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[Genki Yonezawa](https://orcid.org/0009-0006-1271-2170), [Jun-ichi Fukuda](https://orcid.org/0000-0003-3552-8406), [Toshikaze Kariyado](https://orcid.org/0000-0002-3746-6803)

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[Realization of Topological Phase in a Chiral Honeycomb Lattice Model](https://mdr.nims.go.jp/datasets/b52a2a0e-1690-456e-8d29-6f7285c2b8fa)

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Realization of Topological Phase in a Chiral HoneycombLattice ModelGenki Yonezawa,1, ∗ Jun-ichi Fukuda,1, 2 and Toshikaze Kariyado31Department of physics, Faculty of Science,Kyushu University, Fukuoka, Japan2International Institute for Sustainability withKnotted Chiral Meta Matter (WPI-SKCM2),Hiroshima University 1-3-1 Kagamiyama,Higashi-Hiroshima, Hiroshima 739-8526, Japan3Reserch Center for Materials Nanoarchitectonics(MANA),National Institute for Materials Science, Tsukuba, Japan1AbstractWe investigate topological properties of a chiral honeycomb lattice model with next-nearest-neighbor hoppings characterized by the reflection symmetry breaking. Topological nontrivi-ality is detected by analyzing effective Dirac Hamiltonian, and confirmed by numerical andanalytical study of the emergence of topological edge states at the boundaries between topo-logically distinct regions. We have also discovered that a novel asymmetric edge currentattributable to chirality can be excited without any involved phase shifts in input sources topick up one of the pseudospin components.I. IntroductionBand theory, which specifies band energies and wave functions by momentum inBrillouin zone, has been traditionally used in solid state physics. Historically, bandgaps or effective masses, which are encoded in band energies, play important roles inrelation to the semiconductor technology. The study of quantum Hall effect (QHE)[1, 2] then brought a new approach to classification of band structures, namely theuse of topological invariants like Chern numbers that are encoded in the connection ofthe wave functions in the Brillouin zone. Although the QHE requires the time-reversalsymmetry breaking, it has been recognized that various systems with time-reversalsymmetry can also exhibit topological phases, realized by spin-orbit coupling [3–5],crystalline symmetry [6], and so on.As an ideal platform for investigating topological phases, tight-binding models onhoneycomb lattice have been attracting interest as a system exhibiting the Dirac bandstructure. One of the most interesting model was proposed by Haldane [7], and aquantum anomalous Hall effect (QAHE) can be realized by introducing next-nearest-neighbor (NNN) hopping with complex value. These complex hoppings can also beinduced by considering the intrinsic spin-orbit coupling (SOC) in the honeycomb lattice,and spinful electron systems show a quantum spin Hall effect (QSHE) [8–10]. After∗ g.yonezawa@cmt.phys.kyushu-u.ac.jp2these theoretical predictions of QAHE and QSHE, many studies have been establishedto realize topological phases on the honeycomb lattice [11–18]. QAHE has also beenexplored to support topological states in metamaterial settings, such as Floquet systems[19–21] or photonic crystals [22, 23].As another direction of study, a modulated honeycomb lattice model proposed byWu and Hu [24] should be mentioned as a system exhibiting a quantum psuedospinHall effect (QpSHE). They take a hexagonal unit cell and treat the honeycomb latticeas a triangle network of hexagons. By tuning the ratio of intra-hexagon hopping tointer-hexagon hopping, a topological phase transition can be realized, accompanied bya band inversion at Γ point. This model has wide application to metamaterials, becauseit does not require SOC. In fact, their setup is also applicable to a photonic crystal madeof only a dielectric medium [25, 26].In the studies of honeycomb lattice models, the sublattice symmetry often playsan important role. Only with the nearest-neighbor (NN) hoppings (which is supposedto give a minimal model for graphene), a