Topological characterization of modified Kane-Mele-Rashba models via local spin Chern marker
Sebastião dos Anjos Sousa Júnior Marcus V. de S. Ferraz José P. de Lima Tarik P. Cysne







sebastiao.a.s.junior@gmail.com
Department of Physics, University of Houston, Houston, Texas 77004


Departamento de Física, Universidade Federal do Piauí, 64049-550 Teresina, Piauí, Brazil


Departamento de Física, Universidade Federal do Piauí, 64049-550 Teresina, Piauí, Brazil


tarik.cysne@gmail.com
Instituto de Física, Universidade Federal Fluminense, 24210-346 Niterói RJ, Brazil



§ ABSTRACT 
In this work, we use the local spin Chern marker (LSCM) recently introduced by Baù and Marrazzo [Phys. Rev. B 110, 054203 (2024)] to analyze the topology of the ground-state electronic wave functions in a finite honeycomb lattice flake described by three distinct models. The models considered here are characterized by strong Rashba spin-orbit interaction, which leads to non-conservation of the spin operator, i.e., [ℋ,ŝ_z]≠ 0. The three spin-orbit couplings associated with the topological aspects of the models are: 1) Standard Kane-Mele coupling, 2) Sublattice-dependent Kane-Mele coupling, and 3) In-plane (ŝ_y) polarized Kane-Mele coupling. These couplings occur in graphene grown on suitable substrates and are relevant for modeling its van der Waals heterostructures. A particular topological phase diagram characterizes each of these spin-orbit interactions, and our calculations of LSCM successfully capture its general features. We also performed a detailed analysis of the spectral properties of the energy and valence-projected spin matrix eigenvalues, which shows that both exhibit a gap that protects the marker. Our results expand the applicability of the spin Chern number method to a class of Hamiltonians with experimental relevance and may contribute to future research on the real-space topology of realistic materials. 
  




Topological characterization of modified Kane-Mele-Rashba models via local spin Chern marker
    Tarik P. Cysne
    September 9, 2024 – Version 1.0
============================================================================================






§ INTRODUCTION



Insulating materials in a non-trivial topological phase are characterized by gapped bulk energy spectra, which are indexed by a topological invariant. This invariant remains unchanged under smooth modifications in the electronic Hamiltonian and is robust against weak disorder and electronic interactions <cit.>. Much of our knowledge about the topological nature of matter has emerged from studying systems in the thermodynamic limit with translational symmetry, i.e., in the band theory paradigm. In these systems, the crystalline Hamiltonian can be mapped into a Bloch Hamiltonian ℋ( k) defined in k-space within the Brillouin Zone (BZ), which has the topology of a torus. From this perspective, topological invariants are regarded as global properties of Bloch states and are expressed through integrals of geometric quantities in k-space. For instance, Chern insulators are characterized by a nonzero integer invariant known as the Chern number C, which can be computed by integrating the Berry curvature over the first BZ <cit.>. The occurrence of a Chern insulator phase requires the breaking of time-reversal symmetry and is associated with gapless boundary states that support Hall conductance. On the other hand, quantum spin Hall insulators (QSHI) are characterized by a ℤ_2-index whose definition requires the presence of time-reversal symmetry <cit.>. These topological phases are distinguished by boundary modes that enable spin-Hall conductance. 

In Ref. <cit.>, Prodan introduced the spin Chern number as an alternative topological invariant to characterize the QSHI phase. Unlike the ℤ_2-invariant, the definition of the spin Chern number does not explicitly rely on the presence of time-reversal symmetry, but it does require a gap in the eigenvalue spectrum of the valence-projected spin matrix. Although such a gap is not a universal feature of spin-orbit coupled Hamiltonians, the spin Chern number is a genuine topological index. In time-reversal symmetric systems, it is widely accepted that it essentially contains the information of the ℤ_2-invariant <cit.>. This topological invariant was introduced within the framework of band theory (thermodynamic limit) and assumes a quantized value even for Hamiltonians where spin is not a conserved quantity, i.e., [ℋ( k), ŝ_z]≠ 0. Prodan's method has also been applied to define Chern numbers related to other observables, such as the isospin Chern number <cit.> and the orbital Chern number <cit.>, demonstrating the usefulness of this concept. An interesting aspect of this topological index is its direct connection to response functions, such as orbital Hall conductivity <cit.> and isospin Hall conductivity <cit.>.

