3D Su–Schrieffer–Heeger Model

• The three-dimensional SSH model is a topological framework defined by alternating strong and weak bonds along multiple spatial directions, leading to protected boundary states.
• It manifests in materials like bulk α-Bi and in bipartite chiral cubic-lattice models, realizing midgap surface and domain-wall states through dimerization and winding invariants.
• Key diagnostics include parity eigenvalues, Zak phases, and direction-resolved winding numbers, which provide concrete methods to identify and characterize topological boundary phenomena.

Searching arXiv for papers on the three-dimensional Su–Schrieffer–Heeger model to ground the article in published work. The three-dimensional Su–Schrieffer–Heeger model denotes a class of topological lattice systems that generalize the one-dimensional SSH chain to three spatial dimensions while retaining dimerization-driven boundary or interface phenomena. In one realization, pristine bulk $\alpha$-Bi provides a three-dimensional electronic analogue of the SSH model because alternating covalent intra-bilayer and van der Waals inter-bilayer bonding along $[111]$ reproduces the alternation of strong and weak links familiar from the one-dimensional chain, with antiphase domain walls acting as higher-dimensional counterparts of SSH solitons (Kim et al., 2023). In another formulation, a bipartite chiral tight-binding model on a strained-and-staggered simple cubic lattice realizes a fully three-dimensional SSH system characterized by direction-resolved winding numbers and by nearly flat or absolutely flat midgap surface and hypersurface bands without parameter tuning (Ahn, 19 Jun 2025). Taken together, these formulations establish the three-dimensional SSH model as both a materials-based and a model-Hamiltonian framework for studying dimerization-sensitive topology, Zak phases or chiral windings, and boundary-localized midgap states in three dimensions.

1. Conceptual origin and defining structure

The one-dimensional SSH model describes soliton excitations in polyacetylene arising from antiphase domain walls between two dimerization patterns. Its defining feature is an alternation of strong and weak bonds, together with a topological distinction between the two dimerized configurations. The three-dimensional SSH model preserves this logic but embeds it in a crystal or lattice with three-dimensional connectivity (Kim et al., 2023).

In bulk $\alpha$-Bi, the structural motif is explicitly layered. The crystal has space group $R\bar{3}m$ and is built from $(111)$ atomic layers stacked along $[111]$ into bilayers. Within each bilayer there are strong covalent intra-bilayer bonds, while between bilayers the bonding is weak van der Waals. Along $[111]$, these two bond types alternate exactly as the alternating strong and weak hopping in the one-dimensional SSH chain. The alternation is controlled by a dimerization parameter $\delta = \Delta/\Delta_0 = \pm 1$, where $\Delta$ is the Bi sublattice displacement along $[111]$ in units of the diagonal lattice vector $[111]$0 and $[111]$1 is its equilibrium magnitude. Reversing $[111]$2 swaps which links are strong and weak along the stacking direction (Kim et al., 2023).

A complementary, explicitly model-based three-dimensional SSH construction starts from a simple cubic lattice with a single-site basis and unit lattice constant, deforms it by uniform strains $[111]$3, and introduces 3D checkerboard-style staggered distortions $[111]$4 that double the unit cell and produce a two-site basis with sublattices $[111]$5 and $[111]$6. The nearest-neighbor hoppings are parameterized by

$[111]$7

so that each A–B bond carries either $[111]$8 or $[111]$9 depending on the local staggering pattern (Ahn, 19 Jun 2025). This formulation generalizes dimerization to three independent spatial directions.

These two realizations emphasize different aspects of the same theme. The Bi construction is a three-dimensional manifestation tied to crystallographic layering, inversion symmetry, parity eigenvalues, and domain-wall topology. The cubic-lattice model instead foregrounds chiral symmetry, weakly coupled SSH chains, and surface or hypersurface flat bands (Kim et al., 2023, Ahn, 19 Jun 2025).

2. Bulk topological characterization

In the Bi realization, the relevant bulk topological quantity is the Zak phase associated with one-dimensional momentum-space strings parallel to the stacking direction. For an inversion-symmetric one-dimensional insulator,

$\alpha$0

with

$\alpha$1

It is related to the electronic polarization $\alpha$2 by

$\alpha$3

where $\alpha$4 are Wannier charge centers along the one-dimensional direction and $\alpha$5 is the lattice constant (Kim et al., 2023).

