Truncated Trihexagonal Lattice

• The truncated trihexagonal lattice is an Archimedean tiling characterized by dual vertex configurations, with either (4,6,12) or (3·12²) arrangements defining its structure.
• Its quantum-graph formulation yields an exact, factorized dispersion relation that differentiates flat band states from propagating modes through Bloch periodicity and Neumann–Kirchhoff vertex conditions.
• The tight-binding model demonstrates nearly flat bands, Dirac-like crossings, and a quantum spin Hall phase induced by intrinsic spin–orbit coupling and robust helical edge modes.

Searching arXiv for the cited papers and closely related work on truncated trihexagonal / Archimedean lattices. The truncated trihexagonal lattice is treated in the cited literature as an Archimedean planar network whose periodicity supports both spectral analysis on quantum graphs and multi-orbital or single-orbital tight-binding descriptions. In the quantum-graph treatment of Archimedean tilings, the truncated trihexagonal tiling is identified with the vertex configuration $(4,6,12)$, meaning that a square, a hexagon, and a dodecagon meet cyclically at each vertex, and its dispersion relation can be written in closed form (Luo et al., 2019). In a later tight-binding study of planar Archimedean lattices, the label “truncated trihexagonal lattice” is used for a lattice with the $3\cdot 12^2$ vertex configuration, with one triangle and two dodecagons meeting at each site, and this model is analyzed in the context of quantum spin Hall physics (Pham et al., 1 Sep 2025). Accordingly, the term is not used uniformly across the cited sources.

1. Geometric realizations and nomenclature

In the quantum-graph formulation, the truncated trihexagonal tiling is a uniform Archimedean tessellation in which each vertex is surrounded in cyclic order by a square, a hexagon, and a dodecagon, with all edges of the same length $a$. A convenient fundamental domain $W$ is the parallelogram spanned by

$$\vec k_1=\Bigl(\tfrac{3+\sqrt3}2\,a,\;\tfrac{3+\sqrt3}2\,a\Bigr),\qquad \vec k_2=\Bigl(\tfrac{3+\sqrt3}2\,a,\;-\tfrac{3+\sqrt3}2\,a\Bigr),$$

so that the full graph is recovered by

$\{\,W+p_1\vec k_1+p_2\vec k_2:p_i\in\Bbb Z\}.$

Inside $W$ there are $18$ edges $e_1,\dots,e_{18}$ and $12$ vertices $3\cdot 12^2$0 (Luo et al., 2019).

In the tight-binding study, the lattice is instead described by a Bravais lattice generated by

$3\cdot 12^2$1

with a unit cell containing $3\cdot 12^2$2 atomic sites. One convenient set of basis positions $3\cdot 12^2$3, in units of $3\cdot 12^2$4, is

$3\cdot 12^2$5

These positions reproduce the $3\cdot 12^2$6 vertex configuration, with all equivalent positions taken modulo the Bravais translations $3\cdot 12^2$7 (Pham et al., 1 Sep 2025).

The coexistence of these two descriptions is a substantive terminological issue rather than a minor notational variation. Within the cited sources, the same label is attached to two distinct Archimedean vertex configurations. A plausible implication is that any comparison of spectral, topological, or transport results must begin by fixing which geometry is intended.

2. Periodic quantum-graph formulation

For the $3\cdot 12^2$8 realization, the lattice is modeled as a periodic quantum graph. On each edge $3\cdot 12^2$9, parametrized by $a$0, the wavefunction satisfies the one-dimensional Schrödinger equation

$a$1

with a common even potential $a$2. At each vertex one imposes the standard Neumann–Kirchhoff conditions: continuity of $a$3 and vanishing sum of outgoing derivatives (Luo et al., 2019).

Bloch periodicity is introduced through the quasi-momentum $a$4. Across opposite sides of the fundamental domain,

$a$5

for the edges identified by the periodic structure. Writing $a$6, each edge solution is expanded as

$a$7

where $a$8 and $a$9 are the standard cosine- and sine-type solutions on $W$0 with Wronskian $W$1.

The $W$2 vertex conditions together with the Bloch phases $W$3 and $W$4 produce a homogeneous linear system for the $W$5 coefficients $W$6. The dispersion relation is the vanishing of the determinant of the resulting $W$7 matrix. After applying the Lagrange identity

$W$8

and using the even-potential assumption so that $W$9, the determinant reduces to a factorized explicit form (Luo et al., 2019).

This formulation separates local differential dynamics on each edge from global periodic constraints. In that sense, the spectral problem is governed jointly by the edge ODE, the Neumann–Kirchhoff coupling, and the Bloch phases.

