Multilayer Kane–Mele Model

• The multilayer Kane–Mele model is a class of systems defined by stacking or mimicking multiple quantum spin Hall layers, with its topology determined by a parity-sensitive Z2 invariant.
• It exhibits key phenomena such as doubled spin Hall conductivity, correlation-induced edge gaps, and a mod-2 stacking law, highlighting how odd versus even layers yield distinct topological phases.
• Methodological approaches include tight-binding Hamiltonians, quantum Monte Carlo simulations, and effective field theory to analyze interaction effects, disorder, and electromagnetic response.

Searching arXiv for the cited Kane–Mele and multilayer-related papers to ground the article in the literature. arXiv search query: (Bercx et al., 2014) OR (Goryo et al., 2011) OR (Bachmann et al., 2024) OR (Wang et al., 2022) OR (Dai et al., 2024) OR (Orth et al., 2015) OR (Zheng et al., 2010) The multilayer Kane–Mele model denotes a class of systems built by stacking, duplicating, or otherwise coupling copies of the two-dimensional Kane–Mele quantum spin Hall insulator, and more broadly includes effective multi-channel realizations in which a single physical layer emulates multiple Kane–Mele copies. Across these realizations, the central organizing principles are the parity-sensitive $\mathbb{Z}_2$ structure of time-reversal-invariant topological bands, the possibility of higher spin-sector Chern numbers such as $|C|=2$, and the nontrivial role of interactions, disorder, flux insertion, and crystalline symmetries in determining whether edge or surface modes remain gapless. In the noninteracting limit, multilayer stacking reproduces the familiar mod-2 addition law of Kane–Mele topology; in interacting settings, this structure persists through the many-body Fu–Kane–Mele index, while additional phenomena such as correlation-induced edge gaps and Meissner-like electromagnetic response emerge in specific layered constructions (Bachmann et al., 2024, Goryo et al., 2011, Bercx et al., 2014).

1. Concept and scope

In its most literal sense, a multilayer Kane–Mele model is a stack of two-dimensional Kane–Mele quantum spin Hall layers, each defined on a honeycomb lattice with intrinsic spin–orbit coupling and, in some variants, Hubbard interaction. In the layered Kane–Mele–Hubbard construction of magnetic response, the system is a stack of decoupled quantum spin Hall layers separated by an interlayer distance $d$; the microscopic Hamiltonian is written per layer, with no explicit interlayer hopping, so layering enters through the effective three-dimensional response coefficients rather than through band hybridization (Goryo et al., 2011). In a different but closely related sense, multilayer Kane–Mele physics may also be realized effectively within a single layer when symmetry and enlarged unit cells generate multiple helical channels that behave like stacked Kane–Mele copies. The $\pi$-flux Kane–Mele–Hubbard model is the clearest such example: each spin sector realizes an effective doubled Chern insulator, and the edge hosts two Kramers doublets analogous to two decoupled Kane–Mele layers (Bercx et al., 2014).

This broader usage reflects a structural rather than merely geometric definition. What identifies a system as “multilayer Kane–Mele” is not only physical stacking, but the presence of multiple Kane–Mele-like channels whose topological content combines according to the same $\mathbb{Z}_2$ parity law. The many-body Fu–Kane–Mele framework makes this explicit by treating stacking as a tensor-product operation on symmetric short-range-entangled states and showing that the topological index is additive mod 2 under stacking (Bachmann et al., 2024).

A common misconception is that multilayer Kane–Mele systems are automatically nontrivial whenever each constituent layer is nontrivial. The literature instead shows that only the parity of the number of nontrivial layers is topologically stable in class AII with charge conservation and time-reversal symmetry: odd stacks remain nontrivial, whereas even stacks are trivial in the interacting $\mathbb{Z}_2$ classification (Bachmann et al., 2024). This parity structure coexists with richer band-theoretic quantities, including $|C|=2$ per spin sector and doubled spin Hall conductivity in effective two-copy realizations (Bercx et al., 2014).

