i ( ˆ T i ⊗ I ) f R + H . c . + √ λ i X R f † R ( σ y ⊗ s z ) f R , (17)where f † R = ( f † R ,a, ↑ , f † R ,a, ↓ , f † R ,b, ↑ , f † R ,b, ↓ ), a, b = 1 , f † R ,a,s ( f † R ,b,s ) is the appropriate linearcombination of c † R ,α,s ( α = A, B, C ) corresponding tothe a ( b ) band. The complexity of the model is hiddenin the hopping matrices ˆ T i , which are different for thethree directions a , a , and a , and break the inversionsymmetry on each bond (i.e., ˆ T † i = ˆ T i ):ˆ T = t ′ (cid:18) −√
30 0 (cid:19) , ˆ T = t ′ (cid:18) − −√ √ (cid:19) , ˆ T = t ′ (cid:18) √ (cid:19) . (18) FIG. 5. Energy dispersion of the effective Hamiltonian in thenearly-trimerized limit with SOC given in Eqs. (14) and (15)near the phase transition: λ i . √ t ′ / The band dispersion of this effective model is shownin Fig. 5 near the band-closing-band-opening transition.Note that in the topological phase near the trimerizedlimit, there is no overall gap between the two pairs ofband when including only linear terms in t ′ /t (the thirdband in Fig. 5 has exactly the same energy at k = K ± as the lower bands at k = Γ), but higher order terms in t ′ /t will gap the two pairs of band completely (i.e., thereis a full gap in the full model with all bands).Finally, we note that in the unphysical δ/t / ∈ [ − , t and t ′ with different signs), we can still define the topologicalindex, but the system is in a semimetal phase with nooverall gap, even in the full model. IV. SPIN-POLARIZED SYSTEM
Let us finally consider a ferromagnetically ordered sys-tem where, on each site, the magnetic dipole points alongthe same unit vector ˆe = (sin Θ cos Φ , sin Θ sin Φ , cos Θ).In the following, we show that the system can host non-trivial Chern numbers for the three bands, as shown inRef. [31] for the δ = 0 case, and draw the resulting phasediagram for δ = 0 in Fig. 6.Such a constraint can be enforced in our systemthrough the double-exchange model H DE = H − A X R X α = A,B,C c † R α ( ˆe · σ ) c R ,α , (19)where H is given in Eq. (4). The coupling to the localizedmagnetic moments (with A >
0) typically originates fromHund’s coupling, but it could also originates from an on-site repulsion U , based on the Hubbard model at themean field level. For large A , we effectively only keepthe component of the spinor parallel to ˆe , | χ + i , suchthat we are left with a three-band model with effectivehopping t eff i = h χ + | ˆ t i | χ + i ≡ e iϕ i | t eff i | ,t ′ eff i = h χ + | ˆ t ′ i | χ + i ≡ e iϕ i | t ′ eff i | . (20) FIG. 6. Diagram of the different phases characterized by theChern numbers of the three bands. The Chern numbers areindicated as ( c , c , c ), where c , c , and c are the Chernnumbers of the lowest, middle, and top bands, respectively.The dotted and hashed regions are topological at 1 / / We then define ϕ = ϕ + ϕ + ϕ , and choose the gaugewhere ϕ = ϕ = ϕ = ϕ/ | t eff1 | = | t eff2 | = | t eff3 | , so that the C sym-metry is preserved. For intrinsic SOC, this is always thecase and we have the relation tan( ϕ/
3) = cos(Θ) tan( φ )[where φ quantifies the SOC, as defined in Eq. (7)], sothat ϕ is finite unless ˆe lies in the plane. For RashbaSOC, we can only have the C symmetry for Θ = 0 , π ,in which cases t eff i = t cos( φ ), and ϕ = 0. The resultspresented in the following are thus only directly relevantfor a system with intrinsic SOC. A. Results
Because of the C symmetry, the gap-closing-gap-opening transitions only happen at k = K + or k = K − ,where a pair of bands forms Dirac cones. Additionally,when ϕ = 0 or π , the perfectly flat band touches one ofthe dispersive bands at k = Γ. In Fig. 6, the differentregions are delimited by lines in parameter space alongwhich a pair of bands are touching (calculated analyt-ically). The three Chern numbers, one for each band,are indicated in the different regions. The Chern num-bers in each region were calculated numerically using thealgorithm described in Ref. [32]. B. Discussion
In Ref. [31], the same effective model with δ = 0has already been studied. They have shown that unless ϕ = 0 , π , the top and bottom bands have a finite Chernnumbers. The same result is deduced from the δ = 0 linein Fig. 6.The effect of trimerization is very different for 1 / / Z index. At 1 / δ (of the order ofthe SOC) will drive the system in a trivial phase, after agap-closing-gap-opening transition at either K + or K − ,as is usually the case for inversion-symmetry breakingterms in Chern insulators (e.g., for the Haldane model).However, we see that the weakly dispersing band staystopological in a large region around δ = 0 and ϕ = 0.Just like in Sec. III, its topological properties are robustto trimerization. This can be seen in Fig. 6; the systemis topological at 2 / k =Γ, the weakly dispersing band hosts a nontrivial Chernnumber for any values of δ , except close to the fully-trimerized limit.Finally, for Rashba SOC, we expect a similar robust-ness of the topology of the flat bands (as in Sec. III).However, we also expect a more complex phase diagrambecause of the lack of C symmetry, and the additionalpotential gap-closing-gap-opening transitions. We do notexplicitly calculate this phase diagram here, which woulddepend on extra parameters (the direction of ˆe ), in ad-dition to ϕ and δ . V. CONCLUSION