3. Explicit dispersion relation

Denoting

$$\vec k_1=\Bigl(\tfrac{3+\sqrt3}2\,a,\;\tfrac{3+\sqrt3}2\,a\Bigr),\qquad \vec k_2=\Bigl(\tfrac{3+\sqrt3}2\,a,\;-\tfrac{3+\sqrt3}2\,a\Bigr),$$0

the dispersion relation for the quantum graph takes the form

$$\vec k_1=\Bigl(\tfrac{3+\sqrt3}2\,a,\;\tfrac{3+\sqrt3}2\,a\Bigr),\qquad \vec k_2=\Bigl(\tfrac{3+\sqrt3}2\,a,\;-\tfrac{3+\sqrt3}2\,a\Bigr),$$1

$$\vec k_1=\Bigl(\tfrac{3+\sqrt3}2\,a,\;\tfrac{3+\sqrt3}2\,a\Bigr),\qquad \vec k_2=\Bigl(\tfrac{3+\sqrt3}2\,a,\;-\tfrac{3+\sqrt3}2\,a\Bigr),$$2

$$\vec k_1=\Bigl(\tfrac{3+\sqrt3}2\,a,\;\tfrac{3+\sqrt3}2\,a\Bigr),\qquad \vec k_2=\Bigl(\tfrac{3+\sqrt3}2\,a,\;-\tfrac{3+\sqrt3}2\,a\Bigr),$$3

where

$$\vec k_1=\Bigl(\tfrac{3+\sqrt3}2\,a,\;\tfrac{3+\sqrt3}2\,a\Bigr),\qquad \vec k_2=\Bigl(\tfrac{3+\sqrt3}2\,a,\;-\tfrac{3+\sqrt3}2\,a\Bigr),$$4

The prefactor $$\vec k_1=\Bigl(\tfrac{3+\sqrt3}2\,a,\;\tfrac{3+\sqrt3}2\,a\Bigr),\qquad \vec k_2=\Bigl(\tfrac{3+\sqrt3}2\,a,\;-\tfrac{3+\sqrt3}2\,a\Bigr),$$5 corresponds to flat bands, interpreted as Dirichlet eigenmodes on individual edges, and is independent of $$\vec k_1=\Bigl(\tfrac{3+\sqrt3}2\,a,\;\tfrac{3+\sqrt3}2\,a\Bigr),\qquad \vec k_2=\Bigl(\tfrac{3+\sqrt3}2\,a,\;-\tfrac{3+\sqrt3}2\,a\Bigr),$$6. The remaining bracket determines the absolutely continuous band spectrum (Luo et al., 2019).

This factorization is structurally important. It isolates a momentum-independent contribution from the Bloch-dependent part of the spectrum, thereby distinguishing flat-band states from propagating states. The explicit polynomial dependence on $$\vec k_1=\Bigl(\tfrac{3+\sqrt3}2\,a,\;\tfrac{3+\sqrt3}2\,a\Bigr),\qquad \vec k_2=\Bigl(\tfrac{3+\sqrt3}2\,a,\;-\tfrac{3+\sqrt3}2\,a\Bigr),$$7 also makes the band analysis reducible to a finite algebraic problem once the edge equation has been solved.

4. Band families, gaps, and high-symmetry points

The absolutely continuous spectrum $$\vec k_1=\Bigl(\tfrac{3+\sqrt3}2\,a,\;\tfrac{3+\sqrt3}2\,a\Bigr),\qquad \vec k_2=\Bigl(\tfrac{3+\sqrt3}2\,a,\;-\tfrac{3+\sqrt3}2\,a\Bigr),$$8 is obtained by solving the bracketed equation as $$\vec k_1=\Bigl(\tfrac{3+\sqrt3}2\,a,\;\tfrac{3+\sqrt3}2\,a\Bigr),\qquad \vec k_2=\Bigl(\tfrac{3+\sqrt3}2\,a,\;-\tfrac{3+\sqrt3}2\,a\Bigr),$$9 vary over $\{\,W+p_1\vec k_1+p_2\vec k_2:p_i\in\Bbb Z\}.$0. Equivalently, it is the set of those $\{\,W+p_1\vec k_1+p_2\vec k_2:p_i\in\Bbb Z\}.$1 for which there exists $\{\,W+p_1\vec k_1+p_2\vec k_2:p_i\in\Bbb Z\}.$2 and phases such that the bracket vanishes. The detailed analysis yields three disjoint allowed intervals for $\{\,W+p_1\vec k_1+p_2\vec k_2:p_i\in\Bbb Z\}.$3: $\{\,W+p_1\vec k_1+p_2\vec k_2:p_i\in\Bbb Z\}.$4 Since $\{\,W+p_1\vec k_1+p_2\vec k_2:p_i\in\Bbb Z\}.$5, these intervals define three spectral families of bands (Luo et al., 2019).

For each $\{\,W+p_1\vec k_1+p_2\vec k_2:p_i\in\Bbb Z\}.$6, the condition

$\{\,W+p_1\vec k_1+p_2\vec k_2:p_i\in\Bbb Z\}.$7

implies

$\{\,W+p_1\vec k_1+p_2\vec k_2:p_i\in\Bbb Z\}.$8

and analogous statements hold for the other intervals. The band spectrum therefore has an infinite number of open gaps. Those gaps shrink as $\{\,W+p_1\vec k_1+p_2\vec k_2:p_i\in\Bbb Z\}.$9 because $W$0 is highly oscillatory.