The parity criterion gives a compact diagnosis:

$\alpha$6

where $\alpha$7 are inversion eigenvalues at the one-dimensional TRIM. For three-dimensional Bi, the construction is applied to fixed transverse momentum $\alpha$8, so that

$\alpha$9

with $R\bar{3}m$0 the pair of bulk TRIM projected onto a given surface TRIM along the $R\bar{3}m$1 string (Kim et al., 2023).

For the $R\bar{3}m$2 surface, the projected pairs are $R\bar{3}m$3 and $R\bar{3}m$4. The reported result is that the two dimerized Bi structures realize distinct Zak phases: $R\bar{3}m$5 has $R\bar{3}m$6, while $R\bar{3}m$7 has $R\bar{3}m$8. Wilson-loop and Wannier-charge-center analysis confirms this through odd versus even boundary crossings per spin channel (Kim et al., 2023).

The same work also shows that this dimerization-sensitive Zak-phase distinction is not equivalent to a change in the previously reported bulk topological indices of Bi. The strong TI index remains $R\bar{3}m$9 for both $(111)$0. The HOTI indices protected by $(111)$1 remain $(111)$2 and $(111)$3 in both structures, and the CTI indices protected by $(111)$4 remain odd for both dimerizations. This separation between SSH-type polarization topology and previously reported strong or crystalline topological indices is central to the three-dimensional SSH interpretation of Bi (Kim et al., 2023).

In the chiral cubic-lattice model, by contrast, the key topological invariants are winding numbers. The Bloch Hamiltonian is

$(111)$5

with chiral operator $(111)$6 satisfying $(111)$7. The spectrum is

$(111)$8

The three direction-resolved winding numbers are defined by

$(111)$9

Nonzero $[111]$0 indicate nontrivial chiral topology and predict midgap surface bands on surfaces perpendicular to the corresponding reciprocal vector $[111]$1 (Ahn, 19 Jun 2025).

3. Lattice Hamiltonians and dimensional generalization

The three-dimensional SSH model is not a single Hamiltonian but a family of constructions unified by alternating couplings and topological distinctions between dimerized sectors. The Bi case implements this through real crystal chemistry. Each $[111]$2 atomic layer can be treated as an SSH “site,” and the stacked bilayers as dimerized bonds. For each transverse momentum $[111]$3 in the plane normal to $[111]$4, the Bloch Hamiltonian along $[111]$5 behaves as a one-dimensional inversion-symmetric band insulator with alternating strong and weak interlayer hoppings, so that the full three-dimensional manifestation arises from a collection of one-dimensional SSH problems indexed by $[111]$6 (Kim et al., 2023).

The cubic-lattice realization is given in real space by a nearest-neighbor bipartite Hamiltonian with no AA or BB hopping. In the notation of the source, the Hamiltonian is $[111]$42 This Hamiltonian implements six distinct A–B bonds per unit cell with amplitudes $[111]$7 (Ahn, 19 Jun 2025).

In momentum space,

$[111]$8

with $[111]$9 obtained by summing the six A–B bonds and their phase factors. In the parameter regimes emphasized in the source, insulating behavior appears when the intrachain dimerization dominates the sum of interchain couplings. The explicit inequalities are

$[111]$0

depending on whether the effective SSH chains are taken along $[111]$1, $[111]$2, or $[111]$3 (Ahn, 19 Jun 2025).

A broader implication is that the three-dimensional SSH model admits multiple symmetry implementations. In Bi, inversion-symmetric Zak phases on projected one-dimensional strings remain central, while in the cubic-lattice model the decisive structure is a bipartite chiral block Hamiltonian. This suggests that “three-dimensional SSH” is best understood as a topological motif rather than a single universal normal form.

4. Boundary and interface states

The hallmark of the one-dimensional SSH model is a midgap soliton state at a domain wall. In three dimensions, this idea bifurcates into surface states and domain-wall states, depending on how the dimerization pattern terminates or reverses (Kim et al., 2023, Ahn, 19 Jun 2025).

For Bi, an antiphase domain wall is defined by a reversal of the dimerization order parameter, with $[111]$4 on one side and $[111]$5 on the other. Two orientations were analyzed. The $[111]$6 domain wall has its plane normal to $[111]$7, directly paralleling the one-dimensional SSH domain wall. The $[111]$8 domain wall contains $[111]$9, so the dimerization axis lies within the wall plane (Kim et al., 2023).