2. Microscopic constructions

The standard single-layer Kane–Mele–Hubbard Hamiltonian on a honeycomb lattice provides the basic building block for multilayer generalizations. In the particle-hole-symmetric formulation,

$$\begin{aligned} H_0 &= -t\sum_{\langle i,j\rangle,\sigma} c_{i\sigma}^{\dagger} c_{j\sigma} + i\lambda \sum_{\langle\langle i,i'\rangle\rangle,\alpha,\beta} \Big[ c^{\dagger}_{i\alpha}\,\sigma_{z,\alpha\beta}\,c_{i'\beta} - c^\dagger_{i'\alpha}\,\sigma_{z,\alpha\beta}\,c_{i\beta} \Big] - \mu \sum_{i,\sigma} c_{i\sigma}^\dagger c_{i\sigma}, \end{aligned}$$

with on-site interaction

$$H_{\text{int}} = U \sum_i \Big(n_{i\uparrow} - \frac{1}{2} \Big) \Big(n_{i\downarrow} - \frac{1}{2} \Big).$$

The purely imaginary next-nearest-neighbor hopping is central both to the topological structure and to the exact particle-hole symmetry at half filling $\mu=0$ (Zheng et al., 2010).

The layered Kane–Mele–Hubbard model used for electromagnetic response studies begins from the standard per-layer Kane–Mele Hamiltonian

$|C|=2$0

with gap

$|C|=2$1

In this formulation the layers are effectively decoupled microscopically, $|C|=2$2, and the multilayer character is encoded via the interlayer distance $|C|=2$3, which enters the effective coupling

$|C|=2$4

and the spin Hall conductivity per sample volume (Goryo et al., 2011).

A distinct route to multilayer Kane–Mele physics is furnished by the $|C|=2$5-flux Kane–Mele–Hubbard model, in which each honeycomb plaquette carries a magnetic $|C|=2$6 flux. For fixed spin $|C|=2$7, the spinless $|C|=2$8-flux Haldane-like Hamiltonian is

$|C|=2$9

with Peierls factors satisfying

$d$0

around each hexagon. Because the $d$1 flux doubles the unit cell, the model contains two hexagons and four orbitals per cell, thereby generating an effective two-layer structure in momentum space (Bercx et al., 2014).

The full spinful noninteracting $d$2-Kane–Mele Hamiltonian adds Rashba coupling,

$d$3

and the Hubbard interaction is

$d$4

This model is not a literal stack, but it is repeatedly used as an explicit analogue of a two-layer Kane–Mele system (Bercx et al., 2014).

A material route to Kane–Mele-type multilayers is provided by $d$5N$d$6-embedded graphene, whose low-energy sector is captured by a modified Kane–Mele block

$d$7

$d$8

This monolayer block, with anisotropic velocities and a symmetry-allowed $d$9 term, is explicitly presented as a natural building block for multilayer Kane–Mele constructions with interlayer coupling (Wang et al., 2022).

3. Topological classification under stacking

For free or weakly interacting Kane–Mele layers, topology is commonly described by the Fu–Kane–Mele $\pi$0 invariant. The many-body extension establishes the same classification for interacting, symmetric, stably short-range-entangled states of two-dimensional fermions with U(1) charge conservation and time reversal satisfying $\pi$1. The many-body invariant is defined through $\pi$2-flux insertion: a state is nontrivial precisely when the fluxon transforms under time reversal as part of a Kramers pair (Bachmann et al., 2024).

The construction begins with a symmetric short-range-entangled pure state $\pi$3 and a symmetric parent Hamiltonian. A half-plane U(1) twist is implemented quasi-adiabatically, then restricted to a half-line to create a localized flux defect. Denoting the $\pi$4 defect states by $\pi$5, one proves that they belong to the same superselection sector and differ by an even almost-local unitary. In the GNS representation, time reversal is represented by an antiunitary $\pi$6, and the index is determined by whether

$\pi$7

or

$\pi$8

The latter case defines the nontrivial phase (Bachmann et al., 2024).

