Fractality-Induced Topology

• Fractality-induced topology is a framework where self-similar, non-integer geometries generate and reshape topological boundary states and phase structures.
• Methodologies such as isospectral reduction, real-space invariants, and renormalization techniques reveal effective Hamiltonians that capture fractal-induced topological phenomena.
• The approach distinguishes between fractal-induced generation and diagnostic roles, impacting higher-order boundary modes, transport properties, and phase transitions.

Fractality-induced topology denotes a family of mechanisms and diagnostics in which self-similar geometry, non-integer dimensionality, or fractal observables are inseparable from topological structure. In its strongest form, fractal geometry itself generates symmetry-protected boundary or corner states in systems with otherwise uniform couplings and no conventional topological drivers. In other settings, fractal lattices reorganize topological transport, higher-order boundary phenomenology, or bulk-boundary correspondence in the absence of Bloch momentum. A third usage is diagnostic rather than causal: fractal growth laws, coastline dimensions, or disorder-induced fractal domains reveal an underlying topological phase or topology-changing transition rather than producing the topology themselves (Eek et al., 2024, Tanaka et al., 11 Sep 2025, Salib et al., 24 Apr 2025).

1. Conceptual scope and principal usages

Across recent work, the term does not refer to a single invariant or a single construction. It spans geometry-induced higher-order topology, Floquet and crystalline phases on fractal-dimensional lattices, disorder- and growth-induced fractal diagnostics of topology, and more abstract topological or combinatorial formulations on graphs. The common thread is that self-similarity changes either the effective Hamiltonian, the available boundaries, or the observables used to characterize phase structure.

Usage	Core mechanism	Representative works
Geometry as generator	Isospectral reduction of a fractal graph yields effective breathing couplings and corner states	(Eek et al., 2024, Zhang et al., 22 Jun 2026, Yan et al., 23 Jun 2026)
Topology on fractal lattices	Fractal boundaries host chiral, edge, hinge, or corner modes without translational symmetry	(Yang et al., 2021, Manna et al., 2021, Zheng et al., 2022, Ma et al., 2023)
Fractality as diagnostic	Fractal metrics of growth or disorder reveal a topological phase or critical point	(Tanaka et al., 11 Sep 2025, Salib et al., 24 Apr 2025)
Counterexamples and abstractions	Fractality can suppress gapped topology, or define topology-like structure in graphs and manifolds	(Agarwala et al., 2018, Skums et al., 2019, Porchon, 2012)

This plurality matters. Some papers explicitly argue that fractality is not the cause of topology, but rather a consequence or fingerprint of topology-changing dynamics. Others use the phrase in a literal causal sense, where self-similarity alone induces an effective topological Hamiltonian. The phrase is therefore best read as a research program linking self-similarity and topological organization, not as a single universal theorem.

2. Fractal geometry as an intrinsic topological mechanism

The most direct formulation appears in isospectral-reduction approaches. A nearest-neighbor tight-binding Hamiltonian on a fractal graph is partitioned into retained sites $S$ and auxiliary sites $A$, producing the exact energy-dependent effective operator

$$H_{\mathrm{eff}}(E)=H_{SS}+H_{SA}(E I_A-H_{AA})^{-1}H_{AS}.$$

In the Sierpiński-kagome setting this reduction yields an effective breathing kagome model with energy-dependent intracell hopping $v(E)$ and onsite term $a(E)$. Corner states occur when the self-consistency condition $E_c=a(E_c)$ is satisfied and the effective breathing ratio lies in the higher-order regime $|v(E_c)|<1$. The same reduction strategy extends to Pascal mod $k$ fractals, hexaflake, Vicsek, and the Sierpiński tetrahedron, mapping them to breathing kagome, Kekulé honeycomb, or breathing pyrochlore parents with crystalline higher-order topology (Eek et al., 2024).

