Depleted Triangular Lattices: Geometry & Physics

• Depleted Triangular Lattices are modified triangular structures obtained by systematically removing sites or bonds while retaining triangular motifs with altered connectivity and unit cells.
• They appear in diverse systems such as graphene antidot lattices, vacancy-ordered iridates, and emergent spin liquids, illustrating how depletion drives topological and electronic transitions.
• Energetic and spectral analyses show that depletion modifies band gaps, frustration, and critical behavior, offering practical insights for designing advanced quantum and materials systems.

A depleted triangular lattice is a structure obtained from a triangular parent lattice by a systematic removal of sites, bonds, or material, so that the descendant geometry retains triangular motifs while acquiring a modified unit cell, connectivity, and low-energy theory. Across the literature, the term encompasses several distinct but related constructions: graphene sheets perforated by triangular arrays of antidots, ordered vacancy patterns in triangular layers of edge-sharing octahedra, Archimedean descendants such as kagome and maple-leaf lattices, domination-defined subgraphs of the triangular tessellation, and field-theoretic families of triangular-derived spin liquids. In each case, the central effect of depletion is topological rather than merely dilutive: it changes band structure, frustration, admissible symmetry representations, and the space of energetically competitive ordered or liquid-like states (Petersen et al., 2011, Mehlawat et al., 2019, Makuta et al., 2021, Feuerpfeil et al., 30 Mar 2026).

1. Taxonomy of depletion on triangular parents

The common starting point is a triangular organization of degrees of freedom—atomic sites, spin sites, graph vertices, or reciprocal-space nodes—and the common operation is a periodic subtraction that preserves enough symmetry to define a new regular lattice problem. The resulting object is usually not a random dilution. Instead, it is an ordered depletion with a well-defined unit cell, reciprocal lattice, and symmetry group.

Context	Parent structure	Depletion mechanism
Graphene antidot lattices	Graphene honeycomb sheet	Periodic triangular array of nanometer-scale holes
Vacancy-ordered iridates	Triangular Ir planes	Ordered Ir vacancies leaving honeycomb backbone plus stuffed sites
Maple-leaf and kagome systems	Triangular spin lattice	Regular site depletion into Archimedean descendants
Dominating-set constructions	Triangular tessellation graph $\{3,6\}$	Removal/complement relative to PDS or QPDS subsets

A useful unifying distinction is between material depletion and effective depletion. In graphene antidot systems, atoms are physically removed from the parent network. In $K_xIr_yO_2$, charge balance imposes an ordered reduction of Ir occupancy. In LiZn$_2$Mo$_3$O$_8$, by contrast, the low-temperature “depletion” is emergent: the triangular lattice effectively decomposes into a strongly coupled honeycomb subsystem plus residual orphan spins, even though the starting crystallographic motif is triangular (Flint et al., 2013). This suggests that depleted triangular lattices are best viewed as a broader class of triangular-derived topologies rather than a single crystallographic category.

2. Electronic depleted triangular lattices: graphene antidot superlattices

In graphene, depletion is implemented by carving a periodic array of holes into a semimetallic honeycomb sheet, thereby generating a graphene antidot lattice. For the triangular antidot geometry, the antidot centers form a triangular Bravais lattice with elementary vectors parallel to the carbon–carbon bonds, and the unit cell is denoted $\{L,R\}$, where $L$ counts non-shared hexagons along a cell edge and $R$ is the hole radius in units of the graphene lattice constant $a_0=2.46\,\text{\AA}$. The resulting depleted network is treated in both nearest-neighbor tight-binding,

$$H=-t\sum_{\langle i,j\rangle}(c_i^\dagger c_j+c_j^\dagger c_i),$$

with $K_xIr_yO_2$0, and a quasiparticle tight-binding model including up to three neighbors and overlaps (Petersen et al., 2011).

The electronic result specific to the non-rotated triangular depletion is unusually robust: every triangular antidot lattice tested exhibits a sizable band gap, and the gap sits at the $K_xIr_yO_2$1 point. For fixed hole size $K_xIr_yO_2$2, increasing the unit-cell size $K_xIr_yO_2$3 reduces the depletion fraction and monotonically reduces the gap. The scaling emphasized in the work is

$K_xIr_yO_2$4

so larger depletion at fixed cell size yields larger gaps. By contrast, rotated triangular, rectangular, and honeycomb antidot lattices display a threefold selection rule: large gaps occur only for $K_xIr_yO_2$5 in the rotated triangular and honeycomb cases, and for $K_xIr_yO_2$6 in the rectangular case. The triangular arrangement aligned with the C–C bonds is therefore distinguished not simply by having a gap, but by having one without the periodic exclusions that constrain the other geometries.