honeycomb lattice model has the sublatticesymmetry. The sublattice symmetry is also preserved in the case of QpSHE in thehoneycomb lattice when there is no NNN hoppings. In contrast, the honeycomb modelsfor QAHE and QSHE breaks the sublattice symmetry due to the complex NNN hoppings[27, 28].Although NNN hopping on honeycomb lattice brings exotic phenomena, the previousstudy of the QpSHE did not focus on how the sublattice symmetry breaking affects thetopological phase transition. Moreover, the effect of chirality characterized by thereflection symmetry breaking has not been explored.The aim of the present study is to investigate how chirality affects the classificationof the band topology and the edge transport characteristic of the non-trivial band topol-ogy. We focus on the cases in which time reversal symmetry is preserved. As mentionedabove, a tight-binding model with the honeycomb structure has been regarded as anideal platform for topological phases, and here we propose a chiral honeycomb latticemodel by extending the model proposed by Wu and Hu [24] and introducing chirality inthe NNN hopping. The setup of our model also breaks the above-mentioned sublattice3symmetry, which complements the previous studies of the QpSHE.The rest of this paper is organized as follows. In Sec.II, we start with describingthe setup of our tight-binding model. As we will discuss in detail there, our model isapplicable not only to quantum fermionic system but also a classical system charac-terized by a dynamical matrix instead of a quantum Hamiltonian. Then in Sec.III, weconduct topological classification by using effective Hamiltonian, and calculate energydispersions. The validity of analytically shown topological classification is numericallyconfirmed by calculating interface states. In Sec.V, we calculate interface transport toelucidate the effect of chirality. We conclude this paper and make discussions in Sec.VI.II. ModelIn order to consider how chirality affects the behavior of a topologically non-trivialsystem, we specifically discuss a two-dimensional fermionic system or its correspondingclassical harmonic oscillator system. Generally, band structures is not only for quantumsystems but also for classical systems, such as frequency spectra of dynamical matricesin a spring-mass model [29]. Therefore, topological nontriviality of band structurescould be detected by using both fermionic system and harmonic oscillator system.We first consider a two-dimensional tight-binding model on a honeycomb latticewith NN and NNN hoppings. Figure 1 presents the schematic illustration of the modeland the geometry. Due to the modulation in hoppings, the primitive unit cell is ahexagon containing six sites instead of two in the pristine honeycomb lattice model.Then, the unit vectors for the modulated honeycomb lattice are a1 = (3a0/2,√3a0/2)Tand a2 = (−3a0/2,√3a0/2)T , where a0 is a lattice constant for the pristine case. TheHamiltonian readsH =∑〈i,j〉tijc†icj +∑〈〈i′,j′〉〉ti′j′c†i′cj′ . (1)Here ci (c†i ) is the annihilation (creation) operator for electrons. 〈i, j〉 denotes nearestneighbors, and 〈〈i′, j′〉〉 denotes next nearest neighbors. As NN hoppings, we considerboth intrahexagon hopping t0 and interhexagon hopping t1, as depicted in FIG. 1. On4the other hand, as NNN hoppings, we only consider interhexagon hopping. Further-more, we classify the interhexagon hoppings into two types, and respectively assign t2and t3 to introduce chirality on the system. The two types of hoppings are illustratedin FIG. 1.The reflection symmetry breaking can be introduced by setting t2 unequal to t3.Indeed, as depicted in FIG. 2, the t2 bonds are the mirror image of the t3 bonds, wherethe reflection plane is on the vertical line. This means that the system becomes chiralwhen t2 6= t3, because the roles of t2 and t3 are swapped before and after the reflectionoperation.As mentioned above, this model is an extension of the model proposed by Wu andHu [24], who considered a quantum psuedospin Hall effect on a modulated honeycomblattice. Their setup