A fully experimental realization of this geometry-only mechanism was reported in a third-generation Sierpiński-gasket waveguide lattice with uniform nearest-neighbor coupling $c_0=2\,\mathrm{cm}^{-1}$ and onsite detuning $\epsilon_m=0\,\mathrm{cm}^{-1}$. There the physical array has no imposed staggered hopping, no synthetic gauge field, and no temporal modulation. Yet isospectral reduction maps it onto an effective breathing kagome description, and open-boundary spectra reveal corner-localized modes protected by $A$0 symmetry. Selective excitation with a weakly coupled auxiliary waveguide with $A$1 and $A$2 produces real-space corner localization, while random onsite disorder preserves the corner states up to gap closures at $A$3, $A$4, and $A$5 (Zhang et al., 22 Jun 2026).

A parallel photonic-crystal realization fractalizes a kagome lattice into a Sierpiński pattern and observes the same logic in tight-binding-like band structures. The effective reduction gives

$A$6

with $A$7. At the corner-state eigenenergy $A$8, one obtains $A$9 and $$H_{\mathrm{eff}}(E)=H_{SS}+H_{SA}(E I_A-H_{AA})^{-1}H_{AS}.$$0, exactly the breathing inequality needed to gap the parent kagome Dirac point and produce corner modes. Experimentally, first-generation samples show corner states at $$H_{\mathrm{eff}}(E)=H_{SS}+H_{SA}(E I_A-H_{AA})^{-1}H_{AS}.$$1 and $$H_{\mathrm{eff}}(E)=H_{SS}+H_{SA}(E I_A-H_{AA})^{-1}H_{AS}.$$2, while second-generation samples show corner states at $$H_{\mathrm{eff}}(E)=H_{SS}+H_{SA}(E I_A-H_{AA})^{-1}H_{AS}.$$3, $$H_{\mathrm{eff}}(E)=H_{SS}+H_{SA}(E I_A-H_{AA})^{-1}H_{AS}.$$4, and $$H_{\mathrm{eff}}(E)=H_{SS}+H_{SA}(E I_A-H_{AA})^{-1}H_{AS}.$$5 (Yan et al., 23 Jun 2026).

Fractal hierarchy can also amplify rather than merely create topology. In periodic lattices with fractal unit cells, multi-topological-phase theory predicts exponential proliferation of minigaps and boundary states with generation index $$H_{\mathrm{eff}}(E)=H_{SS}+H_{SA}(E I_A-H_{AA})^{-1}H_{AS}.$$6. In the Koch-chain architecture, $$H_{\mathrm{eff}}(E)=H_{SS}+H_{SA}(E I_A-H_{AA})^{-1}H_{AS}.$$7 topological minigaps and $$H_{\mathrm{eff}}(E)=H_{SS}+H_{SA}(E I_A-H_{AA})^{-1}H_{AS}.$$8 edge states appear in the nontrivial regime, while in the Sierpiński-gasket tiling $$H_{\mathrm{eff}}(E)=H_{SS}+H_{SA}(E I_A-H_{AA})^{-1}H_{AS}.$$9 and $v(E)$0. The multiplicative factor is fixed by symmetry: $v(E)$1 for inversion-related edges in the quasi-1D case and $v(E)$2 for $v(E)$3-related corners in the triangular case (Song et al., 1 Apr 2026). This suggests that fractal self-similarity can function as a structural multiplier of independent topological channels.

3. Higher-order topology and proliferating boundaries on fractals

A distinct but closely related direction treats fractal lattices as hosts for higher-order phases whose codimension-two states populate a hierarchy of outer and inner boundaries. On planar Sierpiński carpets and glued Sierpiński triangles, second-order topological insulators and superconductors can be implemented directly in real space by a Wilson–Dirac mass pattern tailored to the fractal geometry. The real-space quadrupole moment

$v(E)$4

remains quantized in the insulating case, with $v(E)$5 for the higher-order phase. On the Sierpiński carpet, outer corners host near-zero modes, while inner corners are not naked and therefore carry finite-energy corner-like states. On the glued Sierpiński triangle, inner naked corners support exact zero modes, and with periodic boundaries the corner local density of states shifts from the outer corners to these inner naked corners. The superconducting analog supports Majorana corner modes, although the quadrupole response does not become origin independent in the thermodynamic limit, which the work interprets as extrinsic rather than intrinsic higher-order topology (Manna et al., 2021).