The requirement for the existence of a BZ when defining a topological index constrains its application to pristine and uniform solids. In particular, a BZ cannot be defined in a finite system or in a system with a high degree of disorder. This compromises the topological characterization of systems with significant practical interest <cit.>. To overcome these limitations, the tool of topological local markers was introduced. These local markers are mathematical objects that encode information about the topology of the ground state's electronic wave functions as real-space quantities. Currently, markers to characterize distinct topological phases are available <cit.>. For instance, Bianco and Resta introduced in Ref. <cit.> a formula for the local Chern marker that allows the real-space indexation of Chern insulators. Extensions of this scheme to treat periodic superlattices <cit.> and inclusion of thermal effects <cit.> have been made. Local Chern markers have also been applied to study the robustness of the topological phase against disorder <cit.> and the topological Anderson transition <cit.>. 

Recently, Baù and Marrazzo <cit.> introduced a real-space quantity called the Local Spin Chern Marker (LSCM), which combines the Bianco-Resta and Prodan methods to capture the topological information associated with the QSHI phase. The authors validate the method by using the standard Kane-Mele-Rashba model and comparing it to the universal ℤ_2-local marker. In this work, we employ LSCM to analyze the topology of generalized Kane-Mele-Rashba models in real space. The models considered here exhibit a strong Rashba effect. In addition to the standard Kane-Mele coupling, we also consider two generalizations that are relevant for modeling graphene van der Waals heterostructures: the sublattice-resolved Kane-Mele and the in-plane Kane-Mele interactions. The sublattice-dependent Kane-Mele coupling occurs in graphene grown on transition metal dichalcogenides (TMDs) in the 2H structural phase <cit.>. On the other hand, the in-plane polarized Kane-Mele coupling occurs in graphene proximate to a low-symmetry substrate with strong spin-orbit interaction <cit.>. This latter type of interaction has also been predicted to occur in a low-symmetry structural phase of WTe_2 <cit.>. We calculate the topological phase diagram of the three models mentioned above using the LSCM in a flake geometry. The models studied here present very distinct and rich phase diagrams, and we show how the LSCM successfully captures their characteristics. We also discuss the features of the energy and valence-projected spin spectra, showing that both exhibit a gap that protects the topological characterization of each model according to this index <cit.>. Our results expand the applicability of the spin Chern number to describe the topology of electronic Hamiltonians and may represent progress toward describing Chern markers related to other observables <cit.> and to applications in complex materials.




 


§ LOCAL SPIN CHERN MARKER 
  



For Hamiltonians that satisfy [ℋ,ŝ_z  ]=0, the z-component of electronic spin is a good quantum number. In this situation, the Hilbert space of the Hamiltonian can be separated into two decoupled sectors, each associated with an eigenvalue s=↑ and ↓ of the spin operator ŝ_z. One can assign Chern numbers C_↑ and C_↓ to each subspace, and the quantity 1/2(C_↑-C_↓) is an integer quantity, and its non-zero value indicates the QSHI phase. For systems in the thermodynamic limit and with translational symmetry, one can map ℋ into a Bloch Hamiltonian defined in the BZ ℋ( k)=e^i k· rℋe^-i k· r and express the Chern number associated to each sector of Hilbert space as a k-integral of the spin Berry curvature: C_↑,↓=(2π)^-2∫_ BZd^2 kΩ_↑,↓( k).