Band edges are pinned to high-symmetry points in the Brillouin zone,

$W$1

and, for the triangular-lattice corner,

$W$2

At these points the trigonometric terms collapse to simpler values. At $W$3, for example, the reduced equation has real roots in $W$4 exactly at

$W$5

These are the band-edge values. The outermost edges of the lowest band occur at $W$6, while inner band-edge collisions occur at $W$7 and $W$8. At those points the group velocity vanishes and band extremal points are realized (Luo et al., 2019).

In this setting, the truncated trihexagonal quantum graph exhibits a band structure organized into three families separated by infinitely many gaps. The availability of an exact dispersion relation permits continuous tracing of each band as a function of $W$9.

5. Tight-binding Hamiltonian and bulk electronic structure

In the tight-binding treatment, the lattice is modeled with a single $18$0-like orbital per site, two spin states, nearest-neighbor hopping, next-nearest-neighbor hopping, and intrinsic spin–orbit coupling in Kane–Mele form: $18$1 The nearest-neighbor term is

$18$2

the next-nearest-neighbor term is

$18$3

and the intrinsic spin–orbit term is written as

$18$4

with $18$5 determined by the handedness of the next-nearest-neighbor hop around the intermediate site; by convention $18$6 for a left-handed loop. All sums span one unit cell and its periodic images (Pham et al., 1 Sep 2025).

The Bloch Hamiltonian $18$7 is diagonalized in the $18$8 spinful basis $18$9 along the triangular-lattice path $e_1,\dots,e_{18}$0. For the parameter set

$e_1,\dots,e_{18}$1

the model yields $e_1,\dots,e_{18}$2 doubly-degenerate bands. The reported features are:

• Nearly flat bands: one or more nearly flat bands extend across the entire zone, with $e_1,\dots,e_{18}$3.
• Dirac-like crossings: crossings occur at $e_1,\dots,e_{18}$4.
• High-degeneracy points: such points occur at $e_1,\dots,e_{18}$5.
• SOC gaps: these crossings and degeneracies open small spin–orbit gaps proportional to $e_1,\dots,e_{18}$6.

By projection onto the $e_1,\dots,e_{18}$7 subspace, the flat band is found to be purely $e_1,\dots,e_{18}$8, while dispersive bands mix slightly with higher orbitals only if those orbitals are included (Pham et al., 1 Sep 2025).

The flat-band statement is consistent with the broader conclusion of the Archimedean-lattice survey, which reports that several Archimedean lattices, including the truncated hexagonal and truncated trihexagonal lattices, host nearly dispersionless flat bands extending across the Brillouin zone and that these remain robust even in the presence of next-nearest-neighbor hopping and strong spin–orbit coupling (Pham et al., 1 Sep 2025).

6. Topological invariants, spin Hall response, and edge states

For the tight-binding model, inversion symmetry permits evaluation of the $e_1,\dots,e_{18}$9 invariant by the Fu–Kane parity-eigenvalue method. At the four time-reversal invariant momenta

$12$0

each Kramers-degenerate pair contributes a parity product

$12$1

and the invariant is

$12$2

For fillings placing the Fermi level in the main SOC-induced gap, the numerical evaluation yields

$12$3

which identifies a quantum spin Hall phase (Pham et al., 1 Sep 2025).

The intrinsic spin Hall conductivity is computed from the Kubo formula,

$12$4

where

$12$5

with $12$6,

$12$7

A dense $12$8-mesh integration yields a quantized plateau

$12$9

in natural units whenever $3\cdot 12^2$00 lies in the nontrivial gap.

The ribbon spectra provide the edge-state manifestation of the same topology. For wide zigzag or armchair nanoribbons, diagonalization of the one-dimensional Bloch Hamiltonian shows two counterpropagating Kramers pairs of edge modes crossing at the one-dimensional time-reversal momenta $3\cdot 12^2$01. These modes are exponentially localized at opposite edges. When ribbon bands are colored by $3\cdot 12^2$02, the edge branches are nearly pure spin-up and spin-down, with a few avoided crossings attributed to atomic SOC spin-mixing but with

$3\cdot 12^2$03

everywhere on the edge branches. The edge spectrum is further reported to be robust against on-site disorder and nonmagnetic impurities, with the helical modes retaining perfect transmission in two-terminal transport calculations (Pham et al., 1 Sep 2025).

Taken together, these results place the truncated trihexagonal lattice, in the sense of the tight-binding study, at the intersection of flat-band physics and two-dimensional topological transport. This suggests a setting in which nearly dispersionless bulk states and helical edge modes can coexist within the same Archimedean framework.