For multilayer Kane–Mele systems, the crucial structural result is multiplicativity under stacking: $\pi$9 Interpreting $\mathbb{Z}_2$0 as a $\mathbb{Z}_2$1 label, this is addition mod 2. A single quantum spin Hall layer has $\mathbb{Z}_2$2; a stack of two identical nontrivial layers has

$\mathbb{Z}_2$3

hence is topologically trivial in the interacting classification (Bachmann et al., 2024). This is the rigorous many-body version of the statement that only an odd number of Kane–Mele layers yields a nontrivial two-dimensional AII phase.

The $\mathbb{Z}_2$4-flux Kane–Mele realization illustrates the same parity principle from a band-theoretic angle. At half filling, despite a nonzero spin Hall conductivity and two helical edge channels, the two-dimensional $\mathbb{Z}_2$5 invariant is

$\mathbb{Z}_2$6

so the system is $\mathbb{Z}_2$7-trivial. At quarter and three-quarter filling, by contrast, $\mathbb{Z}_2$8, giving a conventional nontrivial quantum spin Hall phase (Bercx et al., 2014). This distinction clarifies that doubled edge structures and doubled spin Hall conductivity do not by themselves imply a nontrivial $\mathbb{Z}_2$9 index.

A frequent misconception is to equate the number of helical channels with the $\mathbb{Z}_2$0 index. The cited results show instead that the $\mathbb{Z}_2$1 invariant detects only parity. Two Kramers pairs per edge, whether realized by two physical layers or by two momentum-distinguished channels in a single layer, are topologically equivalent to the trivial class unless an additional symmetry forbids their hybridization (Bachmann et al., 2024, Bercx et al., 2014).

4. Band topology, higher Chern number sectors, and symmetry-protected edges

The standard single-layer Kane–Mele model is built from two Haldane sectors with opposite Chern numbers, producing a quantum spin Hall insulator with one Kramers pair per edge. In multilayer or effective multi-copy realizations, each spin sector can acquire $\mathbb{Z}_2$2. The $\mathbb{Z}_2$3-flux Kane–Mele–Hubbard model provides an explicit example: for each spin sector, the occupied bands at half filling have total Chern number

$\mathbb{Z}_2$4

depending on the sign and magnitude of $\mathbb{Z}_2$5 (Bercx et al., 2014).

With $\mathbb{Z}_2$6 spin conservation, the two spin sectors carry opposite Chern numbers, leading to

$\mathbb{Z}_2$7

Thus the spin Hall conductivity is quantized to $\mathbb{Z}_2$8, precisely the value expected from two decoupled quantum spin Hall copies (Bercx et al., 2014). This doubled response is central to the analogy between $\mathbb{Z}_2$9-flux Kane–Mele and multilayer Kane–Mele systems.

The associated edge spectrum on a zigzag ribbon contains two Kramers doublets per edge, one crossing at $|C|=2$0 and one at $|C|=2$1. These are helical states, but they are not protected by the bulk $|C|=2$2 invariant. Their protection instead derives from edge translation symmetry: mixing the two channels requires momentum transfer $|C|=2$3, so single-particle backscattering is forbidden as long as translation symmetry along the edge is preserved (Bercx et al., 2014). Only when translation symmetry is broken and spin rotation is also broken, for example by Rashba coupling, can a single-particle gap open in the edge spectrum.

This mechanism is the direct analogue of layer-index conservation in a true multilayer system. If two helical channels belong to distinct layers and interlayer single-particle scattering is absent, they remain gapless even though the overall $|C|=2$4 index is trivial. Once interlayer tunneling is allowed, the even stack can gap without breaking time reversal. The $|C|=2$5-flux model recasts this logic in momentum space by replacing the layer index with crystal momentum $|C|=2$6 versus $|C|=2$7 (Bercx et al., 2014).