The explanatory mechanism is Clar sextet theory. A large gap correlates with the existence of a complete benzenoid Clar pattern, quantified semi-empirically by

$K_xIr_yO_2$7

with $K_xIr_yO_2$8 the number of Clar sextets and $K_xIr_yO_2$9 the number of hexagons in the antidot unit cell. Non-rotated triangular antidot lattices always admit a fully benzenoid pattern except at the hole rim, whereas rotated triangular, rectangular, and honeycomb antidot lattices satisfy the criterion only for roughly one third of parameter choices. The same threefold sensitivity reappears when the triangular array is distorted into a non-isosceles geometry $_2$0: large gaps survive only when $_2$1, with the explicitly listed $_2$2, $_2$3, and $_2$4 cases giving $_2$5, $_2$6, and $_2$7 eV, respectively. In this electronic setting, a depleted triangular lattice is therefore a band-gap engineering platform whose effectiveness is controlled by the compatibility between depletion geometry and aromatic $_2$8-bond resonance.

3. Ordered vacancy depletion and emergent triangular descendants in frustrated magnets

In spin-orbit Mott systems, depletion typically means ordered site removal from an otherwise triangular network of magnetic ions. In $_2$9, layers of edge-sharing IrO$_3$0 octahedra would form full triangular planes if every Ir site were occupied. Instead, increasing $_3$1 introduces ordered Ir vacancies according to

$_3$2

preserving Ir$_3$3 and the Mott-insulating state. For the $_3$4 composition, the refined structure has space group $_3$5, lattice parameters $_3$6, $_3$7, and a planar Ir network that can be described either as a depleted triangular lattice or as a stuffed honeycomb lattice. The honeycomb backbone is fully occupied, while approximately $_3$8 of the honeycomb voids are occupied by additional Ir and about $_3$9 remain vacant (Mehlawat et al., 2019).

The magnetic phenomenology is correspondingly intermediate between triangular and honeycomb limits. Curie–Weiss fitting of

$_8$0

gives $_8$1, $_8$2, and $_8$3, consistent with effective $_8$4 moments and strong antiferromagnetic interactions. Yet no magnetic order or spin freezing is observed down to $_8$5 K. The magnetic heat capacity exhibits a broad maximum near $_8$6 K and a low-temperature form

$_8$7

with $_8$8, while remaining essentially unchanged in $_8$9 T. These observations are stated to be consistent with a gapless quantum spin liquid on the depleted triangular / stuffed honeycomb lattice. The lattice is noteworthy because it interpolates chemically and topologically between triangular and honeycomb Kitaev-relevant limits without changing the Ir valence.

A different form of effective depletion appears in LiZn$\{L,R\}$0Mo$\{L,R\}$1O$\{L,R\}$2. There, the triangular lattice of Mo$\{L,R\}$3O$\{L,R\}$4 cluster spins is proposed to decouple at low temperature into an emergent honeycomb lattice weakly coupled to residual orphan spins, motivated by the disappearance of $\{L,R\}$5 of the spins from the low-temperature Curie response and by the relation $\{L,R\}$6. The distortion is parametrized by

$\{L,R\}$7

so that increasing $\{L,R\}$8 strengthens the honeycomb bonds and weakens the couplings to central spins. The resulting $\{L,R\}$9-$L$0-$L$1 model supports the proposal that the strongly coupled honeycomb subsystem forms a spin liquid or valence-bond state, while orphan spins remain weakly coupled and help stabilize the liquid-like regime (Flint et al., 2013). In this usage, depletion is not literal vacancy formation but an emergent reorganization of a triangular lattice into two weakly coupled lattices with different dynamical roles.

