Eight fold quantum Hall phases in a time reversal symmetry broken tight binding model
QQuantum anomalous spin Hall phases in time reversal symmetry broken quantum spinHall system
Sudarshan Saha,
1, 2, ∗ Tanay Nag, † and Saptarshi Mandal
1, 2, ‡ Institute of Physics, Bhubaneswar- 751005, Odhisa, India Homi Bhabha National Institute, Mumbai - 400 094, Maharashtra, India Institute für Theorie der Statistischen Physik, RWTH Aachen University, 52056 Aachen, Germany (Dated: February 15, 2021)We propose a time reversal symmetry (TRS) broken tight binding model, being an admixture ofKane-Mele and Haldane model where the intrinsic SOC is coupled with a staggered magnetic flux, toinvestigate the fate of quantum anomalous Hall insulator (QAHI) and quantum spin Hall insulator(QSHI) phases. We make resort to topological invariant namely, spin Chern number (C ↑ , C ↓ ) tocharacterize various topological phases. The phase diagram unveils many intriguing features such as,quantum anomalous spin Hall insulator (QASHI) phase, denoted by (0 , C ↓ ) and (C ↑ , , extendedcritical phase where spin Chern number does not work, multicritical points where three / fourtopological phase boundaries coalesce and many more in addition to the QSHI phase [ (C ↑ , C ↓ ) with C ↑ (cid:54) = C ↓ (cid:54) = 0 ] and QAHI phase [ (C ↑ , C ↓ ) with C ↑ = C ↓ (cid:54) = 0 ]. These topological phases are protectedby an effective TRS and a composite anti-unitary particle-hole symmetry leading to remarkablepropertis of edge modes. We find spin-selective, spin-polarized and spin-neutral edge transport inQASHI, QSHI and QAHI phases referring to the validity of bulk-boundary correspondence. Ouranalysis with low energy model gives us the correct understanding of the various topological phases.Moreover, our study further indicates that spin gap plays the key role in determining the robustnessof the topological invariant as it does not necessarily vanish at the Dirac points across a topologicalphase transition. Based on the current experimental advancements in solid state and cold atomicsystems, we believe that our proposals can be tested in near future. The integer quantum Hall effect (QHE), character-izing a two dimensional solid state system from thetopological view point, is one most intriguing phenom-ena in condensed matter community [1, 2]. The manybody Coulomb repulsion between the electrons furtherenriches the physics where fractional QHE is observed[3, 4]. However, remarkable features, associated withnon-interacting QH systems, have been continuouslycoming up in the last few years such as, quantum anoma-lous Hall effect (QAHE) [5, 6] and quantum spin Hall ef-fect (QSHE) [7–9] and many more [10–13]. The breakingof time reversal symmetry (TRS) is a necessary conditionfor the QHE, however, QAH insulator (QAHI) phase ex-hibits the quantized charge currents without the appli-cation of external magnetic field as long as the systembreaks TRS spontaneously [5]. On the other hand, thequanitized spin current is a signature of the TRS pro-tected QSH insulator (QSHI) phase that is immune tononmagnetic scattering. It is important to note that theintrinsic spin orbit coupling (SOC) causes a gap to openin the bulk of the system enabling the spin polarized edgestates known as helical edge modes, to traverse throughthe band gap [7].Coming to the demonstration of bulk boundary corre-spondence [14], it has been shown that the bulk invariantChern number [15] (spin Chern number [16]) can success-fully predict the number of edge states (spin polarized ∗ [email protected] † [email protected] ‡ [email protected] channels) in QAHI (QSHI) phases where quantum Hall(quantum spin Hall) conductivities are quantized [17, 18].Another widely used topological invariant namely, Z in-dex can equivalently classify TR invariant system [7, 19–23]. It may also be noted that mirror symmetry breakingRashba SOC term does not destroy the topological orderof the QSHI state even though the spin conservation nolonger holds. Thereafter it becomes an important ques-tion that what would be the fate of the QSHI phase inthe absence of TRS.In order to seach for the answers, TRS breaking termssuch as, exchange field [24, 25], magnetic doping [26, 27],and staggered magnetic