The $\delta = \Delta/\Delta_0 = \pm 1$0 domain wall is topologically non-trivial. Because $\delta = \Delta/\Delta_0 = \pm 1$1 differs between the two domains at the projected surface TRIM, there is a polarization mismatch of $\delta = \Delta/\Delta_0 = \pm 1$2 per domain-wall unit cell, enforcing an odd number of domain-wall-localized in-gap bands when spin is ignored. Interface Green’s functions reveal three domain-wall-localized bands. One Kramers pair gives a half-filled domain-wall band at $\delta = \Delta/\Delta_0 = \pm 1$3 pinned in the gap. By contrast, the $\delta = \Delta/\Delta_0 = \pm 1$4 domain wall is topologically trivial: the projected parity changes at $\delta = \Delta/\Delta_0 = \pm 1$5 and $\delta = \Delta/\Delta_0 = \pm 1$6 occur in pairs, $\delta = \Delta/\Delta_0 = \pm 1$7 is unchanged across the wall, and the number of domain-wall-localized in-gap bands is even; the interface Green’s function shows eight spin-degenerate domain-wall bands inside the projected bulk continuum (Kim et al., 2023).

The cubic-lattice model instead emphasizes open surfaces. When $\delta = \Delta/\Delta_0 = \pm 1$8, a midgap surface band exists on a surface perpendicular to $\delta = \Delta/\Delta_0 = \pm 1$9. The source distinguishes two types of topological surfaces. Type 1 surfaces are perpendicular to $\Delta$0, or equivalently parallel to the plane spanned by $\Delta$1 and $\Delta$2; each layer parallel to the surface contains only A sites or only B sites. Type 2 surfaces are perpendicular to the SSH chains (Ahn, 19 Jun 2025).

For the Type 1 termination most emphasized, $\Delta$3 yields topological insulating behavior with surface flat bands. For full-unit-cell terminations, there are two surface bands in the gap, one on each surface, and they are nearly flat; the residual dispersion arises from hybridization between opposite surfaces and decays exponentially with slab thickness. For half-unit-cell termination, with an odd number of layers, chiral symmetry and sublattice imbalance force at least one zero-energy eigenstate for each $\Delta$4, producing an absolutely flat midgap band localized on the topological surface (Ahn, 19 Jun 2025).

The boundary modes are sublattice-polarized. For Type 1 terminations, the zero-energy state on one surface lives on one sublattice, while the mode on the opposite surface, when present, lives on the other sublattice. This is the direct three-dimensional generalization of the SSH edge-state structure (Ahn, 19 Jun 2025).

5. Topological criteria, orientation dependence, and common pitfalls

A distinctive feature of the Bi analysis is that not every antiphase domain wall is topological. The source derives an explicit parity-flip criterion for arbitrary domain-wall orientation. For a wall with Miller indices $\Delta$5 relative to reciprocal basis $\Delta$6, the four projected wall TRIM are formed by pairs of bulk TRIM whose labels satisfy

$\Delta$7

These pairs are separated by $\Delta$8 along the wall normal, where

$\Delta$9

For each projected TRIM pair $[111]$0, one compares the parity product

$[111]$1

between the two domains. If an odd number of projected pairs flip sign across the wall, then the wall has $[111]$2 and an odd number of in-gap domain-wall bands ignoring spin. If the number of flips is even, then $[111]$3 and the in-gap bands are even in number (Kim et al., 2023).

Applied to Bi, dimerization reversal flips parity at bulk $[111]$4 and $[111]$5 only. The resulting orientation classification is explicit.

Orientation class in Bi	Topological character	Representative plane
$[111]$6 and exactly one odd index among $[111]$7, $[111]$8, $[111]$9	Non-trivial	$[111]$00
$[111]$01, $[111]$02, $[111]$03	Trivial	$[111]$04

This orientation dependence addresses a common misconception: antiphase reversal of dimerization does not by itself guarantee a topological domain wall in three dimensions. The projected TRIM pairing and parity-flip pattern must be considered explicitly (Kim et al., 2023).

A second common misconception is to identify three-dimensional SSH physics with strong topological-insulator band connectivity. In Bi, the strong TI index and the reported HOTI and CTI indices remain unchanged under dimerization reversal, while the Zak phase changes from $[111]$05 to $[111]$06 depending on the choice of dimerization. The source therefore separates SSH-like domain-wall topology from previously reported strong and crystalline topological orders (Kim et al., 2023).