4. Maple-leaf depletion, dimensional reduction, and Dirac spin-liquid criticality

The maple-leaf lattice is the canonical $L$2-depleted triangular lattice. It has six sites per unit cell and lies geometrically between the triangular and kagome limits. In the quantum spin-$L$3 Heisenberg problem studied on this lattice, uniform antiferromagnetic interactions on the remaining bonds nevertheless produce a ground state with stripe Néel order rather than an isotropic two-dimensional pattern. The temperature dependence of the susceptibility follows that of a one-dimensional XXZ model with a finite spin gap, and the low-energy degenerate manifold can be mapped to a fully packed loop-string model on a dual cluster-depleted honeycomb lattice. The central claim is that the observed dimensional reduction is purely spontaneous: the Hamiltonian retains a spatially isotropic two-dimensional structure, yet magnons fractionalize into two spinons that propagate along the stripe direction, with barriers strongly suppressing transverse motion (Makuta et al., 2021).

The same family of depleted triangular lattices—triangular, kagome, and maple-leaf—has recently been treated in a unified continuum framework for U(1) Dirac spin liquids and Higgs transitions. In that formulation, triangular and kagome lattices both yield QED$L$4-Higgs theories with $L$5 Dirac fermions, while the maple-leaf lattice yields an analogous theory with $L$6. The maple-leaf distinction is not only the larger flavor count but also a different nodal geometry: its six Dirac cones lie on high-symmetry lines and can move in momentum space, generating an additional symmetry-allowed Yukawa coupling not present in the triangular or kagome cases. The generic continuum structure is

$L$7

with a charge-2 Higgs field $L$8 mediating the U(1) to $L$9 transition (Feuerpfeil et al., 30 Mar 2026).

Large-$R$0 analysis shows that Higgs-field fluctuations and large $R$1 both suppress the relevance of the Yukawa coupling, but not enough to render it irrelevant. For $R$2, the reported values are $R$3 and $R$4 for triangular/kagome, versus $R$5, $R$6, and $R$7 for maple-leaf. The Higgs critical point is therefore asymptotically unstable in all three cases, but maple-leaf is pushed closer to stability by its large flavor number, while being partially destabilized again by the extra coupling associated with mobile Dirac cones. Depletion here acts directly on the infrared field theory: it changes flavor multiplicity, the allowed projective symmetry representations, and the operator content of the critical point.

5. Spectral and combinatorial formulations

In spectral graph theory, weak periodic depletion of a triangular lattice can be modeled as a periodic perturbation of the free triangular adjacency operator. For the discrete Schrödinger operator $R$8 with $R$9-periodic potential $a_0=2.46\,\text{\AA}$0, the free Floquet eigenvalues are

$a_0=2.46\,\text{\AA}$1

and the undepleted triangular spectrum is $a_0=2.46\,\text{\AA}$2. In the perturbative regime, the discrete Bethe–Sommerfeld result is sharp: for sufficiently small $a_0=2.46\,\text{\AA}$3, the spectrum consists of at most two intervals; if at least one of $a_0=2.46\,\text{\AA}$4 is odd, no gap opens at all; and any perturbative gap can only occur at the exceptional energy $a_0=2.46\,\text{\AA}$5. An explicit $a_0=2.46\,\text{\AA}$6-periodic potential opens the gap

$a_0=2.46\,\text{\AA}$7

demonstrating that weakly depleted triangular superstructures have highly constrained gap-opening possibilities (Fillman et al., 2018).

