AB-Stacked Bilayer Haldane Lattice

• The AB-stacked bilayer Haldane lattice is a bilayer honeycomb system combining Bernal stacking with complex next-nearest-neighbor hopping to produce nontrivial topological phases.
• It features a four-band structure with interlayer hybridization, anisotropic hopping, and higher-Chern phases (e.g., C = ±2) arising from band touchings and Floquet-driven mass competition.
• The model exhibits critical phenomena such as semi-Dirac point merging and modified edge state behavior, underpinning applications in quantum spin Hall insulators and magnonic systems.

The AB-stacked bilayer Haldane lattice denotes a family of Bernal-stacked honeycomb bilayer systems in which the Haldane mechanism—complex next-nearest-neighbor hopping with zero net flux through the unit cell—is combined with interlayer hybridization. In the electronic versions, the basis is typically $\{A_l,B_l,A_u,B_u\}$, with the $B$ sublattice of the upper layer directly above the $A$ sublattice of the lower layer and interlayer tunneling retained only on that vertical pair. Relative to the monolayer Haldane model, the bilayer introduces a four-band structure, interlayer splitting, higher-Chern phases such as $C=\pm 2$, and additional critical phenomena associated with band touchings at $\mathbf K$, $\mathbf K'$, and the semi-Dirac merger point $\mathbf M$ (Mondal et al., 2023, Huang et al., 2012, Mannaï et al., 2022, Khan et al., 25 Mar 2026).

1. Structural definition and model space

AB stacking, also called Bernal stacking, is the defining geometric ingredient. In one common convention, the $B$ sublattice of the upper layer lies directly above the $A$ sublattice of the lower layer, and the only retained interlayer hopping is $t_\perp$ on the vertical bond $B$0 (Mondal et al., 2023, Khan et al., 25 Mar 2026). In another notation used for the bilayer generalization of Haldane’s model, the first layer contains sites $B$1, the second layer contains $B$2, and $B$3 sites couple vertically to the $B$4 sites; this is again Bernal stacking in the manner of bilayer graphene (Huang et al., 2012). The common content of these notations is an asymmetric dimerization pattern that distinguishes AB from AA stacking.

The intralayer Haldane ingredient is the complex next-nearest-neighbor term $B$5, which breaks time-reversal symmetry while maintaining zero total flux per unit cell. In the anisotropic bilayer electronic model, the real-space Hamiltonian is written as

$B$6

with $B$7 on one nearest-neighbor bond direction $B$8 and $B$9 on $A$0 (Mondal et al., 2023). The anisotropy $A$1 is the “band engineering” that moves the Dirac points through the Brillouin zone.

A broader usage of the term encompasses closely related constructions. One line of work studies a bilayer of the modified Haldane model, where each isolated layer is semimetallic and AB stacking alone opens a chiral insulating gap with $A$2 (Mannaï et al., 2022). Another studies AB-stacked moiré bilayers in which the honeycomb sublattices reside in different layers and a generalized Kane-Mele Hamiltonian realizes a Haldane Chern insulator under a small magnetic field (Zhao et al., 2022). A further extension treats AB-stacked bilayer honeycomb quantum magnets, where alternating next-nearest-neighbour Dzyaloshinsky-Moriya interaction plays the role of the Haldane term for magnons (Owerre, 2016).

2. Canonical Hamiltonians and low-energy descriptions

In momentum space, the anisotropic electronic bilayer Haldane model is expressed in the basis $A$3 as

$A$4

with

$A$5

and

$A$6

The explicit functions $A$7 contain the Haldane phase $A$8, the anisotropic nearest-neighbor amplitudes $A$9, and the honeycomb trigonometric structure factors (Mondal et al., 2023).

For equal fluxes $C=\pm 2$0, the four bands are two conduction bands and two valence bands,

$C=\pm 2$1

$C=\pm 2$2

showing explicitly how interlayer hybridization splits the monolayer spectrum into $C=\pm 2$3 sectors (Mondal et al., 2023).

The bilayer generalization analyzed from the viewpoint of edge states, entanglement spectra, and Wannier functions adopts a block form

$C=\pm 2$4

where each layer is a Haldane monolayer and $C=\pm 2$5 encodes the Bernal interlayer hopping (Huang et al., 2012). In the special case $C=\pm 2$6, the Hamiltonian satisfies an extended inversion-related symmetry

$C=\pm 2$7

denoted $C=\pm 2$8, which is not ordinary inversion and becomes central for the bilayer’s non-edge topological signatures (Huang et al., 2012).