The acoustic Sierpiński carpet realizes the same theme in a 2D SSH setting. The carpet has Hausdorff dimension

$v(E)$6

and the crucial observation is that bulk, edge, and corner state counts all scale with this same non-integer exponent. For the $v(E)$7 structure, the spectrum contains $v(E)$8 bulk states, $v(E)$9 edge states, and $a(E)$0 corner states, summing to $a(E)$1. Spectrally isolated outer-corner, inner-corner, and edge bands were observed around $a(E)$2, $a(E)$3, and $a(E)$4, respectively. In this sense, the fractal boundary hierarchy renders “bulk,” “edge,” and “corner” equally extensive under state-count scaling (Zheng et al., 2022).

Elastic fractal metamaterials extend this boundary hierarchy into a platform where inner and outer edge states can be separated spectrally. In Sierpiński and rhombus elastic plates, the topology is characterized by the real-space quadrupole marker

$a(E)$5

with a topological transition at $a(E)$6. In the nontrivial Sierpiński $a(E)$7 plate, the reported mode counts are $a(E)$8 bulk, $a(E)$9 inner-corner, $E_c=a(E_c)$0 inner-edge, $E_c=a(E_c)$1 outer-edge, and $E_c=a(E_c)$2 outer-corner states. The rhombus fractal introduces inequivalent $E_c=a(E_c)$3 and $E_c=a(E_c)$4 corners, with topology index $E_c=a(E_c)$5 at the $E_c=a(E_c)$6 corners and $E_c=a(E_c)$7 at the $E_c=a(E_c)$8 corners. The $E_c=a(E_c)$9 corner states are reported to be more robust than the Sierpiński corner states under geometric perturbations (Ma et al., 2023).

Taken together, these results shift higher-order topology away from the conventional picture of a small number of isolated corners on a Euclidean boundary. Fractalization creates a nested family of domain-wall terminations, hole perimeters, and corner singularities, so the number, localization pattern, and robustness class of higher-order states become explicitly generation dependent.

4. Real-space invariants, Floquet phases, and construction principles

Because translational symmetry is absent on genuine fractal lattices, real-space invariants become central. In a helical-waveguide Sierpiński gasket, light propagation is governed by a Floquet tight-binding model with synthetic gauge field

$|v(E_c)|<1$0

and topology is characterized by a real-space Chern number constructed from a projector $|v(E_c)|<1$1 and a tripartition $|v(E_c)|<1$2. For the driven $|v(E_c)|<1$3 gasket with $|v(E_c)|<1$4 sites, the central quasienergy window $|v(E_c)|<1$5 has $|v(E_c)|<1$6, with chiral states localized on both outer and inner boundaries. Wavepackets traverse the outer and inner edges without backscattering, remain robust against disorder up to $|v(E_c)|<1$7, and pass seamlessly across a hybrid interface between a honeycomb edge and a fractal edge when both halves share the same real-space Chern number $|v(E_c)|<1$8 (Yang et al., 2021).

A complementary construction program begins from a parent Bloch Hamiltonian on a periodic crystal and maps it to a fractal lattice in real space. Three methods were formulated for the Sierpiński carpet: a symmetry-guided replacement of momentum harmonics by symmetry-equivalent real-space couplings, a site-elimination method that restricts the parent Hamiltonian to the retained fractal subgraph, and a renormalization method based on the Schur complement

$|v(E_c)|<1$9

Applied to a generalized square-lattice QWZ Chern insulator, these constructions reproduce strong Chern phases with Bott index $k$0 and $k$1, as well as a crystalline Valley $k$2 phase with $k$3. For Methods 1 and 2, the phase diagrams agree already between the third and fourth carpet generations, while Method 3 mirrors the parent square-lattice phase diagram at the same parameter values (Salib et al., 2024).