is also applicable to a photonic crystal made of only a dielectricmedium [25], which yields a venue for experimental validations [26].In the following, we limit ourselves to the case with the real-valued hoppingt0, t1, t2, t3, unlike Haldane model. This makes it straightforward to realize the modelin any artificial systems like photonic crystals. Without complex-valued hoppings, thetime-reversal symmetry is preserved in our setup. We emphasize that we handle thecase of t2 6= t3, in which the system becomes chiral.In principle, the Hamiltonian Eq. (1) can be mapped to a dynamical matrix Γ of aclassical system. Here we specifically consider a mass-spring system in which potentialenergy can be written asV {x} =12∑i∑j>ikij(xi − sijxj)2 +12∑iεix2i , (2)as a function of dynamical variables x. The number of components in x corresponds tothe number of degrees of freedom in a given system. The first term represents couplingsbetween different degrees of freedom, with spring constants kij > 0 and sij being +1or −1. The way to choose ±1 for sijs in a spring-mass model is explained in FIG. 3.When a spring stores elastic energy for anti-phase motion of two connected mass points,sij = 1 for this spring, while when it stores elastic energy for in-phase motion of twoconnected mass points, sij = −1 for this spring. The second term, introduced for5yxza2 a16542 31t0 t1 t2 t3FIG. 1. (color online) Schematic illustration of the tight-binding model treated in this article.The NN hoppings inside unit cells are denoted by t0 (black solid lines), and the NN hoppingsbetween unit cells are denoted by t1 (red solid lines). The NNN hoppings are also introducedas t2 (green solid lines) and t3 (yellow solid lines).later convenience, is a local term that, in a spring-mass model, can be understood asa connection between the mass and the ground. Here εi is positive. The dynamicalmatrix of the system is given byΓij =∂2V∂xi∂xj=(εi +∑lkil)δij − sijkij. (3)By appropriately choosing εis, kijs and sijs such that εi +∑l kil is equal to a constantε independent of i, and that sijkij, which can be positive or negative, is equal to thehopping energies of the quantum counter part, the dynamical matrix Γ can be writtenasΓij = εδij − hij, (4)where hij is hopping energies in the Hamiltonian. By setting ε to be sufficiently large, Γ6operationReflection�FIG. 2. (color online) The system after the reflection operation. Due to the existence of NNNhoppings t2 and t3 the reflected system can not be superposed onto the original one.can be positive definite. Thus, one can construct a classical system where its dynamicalmatrix Γ is equal to H (with a constant shift of ε).After establishing the mapping of Eq. (4), the band structure of the Hamiltonian (1)has two interpretations: the energy spectra in the quantum system and the spectra ofthe square of the frequency (modified by the constant shift ε) in the classical system.For convenience, we investigate the topological properties (Sec. III and IV) by usingthe Hamiltonian H, whereas we discuss the interface transport on a classical ribbonstructure with the dynamical matrix Γ (Sec. V), since the coupling with external forcescan be understood more intuitively in classical systems.(a) (b)FIG. 3. (color online) The physical meaning of sij in a spring-mass model. (a)sij = +1.The masses are connected by a spring directly, then the spring acquires elastic energy foranti-phase oscillation. (b)sij = −1. The pulleys change the coordinate axes, and sij is nownegative. In this case the spring acquires elastic energy for in-phase oscillation.7III. Band structures and effective HamiltonianTo elucidate the topological properties of our chiral model given by HamiltonianEq. (1), we calculate its energy dispersions. We first consider the case where the systemis periodic with respect to a1, a2. By Fourier transforming Eq. (1), the Hamiltonian asa function of momentum k is written asH(k) = F (k) D(k)D(k)† F (k)T ,D(k) =t1e∗1(k)e∗2(k) t0 t0t0 t1e1(k) t0t0 t0 t1e2(k) ,F (k) =0 t2e∗1(k) + t3e∗1(k)e∗2(k) t2e∗1(k)e∗2(k) + t3e∗2(k)t2e1(k) + t3e1(k)e2(k) 