In models where [ℋ ( k),ŝ_z  ]≠ 0, the Hilbert space cannot be decoupled into two independent sectors, and the quantity 1/2( C_↑-C_↓) is no longer an integer. Then, a more elaborate procedure must be used to define an integral Chern number that represents information about the topology of the QSHI phase. The spin Chern number, first introduced in Ref. <cit.> and later mathematically formalized in thermodynamic limit by Prodan in Ref. <cit.>, allows for the definition of a quantized invariant even in cases where [ℋ( k),ŝ_z  ]≠ 0. The method consists of constructing the valence-band projector 𝒫( k) and diagonalizing the ground-state projected spin matrix M^s_z_v.b.( k)=𝒫( k)ŝ_z𝒫( k). For time-reversal symmetric Hamiltonians the eigenvalues ξ^s_z( k) of the matrix M^s_z_v.b.( k) are symmetrically distributed around zero, within the interval ξ^s_z( k)∈ [-1,1]. Given that σ=sign(ξ^s_z( k)) is the signal of ξ^s_z( k) [σ=+ for positive ξ^s_z( k) and σ=- for negative ξ^s_z( k)] one defines the projector 𝒫_σ( k)=∑_ξ( k)| σ|ϕ_ξ, k⟩⟨ϕ_ξ, k| where |ϕ_ξ, k⟩ are the eigenvectors of M^s_z_v.b.( k) with eigenvalue ξ^s_z( k). Using 𝒫_σ( k), one can assign an integer Chern number C_σ to each projected subspace, provided that there is a spectral gap Δ^s_z in the set of eigenvalues ξ^s_z( k). The spin Chern number is defined by C_s=1/2( C_+-C_-). A non-zero spin Chern number indicates the existence of a QSHI phase and, in time-reversal symmetric systems, is equivalent to the universal ℤ_2-invariant <cit.>. This invariant is topologically protected by the energy spectrum gap Δ^ E and the gap Δ^s_z. This method has been successfully applied to study the topological properties of uniform insulating materials and other systems from their band structures <cit.>. 



As discussed in the introduction, it is not possible to construct a BZ in finite systems. Instead, real-space markers must be used to quantify the topological properties of the electronic wave functions. Recently, the Ref. <cit.> introduced a combination of Prodan and Bianco-Resta methods to define the LSCM,

    ℭ_s( r)=ℭ_+( r)-ℭ_-( r)/2,

where,

    ℭ_σ( r)=4πIm⟨ r|𝒫_σX̂𝒬_σŶ𝒫_σ - 𝒫_σŶ𝒬_σX̂𝒫_σ| r⟩.

In Eq. (<ref>), X̂ and Ŷ are the Cartesian component of the position operator r. Analogous to the procedure described in the previous paragraph for k-space, one defines the projector into the valence subspace (occupied states) by 𝒫=∑^N_occ_n=1|u_n⟩⟨u_n|, where |u_n⟩ is the eigenstate of real-space Hamiltonian ℋ with state index n. Then, we construct the spin matrix projected onto the valence subspace M^s_z_v.s.=𝒫ŝ_z𝒫 and perform its diagonalization M^s_z_v.s.|ϕ_v⟩=ξ_v^s_z|ϕ_v⟩, where (v=1,..., N_occ). For time-reversal symmetric models, the N_occ eigenvalues ξ_v^s_z are symmetrically distributed around zero within the interval [-1,1]. The branches of this spectrum can be separated by the sign of their eigenvalues [σ=+ for positive ξ_v^s_z and σ=- for negative ξ_v^s_z] if there is a finite gap Δ^s_z. The vector |ϕ_v⟩ has dimension N_occ and can be written as |ϕ_v⟩ = ∑^N_occ_α=1β_v,α|α⟩, where |α⟩ is the basis of N_occ-dimensional matrix M^s_z_ v.s..  We then define the states,

    |ψ_v⟩ =∑^N_occ_α=1β_v,α|u_α⟩.

in terms of the valence eigenstates |u_α⟩ of the flake Hamiltonian ℋ. With these states, the projectors used in Eq. (<ref>) are given by

    𝒫_σ=∑_ξ^s_z_v|σ|ψ_v⟩⟨ψ_v|.