The modified Kane–Mele model with staggered intrinsic spin–orbit coupling provides another route to multichannel behavior. Without Rashba coupling, and with sublattice-dependent intrinsic SOC,

$|C|=2$8

the system enters a two-dimensional Weyl nodal-line semimetal regime when $|C|=2$9. In that regime $$\begin{aligned} H_0 &= -t\sum_{\langle i,j\rangle,\sigma} c_{i\sigma}^{\dagger} c_{j\sigma} + i\lambda \sum_{\langle\langle i,i'\rangle\rangle,\alpha,\beta} \Big[ c^{\dagger}_{i\alpha}\,\sigma_{z,\alpha\beta}\,c_{i'\beta} - c^\dagger_{i'\alpha}\,\sigma_{z,\alpha\beta}\,c_{i\beta} \Big] - \mu \sum_{i,\sigma} c_{i\sigma}^\dagger c_{i\sigma}, \end{aligned}$$0, but the bulk is gapless, so the phase is a $$\begin{aligned} H_0 &= -t\sum_{\langle i,j\rangle,\sigma} c_{i\sigma}^{\dagger} c_{j\sigma} + i\lambda \sum_{\langle\langle i,i'\rangle\rangle,\alpha,\beta} \Big[ c^{\dagger}_{i\alpha}\,\sigma_{z,\alpha\beta}\,c_{i'\beta} - c^\dagger_{i'\alpha}\,\sigma_{z,\alpha\beta}\,c_{i\beta} \Big] - \mu \sum_{i,\sigma} c_{i\sigma}^\dagger c_{i\sigma}, \end{aligned}$$1 topological metal rather than an insulator (Dai et al., 2024). The associated antihelical edge states are not a canonical multilayer phenomenon, but they illustrate how modified Kane–Mele blocks can furnish unconventional spin-channel structures that may serve as ingredients for layered generalizations.

5. Interaction effects

Interaction effects in Kane–Mele systems appear both in the bulk and at boundaries, and their importance increases in multichannel settings. In the single-layer Kane–Mele–Hubbard model with purely imaginary next-nearest-neighbor hopping, determinant quantum Monte Carlo is sign-problem-free at half filling because the spin-up and spin-down fermion matrices are complex conjugates for every Hubbard–Stratonovich configuration. This allows high-precision analysis of the evolution from a topological band insulator to an antiferromagnetic Mott insulator (Zheng et al., 2010).

As $$\begin{aligned} H_0 &= -t\sum_{\langle i,j\rangle,\sigma} c_{i\sigma}^{\dagger} c_{j\sigma} + i\lambda \sum_{\langle\langle i,i'\rangle\rangle,\alpha,\beta} \Big[ c^{\dagger}_{i\alpha}\,\sigma_{z,\alpha\beta}\,c_{i'\beta} - c^\dagger_{i'\alpha}\,\sigma_{z,\alpha\beta}\,c_{i\beta} \Big] - \mu \sum_{i,\sigma} c_{i\sigma}^\dagger c_{i\sigma}, \end{aligned}$$2 increases, three regimes are identified: a topological band insulator with stable helical edges, a bulk paramagnetic phase with unstable edges, and a bulk antiferromagnetic phase (Zheng et al., 2010). In the weak-coupling regime the edge is described by a helical Luttinger liquid. At stronger coupling, edge spin correlations increase and the effective Luttinger parameter falls below $$\begin{aligned} H_0 &= -t\sum_{\langle i,j\rangle,\sigma} c_{i\sigma}^{\dagger} c_{j\sigma} + i\lambda \sum_{\langle\langle i,i'\rangle\rangle,\alpha,\beta} \Big[ c^{\dagger}_{i\alpha}\,\sigma_{z,\alpha\beta}\,c_{i'\beta} - c^\dagger_{i'\alpha}\,\sigma_{z,\alpha\beta}\,c_{i\beta} \Big] - \mu \sum_{i,\sigma} c_{i\sigma}^\dagger c_{i\sigma}, \end{aligned}$$3, rendering two-particle backscattering relevant when symmetry permits. The bulk ultimately develops easy-plane antiferromagnetic order. This sequence is significant for multilayer systems because coupling multiple helical channels generally enhances the space of relevant interaction processes.