0 t2e∗2(k) + t3e1(k)t2e1(k)e2(k) + t3e2(k) t2e2(k) + t3e∗1(k) 0 ,(5)where el(k) = eik·al (l = 1, 2).Figure 4 shows plots of the energy dispersion for the Hamiltonian Eq. (5) by settingthe hopping energies t0, t1, t2, t3 to several typical values. Importantly, a band inversionoccurs by changing the value of t1 appropriately. This can be confirmed by plottingthe values of | 〈unk|d+〉 | as line colors, where |unk〉 is the periodic part of Bloch functionlabeled by index n and Bloch wave vector k. We see that the colors of the band edgesnear E = 0 at the Γ-point are exchanged between Figs. 4(a) and 4(c). For the systemwith t1 = t0 [Fig. 4(b)], double Dirac cones appear at E = 0.In the low energy region around the Γ point, the effective Hamiltonian can be derivedasH(eff)(kx, ky) 'H+(kx, ky) 00 H−(kx, ky) , (6)where8−3−2−10123MΓK MΓK MΓKd±p±(a) (c)KMΓ(b)FIG. 4. (color online) Energy dispersions for the system given by Eq.(1). In all panels, we sett2 = 0.3t0 and t3 = −0.3t0: (a) t1 = 0.9t0, (b) t1 = t0, (c) t1 = 1.1t0. The color maps are forthe values of | 〈unk|d+〉 |. |p±〉 and |d±〉 are pseudo-spin modes, and their definitions are givenin Appendix A.H±(kx, ky) = −(t2 + t3)I + (t0 − t1)σz +t1|a1|2(±kxσx + kyσy). (7)The derivation is given in Appendix A. This Dirac Hamiltonian Eq. (7) clarifies theorigin of the band inversion in FIG. 4, where the band inversion is induced by varingthe hopping energy t1. In the context of the Dirac Hamiltonian, the sign of the Diracmass m := t0 − t1 can be flipped by changing t1, which explains the origin of the gapand infers that the two states in Figs. 4(a) and 4(c) are topologically distinct with eachother.For simplicity, we focus on the sign of the mass term m to pick up topological charac-ters of the system, namely we say two states with the opposite signs of m topologicallydistinct. Strictly speaking, a topological index often requires information of globalstructure of Bloch wave functions in the entire Brillouin zone (as the Chern number),not only information of the band order at a single momentum (Γ-point in this case).9However, a description by a Dirac equation with spatial modulation in its mass termgives a universal understanding of topologically protected edge/interface states. Thiswill be confirmed in the following analysis.We note that a complementary view can be given by further analysis using C2Tsymmetry. It has been shown that in crystals with C2T symmetry, the Wilson loopspectra can be utilized to define a Z2 index [30]. While next-nearest-neighbor hoppingterms break sublattice symmetry in our model, C2T symmetry remains intact. There-fore, rigorous Z2 classification can still be applied. We have included a discussion onWilson loops in Appendix B.IV. Topological edge stateIn this section, we analyze localized states at the boundary between two regionswith distinct topology. We begin with the analytical approach using the low-energyeffective Dirac theory. Let us consider a case where the periodic boundary condition isimposed only in the x direction. There is a boundary normal to the y direction wherethe sign of the mass term switches: t0 − t1 = m0 > 0 for y > 0 and t0 − t1 = −m0 fory < 0. In the x direction, kx is a good quantum number because of the existence ofthe periodic boundary condition. In the y direction, however, we apply a continuousapproximation by replacing ky with −i∂y to take into account the spatial dependenceof m. The eigenvalue equation of H+(kx, ky) then becomes−(t2 + t3) +m0sgn(y) v(kx − ∂y)v(kx + ∂y) −(t2 + t3)−m0sgn(y)φ1φ2 = Eφ1φ2 , (8)where v := t1|a1|/2. Rewriting this equation in a new basis φ± = φ1±φ2, the eigenvalueequation yields−(t2 + t3) + vkx m0sgn(y) + v∂ym0sgn(y)− v∂y −(t2 + t3)− vkxφ+φ− = Eφ+φ− . (9)10When m0/v > 0, the solution obtained under the conditions that the wavefunctionconverges at y = ±∞ and is continuous at y = 0 isE+ = −(t2 + t3)− vkx,φ+φ− ∝ 0exp(−(m0/v)|y|) . (10)The eigenvalue equation for H− is obtained by simply replacing kx of Eq.