Finally, the complementary matrices in Eq. (<ref>) are defined as 𝒬_σ = 1- 𝒫_σ. Here, we adopt an equivalent, but slightly different, notation than Baù and Marrazzo. In Ref. <cit.> authors also demonstrate that the LSCM defined in Eqs. (<ref>-<ref>) is compatible with the ℤ_2 marker in the standard Rashba-Kane-Mele model with sublattice potential. 









§ STANDARD KANE-MELE-RASHBA HAMILTONIAN 






In this section, we review the application of the LSCM to the standard Kane-Mele-Rashba model in a finite flake <cit.>, complementing this with an analysis of the energy and valence-projected spin spectra. The standard Kane-Mele-Rashba Hamiltonian can be cast as,

    ℋ_ 𝓈𝓉𝒹=ℋ_t+ℋ_ AB+ℋ_ R+ℋ_ KM.

We follow Ref. <cit.>, considering this Hamiltonian within a flake of linear size L subjected to periodic boundary conditions. ℋ_t represents the nearest neighbor hopping term of electrons in the honeycomb structure of Fig. <ref>, as given by

    ℋ_t=t∑_s=↑,↓∑_⟨ i,j ⟩c^†_i,sc_j,s,

Here, c^†_i,s and c_j,s are the fermionic creation and annihilation operators for electrons at sites i and j of the flake and spin s=↑,↓. ⟨ i,j ⟩ indicate that the sum runs over the nearest neighbor sites. In all results presented in this work, we set the hopping amplitude t as the unit of energy [t=1]. ℋ_ AB is the sublattice potential

    ℋ_ AB=V_ AB∑_s=↑,↓∑_iτ_ic^†_i,sc_i,s,

where τ_i=+1 for sites i belonging to sublattice A, and τ_i=-1 for sites i belonging to sublattice B. ℋ_ KM is the standard Kane-Mele spin-orbit coupling <cit.> given by 

    ℋ_ KM=iλ_ KM∑_s,s'∑_⟨⟨ i,j⟩⟩ν_ij c^†_i,s(ŝ_z)_s,s'c_j,s'.

Here ν_ij=sign( d_1× d_2)_z=± 1, where d_1 and d_2 are unit vectors along the two bonds that the electron traverses when hopping from the site j to the next nearest neighbor i [indicated Eq.(<ref>) by ⟨⟨ i,j⟩⟩]. Finally the Rashba spin-orbit coupling ℋ_ R can be cast as,

    ℋ_ R=iλ_ R∑_s,s'∑_⟨ i,j⟩c^†_i,s[( ŝ×𝐝_ij)_z ]_s,s'c_j,s' ,

where 𝐝_ij is the unity vector along the direction connecting site i to site j. ŝ=ŝ_x x+ŝ_y y+ŝ_z z is related to physical spin of electrons (up to a factor ħ/2) and ŝ_x,y,z are Pauli matrices. The Rashba coupling breaks (z→ -z)-symmetry <cit.> and is responsible for the non-conservation of spin in the full Hamiltonian [ℋ_ R, ŝ_z]≠ 0.

We show in Fig. <ref> (a) the LSCM calculated along the line indicated by the black arrow in Fig. <ref>. The blue squares are the LSCM for V_ AB and λ_ R adjusted to match a QSHI phase, while the red squares are the results for parameters in the trivial phase. For the system in the QSHI phase, as shown by the blue curve in Fig. <ref> (a), the LSCM assumes a quantized value +1, except for anomalies near the edges, which also occur in usual Chern markers <cit.>. We emphasize that this quantization of the LSCM [Eq. (<ref>)] occurs in the presence of a finite Rashba coupling, which is set to λ_ R=√(3)λ_ KM in Fig. <ref>. If one simply calculates 1/2(ℭ_↑-ℭ_↓), a non-quantized value is obtained, as indicated by the yellow squares in Fig. <ref> (a). We present in Fig. <ref> (b) and (d) the energy (E_n) and valence-state projected spin spectra (ξ^s_z_v) for the flake in the QSHI phase. It is worth noting that both spectra exhibit a gap, which is responsible for the topological protection of the LSCM. The red squares in Fig. <ref> (a) represent the results for the marker in the topologically trivial phase, which is zero, as expected. Panels (c) and (e) of Fig. <ref> show the corresponding energy and valence-projected spin spectra, which also display a gap, thereby enabling the definition of the marker. We follow Refs. <cit.> and use the central site of the flake in Fig. <ref> [r= 0 in Eq. (<ref>)] to calculate the topological phase diagram of LSCM. We show the results for ℭ_s( 0) in the λ_ R-V_ AB parameter space in Fig. <ref>. Setting aside the finite size effects, which are more pronounced near the topological phase transition, the phase diagram shown in Fig. <ref> aligns with the one obtained using the k-space formula for the ℤ_2-invariant <cit.>.