In the stacking-induced Chern-insulator construction, the starting point is instead the modified Haldane model, where the Haldane mass is replaced by a valley-dependent scalar potential. Two time-reversed copies with $C=\pm 2$9 are stacked in AB geometry, and the interlayer dimer coupling $\mathbf K$0 removes the nodal-line degeneracy by band repulsion (Mannaï et al., 2022). This mechanism differs microscopically from the ordinary bilayer Haldane lattice, but it produces a closely related AB-stacked chiral phase.

3. Topological invariants and Chern-sector organization

The defining topological data are the band Chern numbers. In the band-engineered AB bilayer Haldane model, the valence band closer to the Fermi level, $\mathbf K$1, carries

$\mathbf K$2

whereas the valence band farther from the Fermi level, $\mathbf K$3, typically carries

$\mathbf K$4

The conduction bands have the same magnitudes with opposite signs. A notable feature is that the higher-Chern phases $\mathbf K$5 occur only in the band $\mathbf K$6, the valence band nearest $\mathbf K$7 (Mondal et al., 2023).

Multiple topological phase transitions occur when gaps close and reopen. Reported jumps include

$\mathbf K$8

with the relevant band touchings occurring at $\mathbf K$9, $\mathbf K'$0, and in the anisotropic regime near the $\mathbf K'$1-related merger (Mondal et al., 2023). These transitions are the bilayer analogue of mass-sign reversals in a Dirac description, but the four-band structure permits a larger set of gap-closing channels than in the monolayer.

The stacking-induced modified-Haldane bilayer provides a more explicit higher-Chern formula. Near valley $\mathbf K'$2, the low-energy effective Hamiltonian takes the form

$\mathbf K'$3

and the lowest-band Chern number is

$\mathbf K'$4

For $\mathbf K'$5, this yields

$\mathbf K'$6

so the AB bilayer realizes a Chern insulator with $\mathbf K'$7 (Mannaï et al., 2022).

A more conventional bilayer rule appears when two Haldane monolayers are coupled without closing the bulk gap: the total Chern number is the sum of the two monolayer Chern numbers,

$\mathbf K'$8

This additive statement holds in the bilayer generalization as long as the central gap remains open under interlayer coupling (Huang et al., 2012). The result is conceptually important because it distinguishes ordinary Chern addition from stacking-induced topology, where the uncoupled layers may be semimetallic rather than individually topological (Mannaï et al., 2022).

4. Anisotropy, semi-Dirac criticality, and Floquet mass competition

A characteristic feature of several AB bilayer Haldane models is the motion and merger of Dirac points under anisotropic nearest-neighbor hopping. As $\mathbf K'$9 increases from $\mathbf M$0, the two inequivalent Dirac points move toward each other, and at

$\mathbf M$1

they merge at the Brillouin-zone $\mathbf M$2 point. At this semi-Dirac limit, the dispersion is linear along one direction and quadratic along the other (Mondal et al., 2023, Khan et al., 25 Mar 2026).

The Floquet-driven study makes the low-energy structure explicit. Expanding around the merger point,

$\mathbf M$3

so that

$\mathbf M$4

The semi-Dirac point is therefore an anisotropic critical point rather than an ordinary Dirac crossing (Khan et al., 25 Mar 2026).

In the static band-engineered bilayer, increasing $\mathbf M$5 causes the Berry-curvature “Chern lobes” in the $\mathbf M$6 plane to shrink; at $\mathbf M$7, the gap closes and the Chern numbers collapse to zero; for $\mathbf M$8, the gap reopens but the phases are topologically trivial (Mondal et al., 2023). The Floquet-driven bilayer introduces an additional tunable mass,

$\mathbf M$9

through off-resonant circularly polarized light with vector potential

$B$0

The effective mass entering the low-energy description becomes

$B$1

so topology is controlled by the competition between the intrinsic Haldane mass and the Floquet-induced mass (Khan et al., 25 Mar 2026).

Interlayer coupling is not a passive perturbation in this regime. The Floquet study reports that interlayer hybridization reshapes the band topology by inducing helicity-dependent and valley-selective band inversions at $B$2 and $B$3, stabilizing higher-Chern phases in the valence bands and redistributing the Berry curvature over the Brillouin zone (Khan et al., 25 Mar 2026). This suggests that the post-merger behavior depends sensitively on whether the system is treated as a purely static band-engineered bilayer or as a driven Floquet bilayer with an additional mass channel.

5. Boundary spectra, Wilson loops, and transport signatures

The usual bulk-boundary correspondence survives in many AB bilayer Haldane regimes. In the band-engineered bilayer nanoribbon geometry, for $B$4 there are chiral edge states traversing the bulk gap, and in $B$5 phases there are two edge modes per boundary, with opposite boundaries carrying currents in opposite directions (Mondal et al., 2023). In the AB-stacked modified-Haldane ribbons, finite $B$6 opens a bulk gap and two chiral edge states appear at each boundary; reversing the sign of the phases reverses their propagation and changes $B$7 (Mannaï et al., 2022).