(9) by −kx.The eigenenergy and states becomeE− = −(t2 + t3) + vkx,φ+φ− ∝ 0exp(−(m0/v)|y|) . (11)Thus, by solving the eigenvalue equations of H(eff), one obtain the solutions such that theeigenenergies E± intersect linearly at E0 := −(t2 + t3). In addition, the correspondingeigenstates are exponentially localized at the boundary y = 0. These localized statesare protected by the difference of topology, i.e. the difference of the sign of massterm in Eq. (A15), which confirms the usefulness of the mass-term based topologicalclassificaiton.Next, we move on to the numerical approach using the tight-binding model. Inorder to discuss boundary states, we consider a system where a region of t1 = 1.1t0is sandwiched between two regions of t1 = 0.9t0 as shown in FIG. 5(a). Note that a1direction is horizontal in FIG. 5(a). Then, periodic boundary conditions are imposed ona1 and a2 directions, respectively. With this construction of the interface, a2 remainsto be a unit vector, i.e., there is no superstructure in a2 direction, and we can calculateenergy dispersion as a function of the momentum along the interface k‖. The calculateddispersion is in FIG. 5(b), showing new states in the bulk gap that intersect linearly atk‖ = 0, which is consistent with the analytical discussions above. Plotting the squarenorm of the corresponding wave function (FIG. 5(c)) also shows that it is localized atthe boundaries.11−101π−π 0k||A B··(a) (b)(c) MAX0|ψ|2m0 = −0.1t0m0 = 0.1t0 m0 = 0.1t0FIG. 5. (color online) The hopping enegies of chirality are set to t2 = 0.3t0 and t3 = −0.3t0.(a) Schematic illustration of the system being simulated, with a region of t1 = 1.1t0 boundedby two regions of t1 = 0.9t0. (b) energy dispersions (c) The square of wave functions corre-sponding to A and B in (b). They have the same value because of time reversal symmetry.V. Interface transportSo far we have seen that the effect of chirality does not manifest itself in the topo-logical classification or the existence of topological edge states, because unequal t2 andt3 just lead to the constant shift of the eigenvalues of the effective Hamiltonian. Inthis section, we discuss how chirality affects the interface transport by considering thedynamics of a classical harmonic-oscillator system with boundaries at which energyis injected [31]. One reason for using a classical system rather than a quantum oneis that the meaning of the energy injection can be much more intuitively caught ina classical system as shown below (the energy injection is simply modeled by forcedoscillation). Another reason is that the study of modulated honeycomb lattice modelhas been greatly developed in the context of classical systems like photonic crystals.To calculate the interface transport, we denote dynamical variables by x = {xi}.12The equation of motion with respect to time τ is written asd2x(τ)dτ 2= −Γx(τ) + f c cosΩτ. (12)Here Γ is a positive-definite dynamical matrix. An external force f c cosΩτ is appliedto the system with a frequency Ω, and the phase of the force is set to be the sameat all sites. One can consider a variety of systems including spring-mass systems andLC-circuit systems, and the variables x and τ and the parameters are set dimensionlessafter rescaling of the length and time according to the specific system to be considered.To facilitate the physical interpretation, let us solve Eq. (12) by using the normalmode decomposition [31]. By introducing an orthogonal matrix O that diagonalizes Γ,Eq. (12) becomesd2x̃(τ)dτ 2= −Gx̃(τ) + f̃ (c) cosΩτ, (13)where G = OTΓO, x̃(τ) = OTx(τ) and f̃ (c) = OTf (c). The expression of Eq. (13) incomponent form becomesd2x̃l(τ)dτ 2= −ω2l x̃l(τ) + f̃l(c)cosΩτ, (14)where we label the normal modes by an integer l. The solution of Eq. (14) for the initialcondition x(τinit) = dx(τinit)/dτ = 0 isx̃l(τ) = a(c)l cosΩτ + b(c)l cosωlτ, (15)wherea(c)l =f̃l(c)ω2l − Ω2, b(c)l = −a(c)l . (16)By reconstructing x(τ) from these x̃l(τ), one can understand the dynamics of thesystem.Now, let us return to the interface transport, and consider a specific system il-lustrated in FIG. 6. We consider a ribbon structure of 10 hexagonal unit cells witht1 = 1.2 cladded from both sides by 5 hexagonal unit cells with t1 = 0.8. We also sett2 = −t3 = 0.3 in all region. To analyze the dynamics, we construct the dynamical1312 34 5m0 > 0m0 > 0m0 < 0FIG. 6. (color online) Schematic illustration of the system consisting of regions with differenttopology. we set t2 = −t3 = 0.3 in all region. For the mass term part, we set m0 = −0.2 forthe m0 < 0 region and m0 = 0.2 for the m0 > 0 region. 