§ SUB-LATTICE-DEPENDENT KANE-MELE COUPLING 




We now examine the first generalization of the standard Kane-Mele-Rashba model that we consider in this paper. When a material with strong spin-orbit coupling breaks the C_3v symmetry of the honeycomb lattice, a sublattice-dependent Kane-Mele spin-orbit coupling emerges <cit.>,

    ℋ_ KM(ab)=i∑_s,s'∑_⟨⟨ i,j⟩⟩λ^ij_ KMν_ij c^†_i,s(ŝ_z)_s,s'c_j,s'.

where λ^ij_ KM=λ^ A_ KM for ⟨⟨ i,j⟩⟩∈ sublatice A, and λ^ij_ KM=λ^ B_ KM for ⟨⟨ i,j⟩⟩∈ sublatice B, with λ^ A_ KM≠λ^ B_ KM. This type of coupling occurs in graphene grown on TMDs in 2H structural phase, leading to intriguing spin-transport and topological features <cit.>. The complete Hamiltonian considered in this section is given by 

    ℋ_ 𝒢ℯ𝓃 1=ℋ_t+ℋ_ AB+ℋ_ R+ℋ_ KM(ab).

where, ℋ_t, ℋ_ AB and ℋ_ R are the terms defined in Eqs. (<ref>, <ref>) and (<ref>). The topological phase diagram of the model in Eq. <ref> was calculated in Ref. <cit.> using the k-space formula for the ℤ_2 invariant. Here, we perform analogous calculations to those detailed in the previous section to index the topology of this model in real space using the LSCM defined in Eqs. (<ref>, <ref>). We also study their energetic and spin spectral properties.

Figure <ref> (a) shows the topological phase diagram of the Hamiltonian in Eq. (<ref>), based on calculations of the LSCM at the central site of the flake ℭ_s( 0). In this figure, we set fixed values for V_ AB=0.1 t and λ_ R=0.05 t and calculated the markers for different values in λ^ A_ KM-λ^ B_ KM space. The LSCM-diagram in Fig. <ref> (a) matches the ℤ_2-diagram in Ref. <cit.>. In the two quadrants where sign(λ^ A_ KM)=sign(λ^ B_ KM), a QSHI phase with a quantized LSCM is obtained. Notably, the LSCM in each topological region have opposite signals. The signal of the spin Chern number does not provide additional information about the topology of the ground state electronic wave function; it is simply a matter of definition in the projection procedure <cit.> described in Sec. <ref>. The two regions correspond to the same topological order, and the connection of LSCM with the local ℤ_2 marker is given by ν ( 0)=ℭ_s( 0)  mod 2 <cit.>. The rest of the diagram is characterized by two insulating phases with zero LSCM, in agreement with Ref. <cit.>. These phases exhibit interesting properties due to the presence of what is known as valley Zeeman coupling, including pseudo-helical states <cit.> and unconventional spin-transport phenomena <cit.>. The distinct regions of the phase diagram can be reached by fabricating heterostructures of graphene combined with different members of the 2H-TMD family <cit.>. In panels (b) and (c) of Fig. <ref>, we present the energy (Δ^ E) and valence-projected spin-matrix (Δ^s_z) gaps throughout the parameter space. Except in regions close to the phase transition lines, the two gaps are finite, which ensures the topological protection of LSCM in each phase.