In this paper, we investigated the topological proper-ties of electrons on the (breathing) kagome lattice withsymmetry-allowed SOC between nearest neighbors. Wedrew several phase diagrams with respect to three param-eters: trimerization, Rashba SOC, and intrinsic SOC.We considered the topological phases associated with both the Z index, in the original time-reversal invariantsix-band model, and the Chern numbers, in an effectivespin-polarized three-band model. Interestingly, the over-all topological properties (i.e., the effects of the differentparameters) are similar for those two models.We showed that the effects of Rashba SOC and theinversion symmetry-breaking term (the trimerization) isdifferent than what is usually expected (for instance ingraphene).We showed that the pair of flat bands (or the single flatband for the spin-polarized system) is intrinsically topo-logical. As soon as the quadratic band touching at Γ islifted, e.g., by an infinitesimally small SOC contribution,the now well-defined topological index of the top bandsis nontrivial until a band-closing-band-opening happensclose to the fully-trimerized limit δ = t . We stress thatthe weakly dispersing bands are topologically very robustagainst the perturbation (here the trimerization), unlikethe other dispersive bands.In addition, the inclusion of Rashba SOC was shownto drive the system into a topological phase by itself,in contrast to graphene where it hinders the nontrivialphase.Finally, because of the possibility to host a nontrivialtopological phase close to the fully-trimerized system, weshowed how the same topological phase could be obtainedfrom a two-band system on a triangular lattice , whichcould be relevant for experimental implementations. VI. ACKNOWLEDGMENTS
A. B. acknowledges the Leading Graduate Course forFrontiers of Mathematical Sciences and Physics (FMSP)for the encouragement of the present paper.N. N. was supported by JSPS KAKENHI Grant Num-ber JP26103006, JP18H03676, and ImPACT Program ofCouncil for Science, Technology and Innovation (Cabinetoffice, Government of Japan), and JST CREST GrantNumbers JPMJCR16F1, and JST CREST Grant Num-ber JPMJCR1874, Japan. M. Z. Hasan and C. L. Kane,Rev. Mod. Phys. , 3045 (2010). B. A. Bernevig and T. L. Hughes,
Topological insula-tors and topological superconductors (Princeton UniversityPress, Princeton and Oxford, 2013). D. J. Thouless, M. Kohmoto, M. P. Nightingale, andM. den Nijs, Phys. Rev. Lett. , 405 (1982). C. L. Kane and E. J. Mele,Phys. Rev. Lett. , 146802 (2005). C. L. Kane and E. J. Mele,Phys. Rev. Lett. , 226801 (2005). G. Liu, P. Zhang, Z. Wang, and S.-S. Li,Phys. Rev. B , 035323 (2009). H.-M. Guo and M. Franz, Phys. Rev. B , 113102 (2009). Z. Wang and P. Zhang,New Journal of Physics , 043055 (2010). G. Liu, S.-L. Zhu, S. Jiang, F. Sun, and W. M. Liu,Phys. Rev. A , 053605 (2010). Z.-Y. Zhang, J. Phys. Condens. Matter. , 365801 (2011). E. Tang, J.-W. Mei, and X.-G. Wen,Phys. Rev. Lett. , 236802 (2011). Z. F. Wang, N. Su, and F. Liu,Nano Letters , 2842 (2013). Z. Liu, Z.-F. Wang, J.-W. Mei, Y.-S. Wu, and F. Liu,Phys. Rev. Lett. , 106804 (2013). Z. F. Wang, Z. Liu, and F. Liu,Nature Communications , 1471 EP (2013). Z. F. Wang, Z. Liu, and F. Liu,
Phys. Rev. Lett. , 196801 (2013). H. Maeda, R. Sakamoto, and H. Nishihara,Langmuir , 2527 (2016). L. Du, X. Zhou, and G. A. Fiete,Phys. Rev. B , 035136 (2017). L. Zhang, J. Ren, J.-S. Wang, and B. Li,Phys. Rev. B , 144101 (2013). R. Chisnell, J. S. Helton, D. E. Freedman, D. K.Singh, R. I. Bewley, D. G. Nocera, and Y. S. Lee,Phys. Rev. Lett. , 147201 (2015). P. Laurell and G. A. Fiete,Phys. Rev. B , 094419 (2018). I. Dzyaloshinsky, Journal of Physics and Chemistry ofSolids , 241 (1958). T. Moriya, Phys. Rev. , 91 (1960). T. Neupert, L. Santos, C. Chamon, and C. Mudry,Phys. Rev. Lett. , 236804 (2011). K. Sun, Z. Gu, H. Katsura, and S. Das Sarma,Phys. Rev. Lett. , 236803 (2011). S. Yang, Z.-C. Gu, K. Sun, and S. Das Sarma,Phys. Rev. B , 241112(R) (2012). C.-C. Liu, H. Jiang, and Y. Yao,Phys. Rev. B , 195430 (2011). F. H. Aidoudi, D. W. Aldous, R. J. Goff, A. M. Z.Slawin, J. P. Attfield, R. E. Morris, and P. Lightfoot,Nature Chemistry , 801 EP (2011). J.-C. Orain, B. Bernu, P. Mendels, L. Clark, F. H.Aidoudi, P. Lightfoot, R. E. Morris, and F. Bert,Phys. Rev. Lett. , 237203 (2017). L. Fu and C. L. Kane, Phys. Rev. B , 045302 (2007). This is not observed in Fig. 3 because t and t ′ change signwhen δ goes from ∞ to −∞ . K. Ohgushi, S. Murakami, and N. Nagaosa,Phys. Rev. B , R6065 (2000). T. Fukui, Y. Hatsugai, and H. Suzuki,Journal of the Physical Society of Japan74