flux [28], are introduced in QSHsystem to obtain QAH effect that is experimentally ac-cessible in certain parameter regimes. Remarkably, eventhough Z index fails to characterize the topological na-ture of the phase, spin Chern number persists to be a rele-vant topological invariant distinguishing a TR symmetry-broken QSHI phase from a QAHI phase. The QSH [7]and QAH [5] models have been generalized to varioustheoretical platforms [29, 30] and realized in experiments[31, 32]. All these studies motivate us to consider Kane-Mele model infused with Haldane model such that theTRS is broken by staggered magnetic flux associatedwith next nearest neighbour (NNN) hopping and intrinsicSOC term. To be precise, we ask the following questions:1. How do the Haldane and Kane-Mele phase diagramsmodify? 2. Are there any new topological phases apartfrom QSHI and QAHI phases? 3. Can spin Chern num-ber successfully describe all the phases?Given the above background, we now present our mainresults concisely. We first demonstrate that how do the a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Haldane (Kane-Mele) phases evolve with the introduc-tion of Rashba and SOC terms (NNN hopping and mag-netic flux) [see Fig. 1, Fig. 2 and Fig. 3]. The zeros ofbulk energy gap determine the topological phase bound-aries while these phases are associated with finite spingap [25, 33] and characterized by the spin Chern num-ber (C ↑ , C ↓ ) [19, 34]. The quantum anomalous spin Hallinsulator (QASHI) [QSHI] phases are denoted by (C ↑ , and (0 , C ↓ ) [ (C ↑ , C ↓ ) with C ↑ (cid:54) = C ↓ (cid:54) = 0 ] while QAHIphase is designated by C ↑ = C ↓ (cid:54) = 0 . In order to es-tablish the bulk boundary correspondence, we find spin-selective, -polarized and -neutral transport in QASHI,QSHI and QAHI phases, respectively while studying theband structure in semi-infinite geometry with zig-zagedge (see Fig. 4). We explain all these findings from thelow energy version of the model where we demonstratethe evolution of spin dependent Haldane gap with variousparameters. These topological phases are protected un-der emerging anti-unitary symmetries that couple withthe chirality of the flux. In essence, considering a simpleflux induced TRS broken QSHI model, our study uncov-ers many extraordinary features in a systematic mannerthat to the best of our knowledge, have not been probedso far and thus opens up the possiblity of practical deviceapplications in future.The Hamiltonian we consider here is given below, H = − t (cid:88) (cid:104) ij (cid:105) c † i c j + iV R (cid:88) (cid:104) ij (cid:105) c † i ( (cid:126)σ × (cid:126)d ij ) z c j + M (cid:88) i c † i σ z c i + t (cid:88) (cid:104)(cid:104) ij (cid:105)(cid:105) e iφ ij c † i c j + iV so √ (cid:88) (cid:104)(cid:104) ij (cid:105)(cid:105) e iφ ij ν ij c † i σ z c j (1)where c i represents the fermion spinor ( c i ↑ , c i ↓ ) ; V so and V R represent the SOC and Rashba interaction strength,respectively. The model incorporates a spin-independentNN (NNN) hopping denoted by t ( t ). The phase fac-tor e iφ ij is originated due to the staggered magneticflux as described in the Haldane model [5]. The factor ν ij = ( d ij × d ij ) z is same as mentioned in Kane-Melemodel [8]. The important point to note here is thatthe SOC term acts as spin-dependent magnetic fluxescoupled to the electron momenta in terms of the spin de-pendent NNN hopping of strength V so . M represents theinversion breaking mass term. The model breaks TRS aswell as inversion symmetry while interpolating betweenthe Haldane and Kane-Mele models. One can obtain themomentum space Hamiltonian after Fourier transforma-tion of Eq. (1) as given by [35] H ( k ) = (cid:88) i =0 n i ( k ) Γ i (2)with Γ i = σ i ⊗ τ for i = 1 , , , Γ i +3 = σ i ⊗ τ for i = 1 , , Γ i +5 = σ i ⊗ τ for i = 1 , , Γ = σ ⊗ τ , Γ = σ ⊗ τ and Γ = σ ⊗ τ . Here σ and τ rep-resent orbital and spin degrees of freedom while writingthe Hamiltonian in the basis ( c A ↑ , c A ↓ , c B ↑ , c B ↓ ) . Thecomponents n i are given by n = 2 t f ( k ) cos φ , n = − t (1 + 2 h ( k )) , n = − t sin √ k y cos k x , n = M − t g ( k ) sin φ , n = V R √ sin √ k y cos k x , n = V R √ ( h ( k ) − , n = − V R cos √ k y sin k x , n = V R sin √ k y sin k x , n = V so g ( k ) cos φ , n = V so f ( k ) sin φ , with f ( k ) =2 cos √ k y cos k x + cos k x , g ( k ) = 2 cos √ k y sin k x − sin k x , h ( k ) = cos √ k y cos k x . We note that for V R = V so = 0 , the model (1) reduces to two copies of Hal-dane model (that breaks TRS T H ( k ) T − (cid:54) = H ( − k ) with T = ( I ⊗ τ ) i K , K being the complex conjugation) withspin up and down block while for t = φ = 0 , it reducesto Kane-Mele model (that preserves TRS). In the rest ofthe paper, we consider t = 1 . and t = 0 . .To begin with, we show the phase diagram in M − φ plane by keeping V so = 1 . fixed as shown in Fig. 1 (a)and (b) for V R = 0 and . , respectively. As the modifi-cation over Haldane’s phase diagram, we find that a finite V so in Hamiltonian (1) results in two additional topologi-cal phases namely, QSHI [with (C ↑ , C ↓ with C ↑ (cid:54) = C ↑ (cid:54) = 0 ]and QASHI [with (C ↑ = 0 , C ↓ (cid:54) = 0) or (C ↑ (cid:54) = 0 , C ↓ = 0) ]phases. The size of QAHI phases, characterized by spinChern number (C ↑ , C ↓ ) with C ↑ = C ↓ = ± , gets re-duced as compared to QAHI phases in Haldane model; (1 , and ( − , − phases are respectively encapsulatedby QASHI phases (0 , , (1 , and (0 , − , ( − , frombelow and above. While the two adjacent QAHI phasesare connected by QSHI phases (1 , − and ( − , . Thecolor coded phase boundaries are assigned to the zeros ofthe respective energy gap equations [36]. It is noteworthythat φ → − φ , C ↑ → − C ↓ and C ↓ → − C ↑ for QASHI andQAHI phases. This correspondence holds also for QSHIphase that maps to itself. The underlying reason couldbe the helical edge modes are time reversed partner ofeach other in QSHI phase [37].Strikingly, we encounter an extended critical phase,bounded by violet (red) and blue (black) phase bound-aries vertically (horizontally), within which the spin gapvanishes identically (see Fig. 1 (a)) [33]. This cap likecritical phase can not be characterized by the spin Chernnumber. The vertical height (horizontal width) of thecritical phase decreases (increases) with increasing V R (such that V R ≤ V so ) while the size of QASHI phases re-duces without qualitatively deforming their phase bound-aries. The QASHI phases vanish and critical phase ex-tends between − π < φ < π when V R > V so as depictedin Fig. 2 (a) and (b). In other words, the critical phasesare bounded by violet phase boundaries from outside for V R > V so . This phase becomes the widest when V so = 0 (see Fig. 2 (a)). Upon introduction of V so , the size ofQAHI phases reduces as well as critical phase becomesnarrower (see Fig. 2 (b)). Finally, when V so ≥ V R ,QASHI phases start to appear near φ = 0 and ± π . Theviolet phase boundaries expand with incresing V R and itfully extends − π < φ < π when V R ≥ V so . The gaplesscritical phase, otherwise bounded from outside, will nowbe bounded from inside by the violet phase boundariesas soon as V so exceeds V R . The exact relation between V so and V R can be found from the gap equation corre-sponding to the violet phase boundary [36].We now investigate the phase diagram in V R - M planeto elucidate the modification over Kane-Mele phasesnamely, QSHI phases as shown in Fig. 3 (a) and (b) for φ = 0 and − π/ , respectively. The distinctive featureis that finite φ is able to break the QSHI phase (1 , − into QASHI phase (1 , and QAHI phase (1 , , whileNNN hopping t alone does not affect the existing phasediagram. Notably, QAHI phase originates between thetwo lobes of QASHI phase. We note that TRS break-ing uniform exchange field can lead to QAHI phases [25].The staggered magnetic flux φ associated with NNN hop-ping t and spin dependent hopping V so acts as a keyingredient to generate all the above phases simultane-ously. It is to be noted that QASHI phases appear when V R < V so . The color coded phase boundaries signifiesthat the identical QASHI phases for positive and nega-tive M are bounded by same gap equations.Below, we emphasize a few essential conclusions fromthese phase diagrams. The QAHI (QSHI) lobes of Hal-dane (Kane-Mele) model dismantle into a variety ofphases in presence of V R and V so ( t and φ ) giving riseto multicritical points where multiple topological phaseboundaries coalesce. Across a phase boundary, separat-ing