20 unit cells are lined up in the a1and a2 directions with periodic boundary condition. We apply the external force f (c) at thepoints of the boundary indicated by blue dots with the amplitude f0 = 1.matrix of this ribbon structure by the procedure Eq. (4), and calculate time evolutionsof the intensity at each site i defined asIi =12[Ω2x2i +(dxidτ)2], (17)using Eq. (12). Equation (17) is designed to eliminate the fast oscillation and focus oneffective propagation of energy in a long time scale. We also choose ε = 3.1 and Ω =√ε.This choice makes Γ positive definite, and Ω is in the bulk gap of the dynamical matrixΓ.The obtained results are shown in FIG. 7. In FIG. 7(a), the lines labeled by 1-5correspond to the intensities in the unit cells 1-5 defined in FIG. 6. The energy isinjected at the unit cell 1, and the unit cells 2 and 3 (and also 4 and 5) are locatedat equal distances from the unit cell 1. The obtained difference between the intensitiesat 2 and 3 (4 and 5) reveals an asymmetric energy propagation caused by the effect ofchirality. This novel asymmetry is prohibited in the achiral case t2 = t3 = 0. It shouldbe noted that t2 and t3 are parameters that control the asymmetry. If the values of t2and t3 are interchanged, an asymmetric energy propagation in the opposite direction14TimeIntensity012354321(a) (b)0 20 40 60 80FIG. 7. (color online) (a) Time evolution of Intensity. The labels 1 to 5 correspond to thosein FIG.6. Intensity in this figure is the sum over the six sites in the blue shaded hexagonin FIG.6. (b) Real space plot of the propagation of Intensity Ii. We show snapshots att = 15, 30, 45, 60. An asymmetric propagation is achieved due to chirality.is observed. In addition, as can be seen from Eqs. (15) and (16), the contribution ofthe modes with ωl close to the frequency Ω dominates the energy propagation. Thechirality effects of t2 and t3 manifest themselves in x(τ) and intensity Ii as a linearcombination of these multiple modes.In the achiral case found in the literature, to select right-moving or left-movinginterface states, the phase shifts are required in input terminals [32]. In our case, nophase shift is assumed in the forced oscillation term, which gives another way to controlenergy propagation.VI. Conclusion and discussionWe have proposed a universal tight-binding model for chiral two-dimensional sys-tems. We have shown that the topological classification can be conducted by using theDirac Hamiltonian, and confirmed the emergence of topological edge states. We have15also discovered a novel asymmetric edge current induced without any phase tuning atinput terminals. These consideration could be a building blocks in exploring chiraltopological materials.As noted in Sec.V, experimental realization of the system proposed in the presentstudy would be more promising in classical systems because the fabrication of the systemand the tuning of the interactions will be easier than for quantum systems. Photoniccrystal of a dielectric material [25, 33] could be one direction, and top-down fabricationtechniques now enable the realization of topological photonics in the visible wavelengthregime [34]. Careful fabrication of chiral structures would provide an ideal platformfor the experimental verification of the asymmetric transport discussed in Sec. 5. Self-organization of chiral soft materials could also be used for the preparation of systemsthat allows the investigation of chiral transport phenomena, and their ideal conditionscould be achieved by the tunability of the structural properties of the soft materialby external stimuli. Recently