The $$\begin{aligned} H_0 &= -t\sum_{\langle i,j\rangle,\sigma} c_{i\sigma}^{\dagger} c_{j\sigma} + i\lambda \sum_{\langle\langle i,i'\rangle\rangle,\alpha,\beta} \Big[ c^{\dagger}_{i\alpha}\,\sigma_{z,\alpha\beta}\,c_{i'\beta} - c^\dagger_{i'\alpha}\,\sigma_{z,\alpha\beta}\,c_{i\beta} \Big] - \mu \sum_{i,\sigma} c_{i\sigma}^\dagger c_{i\sigma}, \end{aligned}$$4-flux Kane–Mele–Hubbard model makes this enhancement explicit. At the fine-tuned point $$\begin{aligned} H_0 &= -t\sum_{\langle i,j\rangle,\sigma} c_{i\sigma}^{\dagger} c_{j\sigma} + i\lambda \sum_{\langle\langle i,i'\rangle\rangle,\alpha,\beta} \Big[ c^{\dagger}_{i\alpha}\,\sigma_{z,\alpha\beta}\,c_{i'\beta} - c^\dagger_{i'\alpha}\,\sigma_{z,\alpha\beta}\,c_{i\beta} \Big] - \mu \sum_{i,\sigma} c_{i\sigma}^\dagger c_{i\sigma}, \end{aligned}$$5, the bulk exhibits a quadratic band crossing point with low-energy dispersion

$$\begin{aligned} H_0 &= -t\sum_{\langle i,j\rangle,\sigma} c_{i\sigma}^{\dagger} c_{j\sigma} + i\lambda \sum_{\langle\langle i,i'\rangle\rangle,\alpha,\beta} \Big[ c^{\dagger}_{i\alpha}\,\sigma_{z,\alpha\beta}\,c_{i'\beta} - c^\dagger_{i'\alpha}\,\sigma_{z,\alpha\beta}\,c_{i\beta} \Big] - \mu \sum_{i,\sigma} c_{i\sigma}^\dagger c_{i\sigma}, \end{aligned}$$6

and a finite density of states at the Fermi level. Mean-field theory and sign-problem-free auxiliary-field quantum Monte Carlo show that any $$\begin{aligned} H_0 &= -t\sum_{\langle i,j\rangle,\sigma} c_{i\sigma}^{\dagger} c_{j\sigma} + i\lambda \sum_{\langle\langle i,i'\rangle\rangle,\alpha,\beta} \Big[ c^{\dagger}_{i\alpha}\,\sigma_{z,\alpha\beta}\,c_{i'\beta} - c^\dagger_{i'\alpha}\,\sigma_{z,\alpha\beta}\,c_{i\beta} \Big] - \mu \sum_{i,\sigma} c_{i\sigma}^\dagger c_{i\sigma}, \end{aligned}$$7 triggers transverse antiferromagnetic order at this point, while a finite $$\begin{aligned} H_0 &= -t\sum_{\langle i,j\rangle,\sigma} c_{i\sigma}^{\dagger} c_{j\sigma} + i\lambda \sum_{\langle\langle i,i'\rangle\rangle,\alpha,\beta} \Big[ c^{\dagger}_{i\alpha}\,\sigma_{z,\alpha\beta}\,c_{i'\beta} - c^\dagger_{i'\alpha}\,\sigma_{z,\alpha\beta}\,c_{i\beta} \Big] - \mu \sum_{i,\sigma} c_{i\sigma}^\dagger c_{i\sigma}, \end{aligned}$$8 is needed away from it (Bercx et al., 2014). This supports the broader inference that higher-Chern or multichannel Kane–Mele-like systems are more susceptible to interaction-driven instabilities.