two topological phases, | ∆C ↑ + ∆C ↓ | can only be-come unity where ∆C ↑ ( ∆C ↓ ) measures the differencein C ↑ ( C ↓ ) among the two adjacent topological phasesseparated by a phase boundary. This situation no longerholds generically when we encounter a multicritical point.The most important finding of our work is the emer-gence of QASHI phase where only one spin component istopologically protected leaving the other to be triviallygapped out. Even though, this type of phase has beenfound in magnetically doped QSHI material [24, 26, 27],ours is the first tight binding model hosting these phasesnaturally, to the best of our knowledge. The detail struc-ture of topological gap will become more evident whilewe study the low energy model below. FIG. 1. Here we show the effect of V R for a fixed V so = 1 . in M - φ phase diagram. (a) and (b) are plotted with V R = 0 . and V R = 0 . respectively. The indices within a given phaserefer the values of (C ↑ , C ↓ ): C1 = (0 , , C2 = (1 , , C3 =(1 , , C4 = (1 , − and Cn = − Cn with n = 1 , , , . Thephase boundary is obtained by the zeros of band gap [36] andthe color codes refer to the relevant band gap equations. Having described the bulk properties we would now
FIG. 2. Here we investigate the effect of V so for a fixed V R =1 . in M - φ phase diagram. (a) and (b) are plotted for V so =0 . and V so = 0 . respectively. The definition of Cn and Cn are provided in the caption of Fig. 1. The spin gap vanishesin critical phase, denoted by the assembly of black dots, thatgets narrower with increasing V so .FIG. 3. We here demonstrate that how a QSHI phase givesrise to QAHI and QASHI phase by varying φ in M - V R planekeeping V so = 1 . fixed. (a) and (b)correspond to φ = 0 . and φ = − π/ , respectively. The definition of Cn and Cn areprovided in the caption of Fig. 1. like to shed light on the bulk-boundary correspondenceby investigating the edge states in the semi-infinite ge-ometry (zig-zag edge in y direction while k x remains agood quantum number). It is an established fact thatChern number calculated by integrating the Berry cur-vature over the momentum Brillouin Zone, is identifiedby the difference in the number of chiral edge modes ina given edge: C = N RM − N LM , here N RM ( N LM ) de-notes the number of right (left) moving edge modes withpositive (negative) chirality [5]. This formulation canbe directly extended to the spin system connecting thespin Chern number with the spin polarized edge modesin respective phases [8]. In particular, for QSHI phase,the edge states of different chirality are associated withdifferent spins. Thus it transports a net quantized spincurrent though the charge current is zero unlike the caseof QAHI where quantized charge transport takes place.Our study, on the other hand, remarkably refers to a sit-uation where one can expect a quantized spin as well ascharge current simultaneously.In Fig. 4 (a), (b), (c) and (d), we depict the edgemodes for (1 , , ( − , , (0 , and (0 , − , respectively.The spin Chern number corroborates the bulk bound-ary correspondence by following way: C ↑ = N ↑ RM − N ↑ LM , C ↓ = N ↓ RM − N ↓ LM where N σ RM ( N σ LM ) represents the num-ber of right (left) moving edge mode for spin σ = ↑ , ↓ . Wenote that edge modes are not completely spin polarizedas far as their numerical calculations are concerned. Weassign an edge state to be spin up (down) if it is maxi-mally populated by spin up (down) states. Turning to thehelical edge states in QSHI phase, as shown in Fig. 4(e)and (f), the spin dependent chiral movement is clearlycaptured where up (down) spin traverses in a clockwise(anti-clockwise) manner along the edges of the system.This results in two types of QSHI phases with spin Chernnumber (1 , − and ( − , depending on the chirality ofthe spin-polarized edge state. Finally, we show the chiraledge states of (1 , and ( − , − QAHI phases respec-tively in Fig. 4 (g) and (h). In this case, both the spin upand down edge states share same chirality while travers-ing along the boundaries of the system. Therefore, usingthe bulk-boundary correspondence, we can successfullyexplain that the spin dependent edge states in differentphases are related by the spin Chern number of the un-derlying phases.