self-assembly of bowtie-shaped nanoparticles has beenshown to exhibit tunable chiral photonic properties [35]. Chiral liquid crystals [36] canalso offer a platform for self-organized tunable chiral structures, and a hexagonal latticeof skyrmions exhibited by a chiral liquid crystal [37] could host asymmetric transportphenomena in the visible wavelength regime. Classical mechanics of course provides aclue to the realization of systems for chiral topological transport phenomena, such asmass-spring systems discussed in Sec.V [29], and spinning top systems [38]. We hopethat the present study will stimulate experimental studies towards the realization andobservation of asymmetric topological transport phenomena in a system with time-reversal symmetry.AcknowledgementsG. Y. is supported by the Kyushu University Leading Human Resources DevelopmentFellowship Program. This study is also supported by JSPS KAKENHI No. JP21H01049(J. F.) and No. JP20K03844 (T. K.).16Appendix A: Derivation of the effective HamiltonianTo derive the effective Hamiltonian, we focus on the eigenstates at the Γ point, i.e.kΓ = 0. The following derivation is in parallel with the supplementary materials of theprevious study [39], but any terms with t2 and t3 are new. The eigenstates of H(k = kΓ)are given by|fy(3x2−y2)〉 = (−1,−1,−1, 1, 1, 1)T/√6,|px〉 = (0,−1, 1, 0, 1,−1)T/2,|py〉 = (2,−1,−1,−2, 1, 1)T/2√3,|dx2−y2〉 = (−2, 1, 1,−2, 1, 1)T/2√3,|dxy〉 = (0, 1,−1, 0, 1,−1)T/2,|s〉 = (1, 1, 1, 1, 1, 1)T/√6. (A1)We use the conventional notation of s, p, d and f atomic orbitals. The correspondingeigenenergies are Efy(3x2−y2)= −2t0−t1+2t2+2t3, Epx,py = t0−t1−t2−t3, Edx2−y2 ,dxy=t1 − t0 − t2 − t3 and Es = 2t0 + t1 + 2t2 + 2t3, respectively. In the following, weconsider the case Efy(3x2−y2)< Epx,py ,dx2−y2 ,dxy< Es. Based on these eigenstates, onecan construct a low-energy effective Hamiltonian around the Γ point. Since we focuson the neighborhood of the Γ point where the bands are dominated by p and d states,it is sufficient to use {|px〉 , |py〉 , |dx2−y2〉 , |dxy〉} as the basis in calculating the effectiveHamiltonian. By using these four eigenstates, we define the following pseudo-spinmodes:|p±〉 =1√2(|px〉 ± i |py〉), (A2)|d±〉 =1√2(|dx2−y2〉 ± i |dxy〉). (A3)In order to consider the effective Hamiltonian, it is convenient to introduce thefollowing basis:|u±〉 =1√2(∓i |p±〉 − |d∓〉), (A4)|l±〉 =1√2(i |p±〉 ∓ |d∓〉). (A5)17Here, the explicit expression of {|u−〉 , |u+〉 , |l−〉 , |l+〉} is|u±〉 =|±〉0 , |l±〉 = 0± |±〉 , (A6)where|±〉 =1ω±ω∓ , ω± = −12±√32i. (A7)Now, let us first calculate the low-energy effective Hamiltonian by using {|u−〉 , |u+〉 , |l−〉 , |l+〉}.Indeed, if we expand the Hamiltonian in the basis, the effective Hamiltonian becomesH(eff)(kx, ky) =〈−|F |−〉 〈−|F |+〉 − 〈−|D |−〉 〈−|D |+〉〈+|F |−〉 〈+|F |+〉 − 〈+|D |−〉 〈+|D |+〉− 〈−|D† |−〉 − 〈−|D† |+〉 〈−|F T |−〉 〈−|F T |+〉〈+|D† |−〉 〈+|D† |+〉 〈+|F T |−〉 〈+|F T |+〉 . (A8)Then, we perform the Taylor expansion up to the first order of the wavevectors. Theeffective Hamiltonian is approximated asH(eff)(kx, ky) ' −(t2 + t3)I ⊗ I + (t0 − t1)σx ⊗ σz +t1|a1|2σx ⊗ (k · σ), (A9)where I is the identity matrix and σi(i = x, y, z) is the Pauli matrices. We denotethe Kronecker product as ⊗. It is shown that Eq.(A9) can be written as the DiracHamiltonian by introducing the following new basis:|1〉 = i |p−〉 =|u−〉+ |l−〉√2, (A10)|2〉 = − |d−〉 =|u+〉+ |l+〉√2, (A11)|3〉 = −i |p+〉 =|u+〉 − |l+〉√2, (A12)|4〉 = − |d+〉 =|u−〉 − |l−〉√2. (A13)18Using these new bases, the effective Hamiltonian is rewritten asH(eff)(kx, ky) 'H+(kx, ky) 00 H−(kx, ky) , (A14)whereH±(kx, ky) = −(t2 + t3)I + (t0 − t1)σz +t1|a1|2(±kxσx + kyσy). (A15)Thus, we obtain the Dirac Hamiltonian Eq.(A15) with the constant energy shift −(t2+t3).Let us make a remark about the effect of t2, t3 from the point of topological classi-fication. When t2 = t3 = 0, the original Hamiltonian proposed by Wu and Hu [24] isretrieved. The effective Hamiltonian Eq.(A15) becomesH±(kx, ky) = (t0 − t1)σz +t1|a1|2(±kxσx + kyσy). (A16)Therefore, topological classification by using mass term m := t0 − t1 survive in thiscase. Moreover, the matrix F in Eq.