A more combinatorial notion of depletion arises from domination on the infinite triangular tessellation graph $a_0=2.46\,\text{\AA}$8 of Schläfli symbol $a_0=2.46\,\text{\AA}$9. If $$H=-t\sum_{\langle i,j\rangle}(c_i^\dagger c_j+c_j^\dagger c_i),$$0 is a perfect dominating set, then every vertex of $$H=-t\sum_{\langle i,j\rangle}(c_i^\dagger c_j+c_j^\dagger c_i),$$1 has exactly one neighbor in $$H=-t\sum_{\langle i,j\rangle}(c_i^\dagger c_j+c_j^\dagger c_i),$$2; if $$H=-t\sum_{\langle i,j\rangle}(c_i^\dagger c_j+c_j^\dagger c_i),$$3 is quasiperfect, vertices of $$H=-t\sum_{\langle i,j\rangle}(c_i^\dagger c_j+c_j^\dagger c_i),$$4 have one or two such neighbors. The complement $$H=-t\sum_{\langle i,j\rangle}(c_i^\dagger c_j+c_j^\dagger c_i),$$5 is then a depleted triangular lattice in graph-theoretic form. Up to symmetry there is exactly one proper PDS in $$H=-t\sum_{\langle i,j\rangle}(c_i^\dagger c_j+c_j^\dagger c_i),$$6, and it is a 1-perfect code with minimum mutual distance $$H=-t\sum_{\langle i,j\rangle}(c_i^\dagger c_j+c_j^\dagger c_i),$$7; the vertex set partitions into seven translates of this code, so the depletion density is $$H=-t\sum_{\langle i,j\rangle}(c_i^\dagger c_j+c_j^\dagger c_i),$$8. The same framework yields $$H=-t\sum_{\langle i,j\rangle}(c_i^\dagger c_j+c_j^\dagger c_i),$$9- and $K_xIr_yO_2$00-QPDS families, with complements tiled by triangles, hexagons, or elongated hexagons, and toroidal quotients $K_xIr_yO_2$01 impose arithmetic divisibility conditions such as $K_xIr_yO_2$02 for the perfect-code partition and $K_xIr_yO_2$03 for parallel $K_xIr_yO_2$04-QPDSs (0903.3685). This formulation shows that depletion can be encoded entirely by local adjacency constraints, without recourse to chemistry or band theory.

6. Energetic benchmarks: depletion relative to the undepleted triangular optimum

A recurrent theme is that depleted triangular lattices are often understood relative to the energetically privileged undepleted triangular Bravais lattice. For admissible completely monotone pair potentials, the triangular lattice $K_xIr_yO_2$05 minimizes

$K_xIr_yO_2$06

among Bravais lattices of fixed area $K_xIr_yO_2$07 for every density, and the same paper gives an explicit convex, decreasing, positive counterexample,

$K_xIr_yO_2$08

for which $K_xIr_yO_2$09 is not a minimizer on a finite density interval (Bétermin, 2015). For spatially extended particles with radial mass distribution $K_xIr_yO_2$10, the corresponding lattice energy

$K_xIr_yO_2$11

is again uniquely minimized by the triangular lattice at fixed density when the particles are sufficiently concentrated, and also for any fixed density when both the interaction and the mass distribution are completely monotone Gaussian mixtures (Bétermin et al., 2017).

For Lennard–Jones-type interactions, the status of the triangular reference becomes density-dependent. In the two-dimensional $K_xIr_yO_2$12-$K_xIr_yO_2$13 problem at fixed area, triangular is rigorously the unique minimizer when

$K_xIr_yO_2$14

but it ceases to be optimal at sufficiently low density, with numerical evidence in the cited work indicating that the square lattice overtakes it around $K_xIr_yO_2$15 (Bétermin et al., 2014). A later computer-assisted result proves that for the classical $K_xIr_yO_2$16 potential, the global minimizer among all two-dimensional Bravais lattices is a scaled triangular lattice, and for fixed co-volume $K_xIr_yO_2$17 the triangular lattice is uniquely optimal whenever

$K_xIr_yO_2$18

while explicitly noting that periodic topological defect patterns and other non-Bravais defect structures lie outside the proof (Bétermin, 2021). A complementary analysis beyond complete monotonicity constructs non-completely monotone $K_xIr_yO_2$19 for which triangular remains uniquely optimal at all scales for $K_xIr_yO_2$20, but also exhibits one-well potentials $K_xIr_yO_2$21 for which

$K_xIr_yO_2$22

so that square rather than triangular order is globally preferred (Bétermin et al., 2018).

These energetic results do not by themselves classify depleted triangular lattices, since most proofs are restricted to Bravais lattices or fixed periodic ansätze. They nevertheless establish the natural benchmark against which depletion is assessed. A plausible implication is that whenever depletion is introduced into a model whose undepleted triangular lattice is known to be optimal, the central question becomes not whether triangular order is intrinsically favorable, but how the depletion changes the admissible topology, symmetry, and density so as to compete with—or inherit from—that benchmark. Across graphene band engineering, vacancy-ordered iridates, maple-leaf spin systems, domination tilings, and QED$K_xIr_yO_2$23-Higgs criticality, depletion is thus best understood as a controlled transformation of triangular order rather than its mere destruction.