A third potential confusion concerns flat-band protection. In the cubic-lattice model, the nearly flat or absolutely flat surface bands are protected by chiral symmetry and the winding invariant, not by spin–orbit coupling or fine tuning. Adding on-site potentials that break chiral symmetry would shift or gap the midgap states away from zero and can degrade flatness (Ahn, 19 Jun 2025).

6. Diagnostics, interactions, and experimental realizations

The Bi study combines several diagnostics. Parity eigenvalues were obtained from a DFT-PBE calculation using PAW potentials, a $[111]$07 $[111]$08-mesh, and a plane-wave cutoff of $[111]$09 eV, followed by a VASP–Wannier90 construction of $[111]$10-orbital Wannier functions and a tight-binding model with atomic SOC. Hybrid Wannier charge centers were computed in the conventional hexagonal cell along $[111]$11 and resolved into mirror subspaces $[111]$12 on the $[111]$13–$[111]$14 plane. Domain-wall spectral densities were obtained from an interface Green’s function in which a central domain-wall Hamiltonian, with linearly interpolated hopping parameters across the wall, is sandwiched between two semi-infinite pristine Bi leads of opposite $[111]$15 and solved using Sancho decimation (Kim et al., 2023).

For the cubic-lattice model, the source gives representative numerical results in the parameter regime $[111]$16, $[111]$17, $[111]$18, $[111]$19, with $[111]$20 in the topological phase. For full unit cells and $[111]$21, two midgap surface bands appear with dispersion approximately $[111]$22 and splitting approximately $[111]$23. For $[111]$24, the surface bands are almost completely flat, with bandwidth approximately $[111]$25 and separation approximately $[111]$26. For half-unit-cell termination, a single spin-degenerate surface band exists and is absolutely flat with zero energy (Ahn, 19 Jun 2025).

Interactions enter naturally because narrow surface or interface bands amplify correlation effects. In the cubic-lattice model, the on-site Hubbard term is

$[111]$27

and Hartree–Fock decoupling yields surface-confined magnetism because the surface bands are much narrower than the bulk bands. For the same parameter set, the bulk bands have width approximately $[111]$28 and gap approximately $[111]$29, with bulk antiferromagnetism appearing only above $[111]$30. By contrast, the critical interaction for surface magnetism is much smaller: $[111]$31 for $[111]$32, $[111]$33 for $[111]$34, and any $[111]$35 for the absolutely flat half-unit-cell case (Ahn, 19 Jun 2025). This suggests a strong surface–bulk separation of correlation scales inherent to three-dimensional SSH flat-band realizations.

Experimental considerations differ between the two realizations. In Bi, intense femtosecond laser pump–probe excitation is proposed as a route to dimerization reversal. The mechanism is ultrafast bond softening: excitation of approximately $[111]$36 of valence electrons transiently drives Bi toward the undimerized state, while larger excitation lowers the barrier between the $[111]$37 ground states. Experiments reporting irreversible sample “damage” above approximately $[111]$38 excitation are interpreted as suggesting that permanent local dimerization reversal, and thus antiphase domain-wall formation, may be achievable by controlled femtosecond pulses (Kim et al., 2023).

The proposed probes for Bi are STM and STS to image domain walls and detect midgap resonances, ARPES to resolve domain-wall bands including the half-filled band at $[111]$39 and possible Dirac features at $[111]$40 when domain-wall inversion is present, and time-resolved pump–probe spectroscopy to track $[111]$41, bond-softening dynamics, and domain-wall creation (Kim et al., 2023). For the cubic-lattice model, proposed platforms include 3D quantum dot arrays and metamaterials, with ARPES or STM/STS for flat-band detection and surface-sensitive magnetometry for interaction-induced surface magnetism (Ahn, 19 Jun 2025).

The broader significance is that the three-dimensional SSH model furnishes a topological framework in which bulk dimerization patterns control midgap states on surfaces, hypersurfaces, or domain walls through polarization or chiral winding rather than through conventional strong-TI surface-state connectivity. In Bi, this framework is realized through crystallographic dimerization and parity-controlled Zak phases. In the chiral cubic-lattice construction, it appears through multidirectional dimerization, winding invariants, and flat boundary bands. Both routes demonstrate that SSH physics persists beyond one dimension as a robust organizing principle for three-dimensional topological boundary phenomena (Kim et al., 2023, Ahn, 19 Jun 2025).