FIG. 4. Here we display the edge modes for various topologi-cal phases: QASHI phase in (a), (b), (c), (d) for (1 , , ( − , , (0 , and (0 , − , respectively; QSHI phases in (e) and (f) for (1 , − and ( − , , respectively; QAHI phases in (g) and (h)for (1 , and ( − , − , respectively. The red (green) refersto the localization of edge modes at top (bottom) part of thesemi-infinite zig-zag chain. Another very intriguing finding that our study unveilsis that edge state can generically appear at finite ener-gies. Interestingly, except for φ = ± π/ , where QAHIphases host zero energy chiral edge states as shown inFig. 4 (g) and (h), all the other values of φ (cid:54) = ± π/ support finite energy edge states if there exist a topo-logical phase. The edge modes do not show any avoidedlevel crossing structure that are observed for magneti- cally doped and exchange field induced QSHI [25–27].Therefore, staggered flux induced topological phases areintrinsically different from the above cases even thoughthe TRS is broken in both the situations [37]. An effectiveTRS emerges implying E ( π − k, φ ) = E ( π + k, − φ ) in ourcase. Even more surprisingly, edge modes are further pro-tected by a composite anti-unitary symmetry ensuring E ( π − k, π − φ ) = − E ( π + k, π + φ ) . Thus the twin effectof these anti-unitary symmetries allows one the mapping C ↑ → − C ↓ and C ↓ → − C ↑ under φ → − φ . This furtherguarantees the existence of zero energy chiral edge modesfor any topological phase obtained at φ = ± π/ .Having extensively explored the lattice model, wenow make resort to the low energy model for bet-ter understanding behind the emergence of differentphases. Expanding around the Dirac points α = ± ,we obtain n = − α √ t k x / , n = √ t k y / , n = M + α √ t sin φ (1 − k x / − k y / , n = − V R k y / , n = − V R / √ αV R k x / , n = αV R / √ V R k x / , n = − α V R k y / , n = − α ( √ / V so cos φ (1 − k x / − k y / , n = − ( V so /
2) sin φ (1 − k x / − k y / . Atthe Dirac points, the eigen-energies take the follow-ing form: E = ( w + w + (cid:112) r + ( w − w ) ) / , E = ( w + w + (cid:112) r + ( w − w ) ) / , E = ( w + w − (cid:112) r + ( w − w ) ) / and E = ( w + w − (cid:112) r + ( w − w ) ) / with w = n + n + n , w = n − n − n , w = − n − n + n , w = − n + n − n , r = − n − n and r = n − n . Let us now start witha simple case V R = 0 leading to the enegy gap for spinup ∆ E ↑ AB = w − w , and spin down ∆ E ↓ AB = w − w .In this case, the low energy model closely follows theBernevig-Hughes-Zhang model for HgTe quantum Well[38] enabling us investigate different phases in the sim-ilar spirit. A topological phase is ensured by oppo-site signs of the gap at two Dirac points k and k : ∆ E ↑ ( ↓ ) AB ( k )∆ E ↑ ( ↓ ) AB ( k ) < . The different combinationof the above product can in principle determine the topo-logical phases.For the QASHI phases with (C ↑ , [ (0 , C ↓ ) ], one canfind spin up [down] sector is only topologically gappedout leaving other spin sector to be trivial. For theQSHI phase with (C ↑ = ± , C ↓ = ∓ , one canfind different combinations of topological gap in boththe spin sectors. In the case of QAHI phase with (C ↑ = ± , C ↓ = ± , the same combination of topo-logical gap occur in both the spin sectors. In par-ticular, for topological spin up channel with C ↑ (cid:54) = 0 , ( M + 3 √ t sin φ − ( √ / V so cos φ )( M − √ t sin φ +( √ / V so cos φ ) < ; on the other hand, for topologi-cal spin down channel with C ↓ (cid:54) = 0 , ( M + 3 √ t sin φ +( √ / V so cos φ )( M − √ t sin φ − ( √ / V so cos φ ) < .The phase boundaries across which C ↑ ( C ↓ ) changesare given by M = ∓ √ t sin φ ± √ V so cos φ/ ( M = ∓ √ t sin φ ∓ √ V so cos φ/ ). This further explains theobservation that C ↑ and C ↓ can only jump by unityacross a phase boundary separating two different topo-logical phases. However, there exist multicritical pointsin the phase diagram where more than two phases con-verge including non-topological phases. At these points,spin Chern number can jump by more than unity. With-out Rasbha interaction V R = 0 , one can observe QAHI,QSHI and QASHI phases in various parameter regimesas shown in Fig. 1 and Fig. 3, can be explained by theabove low energy analysis.Now, we extend our analysis for finite V R (cid:54) = 0 us-ing the low energy formulation. We note that phaseboundaries are modified without altering the topolog-ical nature of the phases in presence of V R provided V so (cid:54) = 0 and φ (cid:54) = 0 . This suggests that phases,present in the absence of V R , are adiabatically connectedwhile Rashba interaction is turned on. The principlefor a phase being topological remains unaltered, how-ever, their explicit forms are modified. Denoting x ζ,ξ = M + ζ √ t sin φ + ξ ( V so /