 


§ IN-PLANE KANE-MELE COUPLING 


Now we consider a second generalization of the Kane-Mele coupling that may occur when a honeycomb lattice interacts with a low-symmetry environment. This generalization consists of a sub-lattice-dependent in-plane polarized (ŝ_y) Kane-Mele coupling 

    ℋ_ y(ab)=i∑_s,s'∑_⟨⟨ i,j⟩⟩λ^ij_ yν_ij c^†_i,s(ŝ_y)_s,s'c_j,s',

and λ^ij_ y=λ^ A_ y for ⟨⟨ i,j⟩⟩∈ sublatice A, and λ^ij_ y=λ^ B_ y for ⟨⟨ i,j⟩⟩∈ sublatice B, with λ^ A_ y≠λ^ B_ y. This effect has been predicted to occur when graphene is grown on top of materials with a low-symmetry crystal field and strong spin-orbit interaction <cit.>. A similar type of in-plane polarized coupling has also been predicted in the monolayer of WTe_2 in the low-symmetric 1T_d structural phase <cit.>. The Rashba interaction tends to destroy the QSHI state in the case of ŝ_z-polarized Kane-Mele couplings by closing its energy gap, making it a frequently cited obstacle to observing this phase in experiments <cit.>. On the other hand, the energy gap created by ŝ_y-Kane-Mele coupling is robust against the presence of Rashba interaction <cit.>, making the study of its properties appealing for the practical realization of topological phases. In this section, we focus on the study of the robustness of this topological phase against the Rashba effect, examining the Hamiltonian of the form:

    ℋ_ 𝒢ℯ𝓃 2=ℋ_t+ℋ_ R+ℋ_ y(ab),

where, ℋ_t and ℋ_ R are given by Eqs. (<ref>, <ref>).

The QSHI phase produced by the in-plane Kane-Mele coupling has a spin-Hall current polarized in the y-direction. Therefore, the projection procedure described in Sec. <ref> must be performed using the ŝ_y operator instead of ŝ_z. This means that, for the calculations of the LSCM for the model in Eq. (<ref>), we obtain the states of Eq. (<ref>) by diagonalizing the matrix M^s_y_ v.s.=𝒫ŝ_y𝒫: M^s_y_v.s.|ϕ_v⟩=ξ_v^s_y|ϕ_v⟩, where (v=1,..., N_occ). Using these states, we construct the projectors 𝒫_σ in Eq. (<ref>) and apply them to the calculations of the markers in Eqs. (<ref>) and (<ref>). The remaining discussion follows closely in analogy to the previous case. The LSCM is protected by the gap in the energy spectrum and in the valence-projected ŝ_y-spectrum.





In Fig. <ref> (a), we present the LSCM calculated for the model of Eq. (<ref>), along the line indicated by the black arrow in Fig. <ref>. In the results shown in Fig. <ref>, we use λ^ A_y = 0.19 t, λ^ B_y = 0.21 t, and two different values for λ_ R. For λ_ R = 3√(3)λ̅_ KM/2 [λ̅_ KM=0.2 t], the LSCM shows the expected profile indicative of a QSHI, which means it is quantized in the central region, with anomalies near the edges of the system due to finite-size effects. The quantization of LSCM in the central region for this case is protected by large gaps in both the energy and ξ^s_y spectra [shaded rectangles in Fig. <ref> (b) and (d)]. For λ_ R = 5√(3)λ̅_ KM/2, the finite-size effects are magnified due to the small gap Δ^s_y in the ξ^s_y spectra reported in Fig. <ref> (e). Nevertheless, the gap remains finite, and the central region with a quantized LSCM increases as the size of the flake grows. It is worth mentioning that we used λ^ A_y and λ^ B_y slightly differently because, for this model, the gap in the ξ^s_y spectra closes when λ^ A_y = λ^ B_y. This is a limitation of the method, as there is no symmetry imposition that guarantees the existence of a finite gap in ξ^s_y spectra. 