(5) becomes the zero matrix, and the Hamiltonianhas the sublattice symmetryγH(k)γ† = −H(k), γ2 = 1 (A17)withγ =1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 −1 0 00 0 0 0 −1 00 0 0 0 0 −1. (A18)In this case, thanks to this sublattice symmetry, one can define mirror winding number[40], and topological classification can be conducted mathematically rigorously. In thecase t2 6= 0 and t3 6= 0, however, the sublattice symmetry is broken, and one can notchoose the strategy to use the mirror winding number in classifing topological phase.19Appendix B: Berry bands and Z2 phaseHere, we basically follow the arguments in Ref. [30] to explain the idea of the Berrybands. We reproduce the arguments here since the explicit form of the equations areimportant in our numerics.1. Wilson loopsWe consider a closed path L in momentum space, and define the Wilson loop operatorfor L asŴ{n}L (k1) := P̂ (k1)P̂ (kN) · · · P̂ (k2)P̂ (k1), (B1)where P̂ (ki) is a projection operator onto a chosen subspace{n}:P̂ (ki) :=∑n∈{n}|un(ki)〉 〈un(ki)| . (B2)Here, the sequence {ki}Ni=1 represents closely spaced points along L.The eigenvalues and engenstates of the Wilson loop are obtained as follows. Byusing Bloch wave functions on the initial momentum k1, we can express it as a unitarymatrix:[W {n}(k1)]ij := 〈ui(k1)| Ŵ {n}L (k1) |uj(k1)〉 . (B3)The matrix W {n}(k1) can be diagonalized by a unitary matrix V (k1), which satisfies[V †(k1)W{n}(k1)V (k1)]ij = δij exp(iγi), (B4)where γi is a gauge-invariant geometric phase, also known as a Berry phase. Therefore,the eigenstates of the Wilson loop are|ũi(k1)〉 =∑l[V (k1)]il |ul(k1)〉 , (B5)with the corresponding eigenvalue being exp(iγi).202. Berry bandsLet us now consider Wilson loops around an infinitesimally small path L′ whichenclose a point k. Hereafter, to elucidate the topological nature of our system, wefocus on the valence bands. First, we construct the Wilson loops as:Ŵ valL′ (k) := P̂val(k1)P̂val(kN) · · · P̂val(k2)P̂val(k1). (B6)Here we denote {ki}Ni=1s as closely spaced points along the path L′, and P̂val(ki) is theprojection operator onto the full valence band space.Since the Wilson loop operator is unitary, we can define an associated Hermitianoperator ĤF(k), given byĤF(k) := limA→0[−i log Ŵ valL′ (k)A], (B7)where A is the area enclosed by L′. The matrix form of this Hermitian operator is[HF(k)]ij = 〈ui(k)| ĤF(k) |uj(k)〉= limA→0[−i[logW val(k)]ijA], (B8)where[W val(k)]ij := 〈ui(k)| Ŵ valL′ (k) |uj(k)〉 . (B9)This matrix HF(k) is Hermitian, and can be diagonalized by a unitary matrix U(k)satisfying[U †(k)HF(k)U(k)]ij = δijFi(k), (B10)where Fi(k) is the non-Abelian Berry curvature. Thus, the eigenvalue problem readsĤF(k) |ũ′i(k)〉 = Fi(k) |ũ′i(k)〉 , (B11)and|ũ′i(k)〉 =∑l[U(k)]il |ul(k)〉 . (B12)Following Ref. [30], we refer to these states as Berry bands.213. C2T -protected Z2 phaseWe first consider the spectrum of the Wilson loop built from projectors onto thethree-dimensional valence bands in our model. We have calculated the Wilson loopsalong a sequence of parallel paths, L(t), as depicted in FIG. B.1, where these pathsgradually traverse the Brillouin zone as t progresses from t : 0 → 1.b2b1L′L(t)tb2tb2 + b1FIG. B.1. (color online) Schematic illustration of a path L(t) where Wilson loops are made ona series of these paths. These paths sweep the Brillouin zone (shaded blue region) as t → t+1.The Wilson loop spectra, as shown in FIG.B.2, indicate that the total windings ofthe Wilson loop spectra are zero. As mentioned in ref [30, 41], the total Chern numberof the valence bands is equivalent to the total spectral winding of Wilson loops, andthis result is consistent with the time-reversal symmetry.WilsonspectrumMΓ ΓFIG. B.2. (color online) Wilson loop spectra for the topological region made through thevalence bands. The total windings of these spectra are zero, as expected from time-reversalsymmetry.22Alternatively, we can utilize the Berry bands introduced in the previous section. 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