The same model also yields a detailed bosonized edge theory for two helical channels. The low-energy Hamiltonian,

$$\begin{aligned} H_0 &= -t\sum_{\langle i,j\rangle,\sigma} c_{i\sigma}^{\dagger} c_{j\sigma} + i\lambda \sum_{\langle\langle i,i'\rangle\rangle,\alpha,\beta} \Big[ c^{\dagger}_{i\alpha}\,\sigma_{z,\alpha\beta}\,c_{i'\beta} - c^\dagger_{i'\alpha}\,\sigma_{z,\alpha\beta}\,c_{i\beta} \Big] - \mu \sum_{i,\sigma} c_{i\sigma}^\dagger c_{i\sigma}, \end{aligned}$$9

describes a two-component Tomonaga–Luttinger liquid (Bercx et al., 2014). At half filling, three kinds of umklapp processes are allowed: $$H_{\text{int}} = U \sum_i \Big(n_{i\uparrow} - \frac{1}{2} \Big) \Big(n_{i\downarrow} - \frac{1}{2} \Big).$$0

$$H_{\text{int}} = U \sum_i \Big(n_{i\uparrow} - \frac{1}{2} \Big) \Big(n_{i\downarrow} - \frac{1}{2} \Big).$$1

$$H_{\text{int}} = U \sum_i \Big(n_{i\uparrow} - \frac{1}{2} \Big) \Big(n_{i\downarrow} - \frac{1}{2} \Big).$$2

Their bosonized form contains cosine operators $$H_{\text{int}} = U \sum_i \Big(n_{i\uparrow} - \frac{1}{2} \Big) \Big(n_{i\downarrow} - \frac{1}{2} \Big).$$3, $$H_{\text{int}} = U \sum_i \Big(n_{i\uparrow} - \frac{1}{2} \Big) \Big(n_{i\downarrow} - \frac{1}{2} \Big).$$4, and $$H_{\text{int}} = U \sum_i \Big(n_{i\uparrow} - \frac{1}{2} \Big) \Big(n_{i\downarrow} - \frac{1}{2} \Big).$$5 (Bercx et al., 2014).

The decisive result is that inter-channel umklapp $$H_{\text{int}} = U \sum_i \Big(n_{i\uparrow} - \frac{1}{2} \Big) \Big(n_{i\downarrow} - \frac{1}{2} \Big).$$6 is always relevant at weak coupling, whereas intra-channel umklapp is irrelevant unless repulsion is strong enough to push $$H_{\text{int}} = U \sum_i \Big(n_{i\uparrow} - \frac{1}{2} \Big) \Big(n_{i\downarrow} - \frac{1}{2} \Big).$$7 (Bercx et al., 2014). This distinguishes multi-channel from single-channel helical liquids. In multilayer Kane–Mele language, once two helical edges are present and half filled, interactions can gap all channels without breaking time reversal through interlayer or inter-channel two-particle scattering. Quantum Monte Carlo on ribbons with Hubbard interaction applied only at one edge confirms the opening of a correlation-induced edge gap at strong coupling and half filling (Bercx et al., 2014).

This body of work resolves another common misunderstanding: the absence of a nontrivial $$H_{\text{int}} = U \sum_i \Big(n_{i\uparrow} - \frac{1}{2} \Big) \Big(n_{i\downarrow} - \frac{1}{2} \Big).$$8 index does not imply immediate edge gapping, and conversely the presence of gapless noninteracting edge states does not guarantee many-body stability. Even-number channel structures may be robust at the single-particle level because of translation or layer conservation, yet be destabilized by symmetry-allowed inter-channel interactions (Bercx et al., 2014).