As mentioned above, an interesting feature of the spin-orbit coupling in Eq. (<ref>) is that it leads to the formation of a QSHI phase, which persists even at high Rashba coupling intensities. Figure <ref> (a, b) shows the LSCM at the central site ℭ_s( 0) and the corresponding gaps Δ^ E and Δ^s_y for the model in Eq. (<ref>), a function of λ_ R/λ̅_ KM [λ̅_ KM=0.2 t] and different flake sizes L. By inspecting the curves for L=24 in Fig. <ref> (b), one can observe two distinct behaviors in the gaps Δ^ E and Δ^s_y. As λ_ R increases from 0 to a value close to 2√(3)λ̅_ KM, the gap Δ^ E decreases from a finite value to zero, and the gap Δ^s_y decreases from 2 to ≈ 1. As the λ_ R continues to increase, the gap Δ^ E reappears and then oscillates around a finite value, while the gap Δ^s_y jumps to a small finite value and remains nearly constant even under strong λ_ R. The finite gaps Δ^ E and Δ^s_y in the two regimes are reflected in a quantized LSCM at the central site across a wide range of Rashba coupling intensities. Due to the non-zero values of Δ^ E and Δ^s_y, the LSCM is a well-defined topological invariant for all values of Rashba coupling, except at the specific value [λ_ R≈ 2√(3)λ̅_ KM] where Δ^ E vanishes. The small values of the gaps in the region where λ_ R≳ 2√(3)λ̅_ KM make the LSCM susceptible to finite-size effects. As illustrated by Fig. <ref> (a), quantization of LSCM is achieved only for larger flakes. However, the construction of projectors of Eq. (<ref>) remains formally well-defined <cit.> despite the small values of Δ^ E and Δ^s_y.

To contrast with the physics discussed above, we show analogous results for the model in Eq. (<ref>) with V_ AB=0 in Fig. <ref> (c, d). The difference in behavior for λ_ R≳ 2√(3)λ̅_ KM, compared to the case discussed in the previous paragraph, is evident. The LSCM of the central site deviates from its quantized value and oscillates around zero, and both gaps Δ^ E and Δ^s_z vanish as the size of the flake increases. The distinct behaviors of the energy gaps shown in Fig. <ref> are already present in the band structure of the models in the thermodynamic limit <cit.> and are reflected here in the spectra for the flake geometry and the LSCM.



§ FINAL REMARKS AND CONCLUSIONS 


In summary, we have used the local spin Chern marker to study the real-space topology of electrons described by three distinct Hamiltonians in a honeycomb lattice with finite flake geometry. The Hamiltonians considered here find applications in the field of graphene van der Waals heterostructures and are characterized by strong Rashba spin-orbit interaction, which leads to the non-conservation of the spin operator. We show that the local spin Chern marker successfully describes the topological phase diagram for the three spin-orbit couplings responsible for the manifestation of quantum spin Hall insulator physics considered in this work. The phase diagrams obtained are consistent with those calculated using the k-space formula for the ℤ_2-invariant. Additionally, we calculate the energy spectrum and the valence-projected spin spectrum, demonstrating that in most of the parameter space, both exhibit gaps that protect the marker against smooth variations in the model Hamiltonian.

Among the results presented in this work, we highlight those shown in Figs. <ref> and <ref>. As mentioned in the text, the robustness of the gaps Δ^ E and Δ^s_y produced by in-plane Kane-Mele coupling [Eq. (<ref>)] against the presence of the Rashba effect strengthens the possibility of observing the quantum spin Hall insulator phase in experiments. In this situation, the quantum spin-Hall insulator phase is characterized by an in-plane polarized spin current produced by this unconventional type of spin-orbit coupling. 

Our work broadens the scope of spin Chern numbers <cit.> as an index that accurately captures the topology associated with the quantum spin Hall insulator phase. This is achieved using the real-space version of the index, demonstrating the flexibility of this concept. The results presented here may be useful for future studies on the real-space characterization of the topology of complex materials.






§ ACKNOWLEDGMENTS

The authors are grateful to the Brazilian Agencies Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Coordenação de Aperfeiçoamento de Pessoal de Ensino Superior (CAPES). 