2) sin φ , y η = η √ V so /
2) cos φ and z ζ,ξ = (cid:113) V R / x ζ,ξ , we find ∆ E ↑↓ AB ( k ) = ± y − + | x , − | sgn( x , − ) + z , , ∆ E ↑↓ AB ( k ) = ± y + | x − , | sgn( x − , ) + z − , − . For spin up (down) sec-tor to be topological, the following condition needsto be satisfied [ y − + | x , − | sgn( x , − ) + z , ][ y + | x − , | sgn( x − , ) + z − , − ] < ( [ y + | x , − | sgn( x , − ) + z , ][ y − + | x − , | sgn( x − , ) + z − , − ] < ). In orderto obtain physical phase boundaries z ζ,ξ has to be posi-tive that yields the modification of phase boundaries inpresence of V R [36, 39]. It is important to note thatin addition to the V R = 0 case, the relative strength be-tween V so sin φ and √ V R / terms also play an importantrole in determining the phase boundaries. For example,the phase boundary is substantially modified with newtopological phase once φ becomes non-zero (i.e., TRS isbroken) as shown in Fig. 3 (a) and (b). Relying on thestructure of topological gap at Dirac points, one can de-fine the spin Chern number in an effective manner asfollows C σ = 12 (cid:34) sgn (cid:18) ∆ E σAB ( k ) (cid:19) − sgn (cid:18) ∆ E σAB ( k ) (cid:19)(cid:35) (3)with σ = ↑ , ↓ .Having physically explained the emergence of differ-ent phases through the analysis of topological band gap ∆ E ↑ AB and ∆ E ↓ AB , we now demonstrate some interestingfeatures of the phase diagrams. To this end, we define thespin gap ∆ E ↑↓ A = E ↑ A − E ↓ A , and ∆ E ↑↓ B = E ↑ B − E ↓ B at twoDirac points which plays very important role in revealingunderlying conditions for topological transitions. It is in-deed necessary to have finite spin gap ∆ E ↑↓ A , ∆ E ↑↓ B (cid:54) = 0 , inorder to characterize a phase with (C ↑ , C ↓ ) [33]. There-fore, the robustness and stability of the topological invari-ant is determined by the finiteness of the spin gap. In-terestingly, our numerical calculation with lattice modelsuggests that spin gap can vanish at any arbitrary pointinside the momentum BZ for φ (cid:54) = 0 . This is in contrast to the energy gap, obtained from the lattice model, thatonly vanishes at Dirac points. The above finding makes aclear distinction between the role of energy and spin gapwhile the former being resposible for the phase bound-aries and later characterizes the phase.In conclusion, we consider the TRS broken Kane-Melemodel merged with Haldane model where intrinsic SOCis coupled with staggered magnetic flux to investigate thefate of QSHI phases. We remarkably find new topolog-ical phases namely, QASHI phase and extended criticalregion in addition to the QAHI and QSHI phase whilestudying the spin Chern number (C ↑ , C ↓ ) . The QASHIphase, characterized by (0 , C ↓ ) and (C ↑ , , supportsspin-selective transport where one spin channel is topo-logically gapped out leaving the other component triv-ially gapped. The other two topological phases namely,QSHI and QAHI phases exhibit spin-polarized and spin-neutral edge transport in accordance with spin Chernnumber. The topological phases in this model are pre-served by an effective TRS and a composite particle-holesymmetry. We show that the findings from the latticemodel can be understood from low energy model aroundthe Dirac point. We also provide an effective descriptionof spin Chern number, based on the low energy model,that corroborates with the lattice calculation. Surpris-ingly, the band gap turns out to be decisive in the topo-logical characterization while stability and robustness isdetermined by the finiteness of the spin gap. We notethat in optical lattice platform SOC is theoretically pro-posed [40–44] and experimentally realized [45–49]. Giventhese studies together with experimental advancement inperceiving Haldane model [31], we believe that our pro-posal and findings can be tested in near future. 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