6. Electromagnetic response, disorder, and material realizations

A distinctive aspect of layered Kane–Mele systems is that stacking can alter not only topology but also electromagnetic response. In the layered Kane–Mele–Hubbard model, the low-energy continuum theory couples Dirac fermions to both the electromagnetic field $$H_{\text{int}} = U \sum_i \Big(n_{i\uparrow} - \frac{1}{2} \Big) \Big(n_{i\downarrow} - \frac{1}{2} \Big).$$9 and a spin gauge field $\mu=0$0. The induced effective Lagrangian contains a BF term,

$\mu=0$1

with spin Hall conductivity

$\mu=0$2

Here the factor $\mu=0$3 converts the two-dimensional per-layer response into a three-dimensional density appropriate to the stack (Goryo et al., 2011).

The Hubbard interaction enters via the mass-like term for the time component of the spin gauge field,

$\mu=0$4

and the combined BF-plus-mass theory supports a Meissner-like magnetic response without superconductivity. The Meissner condition is

$\mu=0$5

under which an applied magnetic field decays exponentially,

$\mu=0$6

with penetration depth

$\mu=0$7

This phenomenon is not superconducting Meissner screening; the electromagnetic U(1) symmetry remains unbroken, and the screening originates from the topological BF coupling to a massive spin gauge field (Goryo et al., 2011).

Rashba spin–orbit coupling modifies this response by inducing

$\mu=0$8

which renormalizes the effective coupling to

$\mu=0$9

so that the Meissner criterion becomes

$|C|=2$00

Because $|C|=2$01, Rashba coupling enhances the tendency toward Meissner-like screening within the perturbative regime analyzed (Goryo et al., 2011).

Disorder introduces a different route by which Kane–Mele topology can be altered. In the disordered Kane–Mele model,

$|C|=2$02

a staggered sublattice potential $|C|=2$03 is a necessary condition for the topological Anderson insulator transition (Orth et al., 2015). To lowest order in disorder strength $|C|=2$04, the Born approximation shows that $|C|=2$05 is renormalized while $|C|=2$06 is not, allowing a clean trivial insulator near the boundary $|C|=2$07 to become topological at finite disorder. This indicates that in layered or multi-copy Kane–Mele settings, disorder may drive topology by renormalizing mass-like parameters while leaving SOC terms comparatively intact (Orth et al., 2015).

On the materials side, $|C|=2$08N$|C|=2$09-embedded graphene realizes modified Kane–Mele physics in a concrete monolayer platform, with low-energy bands captured by the anisotropic Kane–Mele Hamiltonian already quoted and with nontrivial $|C|=2$10 confirmed by Wannier charge centers and helical edge states (Wang et al., 2022). Among the reported compounds, PtN$|C|=2$11C$|C|=2$12, IrN$|C|=2$13C$|C|=2$14, RhN$|C|=2$15C$|C|=2$16, and OsN$|C|=2$17C$|C|=2$18 exhibit topological gaps of $|C|=2$19 meV, $|C|=2$20 meV, $|C|=2$21 meV, and $|C|=2$22 meV, respectively (Wang et al., 2022). The authors explicitly note that this effective Hamiltonian is the relevant object for multilayer generalization, since one can stack the layer-resolved Dirac–Kane–Mele blocks and add symmetry-allowed interlayer terms (Wang et al., 2022).

These developments suggest a broad research program rather than a single canonical Hamiltonian. The multilayer Kane–Mele model is now understood as a family of constructions unified by Kane–Mele building blocks, mod-2 stacking topology, and a rich set of interaction and symmetry effects. Literal stacks of quantum spin Hall layers, effective doubled-channel models such as the $|C|=2$23-flux construction, modified Kane–Mele blocks with staggered spin–orbit structure, and realistic heavy-atom graphene derivatives all fall within this framework, provided the central topological and symmetry principles are preserved (Bachmann et al., 2024, Bercx et al., 2014, Goryo et al., 2011, Wang et al., 2022).