The anomalous Hall response provides a complementary bulk diagnostic. For the static band-engineered bilayer,

$B$8

and when the Fermi energy lies in the bulk gap the conductivity is quantized at

$B$9

The plateau width equals the bulk-gap width, narrows as $A$0, disappears at the semi-Dirac point, and vanishes in the trivial regime (Mondal et al., 2023). The Floquet-driven bilayer likewise exhibits quantized anomalous Hall plateaus at

$A$1

together with plateau collapse or sign reversal under changes in anisotropy, helicity, and interlayer coupling (Khan et al., 25 Mar 2026).

A distinct bilayer phenomenon is the breakdown of the monolayer edge–entanglement–Wannier correspondence. In the $A$2-symmetric opposite-Chern case $A$3, the open-boundary edge spectrum is gapped and shows no protected crossing, yet the entanglement spectrum contains protected half-occupancy modes and the non-Abelian Wilson loop exhibits oppositely winding eigenphases crossing at $A$4 (Huang et al., 2012). The Wilson loop is

$A$5

and in the $A$6-symmetric bilayer it reduces to

$A$7

This is not the standard inversion-protected $A$8 mechanism; the protecting symmetry is $A$9, not ordinary inversion, and the protected entanglement feature is not pinned to inversion-invariant momenta (Huang et al., 2012).

6. Realizations, analogues, and conceptual boundaries

The most direct experimental solid-state realization in the supplied literature is the AB-stacked MoTe$t_\perp$0/WSe$t_\perp$1 moiré heterobilayer, whose moiré potential forms a honeycomb lattice with the two sublattices residing in different layers (Zhao et al., 2022). At filling $t_\perp$2, the moiré bilayer is a quantum spin Hall insulator with a tunable charge gap. Under a small out-of-plane magnetic field it becomes a Chern insulator with Chern number $t_\perp$3, described by a generalized Kane-Mele tight-binding Hamiltonian containing nearest-neighbor inter-sublattice hopping, complex next-nearest-neighbor hopping, a tunable layer potential difference $t_\perp$4, and a Zeeman term $t_\perp$5 (Zhao et al., 2022). Experimentally, the Hall plateau approaches

$t_\perp$6

within about $t_\perp$7 for fields between $t_\perp$8 T and $t_\perp$9 T, while $B$00 drops below $B$01, and the line in $B$02 space follows

$B$03

consistent with the Středa relation and $B$04 (Zhao et al., 2022).

The AB-stacked bilayer Haldane lattice also has a bosonic analogue in honeycomb quantum magnets. There, alternating next-nearest-neighbour Dzyaloshinsky-Moriya interaction is the Haldane ingredient, producing topological magnon bands in an AB-stacked bilayer geometry with sublattices $B$05 and interlayer exchanges $B$06 (Owerre, 2016). For ferromagnetic interlayer coupling, the magnon Chern numbers are

$B$07

whereas for antiferromagnetic interlayer coupling they become

$B$08

Edge-state propagation is correspondingly layer-co-propagating in the ferromagnetic case and layer-counter-propagating in the antiferromagnetic case (Owerre, 2016).

A further conceptual extension replaces explicit Haldane hopping by toroidal magnetic order. A simplified two-band excitonic-insulator model with toroidal order is unitarily equivalent, at low energy, to the Haldane model on the honeycomb lattice with zero net flux through the unit cell (Belyavsky et al., 2010). The toroidal moment density is

$B$09

and the ordered phase yields a Haldane-type insulating state with broken time-reversal symmetry, zero total flux per unit cell, nonzero topological index, and chiral edge modes (Belyavsky et al., 2010). This is not an AB-stacked bilayer construction, but it establishes a symmetry-based route to Haldane physics in real magnetic materials.

Several misconceptions are excluded by these results. Zero net flux does not imply topological triviality: the Haldane mechanism, the toroidal-order analogue, and the magnon DMI construction all rely on staggered or periodic internal flux patterns with zero total flux per unit cell (Belyavsky et al., 2010, Owerre, 2016). AB stacking is not interchangeable with AA stacking: in the stacking-induced modified-Haldane bilayer, AA stacking remains gapless when $B$10 and becomes a trivial insulator when Semenoff masses are present, with no Chern-insulating phase induced by AA stacking (Mannaï et al., 2022). The moiré realization likewise states that the mechanism is not expected in AA-stacked bilayers (Zhao et al., 2022). A plausible implication is that the defining content of the AB-stacked bilayer Haldane lattice is not merely “two Haldane layers plus hopping,” but a specific interplay of Bernal geometry, layer-selective hybridization, and Haldane-type mass generation.