These works establish a broad methodological principle: point-group symmetry can survive fractalization even when translation is destroyed. The loss of Bloch momentum does not eliminate topological classification, but relocates it to projector-based, symmetry-resolved, and real-space indices. This is why Bott indices, local Chern markers, and projector-based Chern numbers recur throughout the field.

5. Morphology and disorder as fractal diagnostics of topology

In nonequilibrium growth, fractality can become a readout of an underlying topological phase. A 2D higher-order topological insulator adiabatically connected to two copies of the BBH model develops corner-localized in-gap states that lower the attachment energy for deposition at corners relative to edges. The effective energy change is modeled by

$k$4

and the heat-bath deposition probability is

$k$5

Because corner deposition creates more single and triple Wannier-orbital configurations, growth is biased along $k$6. The resulting morphology is quantified by $k$7 and $k$8, with $k$9. At matched global openness, the nontrivial case has $c_0=2\,\mathrm{cm}^{-1}$0 and $c_0=2\,\mathrm{cm}^{-1}$1, whereas the trivial case has $c_0=2\,\mathrm{cm}^{-1}$2 and $c_0=2\,\mathrm{cm}^{-1}$3. The corresponding perimeter-area slopes are approximately $c_0=2\,\mathrm{cm}^{-1}$4 and $c_0=2\,\mathrm{cm}^{-1}$5. The paper is explicit that topology changes growth kinetics and morphology; fractality does not change the topology itself (Tanaka et al., 11 Sep 2025).

Near a disorder-driven topological quantum phase transition, fractality again appears as a diagnostic rather than a generator. In the disordered square-lattice QWZ Chern insulator, increasing onsite disorder $c_0=2\,\mathrm{cm}^{-1}$6 drives a transition near $c_0=2\,\mathrm{cm}^{-1}$7, where the disorder-averaged Bott index obeys single-parameter scaling with $c_0=2\,\mathrm{cm}^{-1}$8. Local Chern marker maps reveal normal droplets inside the topological phase and topological puddles inside the normal phase. Near the critical point, thresholded domains exhibit fractal geometry with hull and volume dimensions $c_0=2\,\mathrm{cm}^{-1}$9, $\epsilon_m=0\,\mathrm{cm}^{-1}$0 for normal droplets and $\epsilon_m=0\,\mathrm{cm}^{-1}$1, $\epsilon_m=0\,\mathrm{cm}^{-1}$2 for topological puddles, together with correlation lengths $\epsilon_m=0\,\mathrm{cm}^{-1}$3 and $\epsilon_m=0\,\mathrm{cm}^{-1}$4. The work explicitly states that the emergent fractality is a signature of the topology-changing transition, produced by the interplay of disorder, topology, and Anderson localization, rather than a mechanism that induces topology (Salib et al., 24 Apr 2025).

These two examples define an important semantic boundary. Fractal observables may be operationally powerful for phase diagnosis, but they do not automatically imply that the fractal structure is the microscopic origin of the phase.

6. Suppression, abstraction, and extensions beyond band topology

The relationship between fractality and topology is not uniformly constructive. On homogeneous fractals, gapped topological phases can fail altogether. For the homogeneous Sierpiński gasket with coordination $\epsilon_m=0\,\mathrm{cm}^{-1}$5 everywhere, a two-orbital Chern-insulator-like model does not produce a gapped topological phase in the intermediate regime $\epsilon_m=0\,\mathrm{cm}^{-1}$6. Instead it forms a gapless “fractalized metal,” with vanishing Bott index, a finite density of states near $\epsilon_m=0\,\mathrm{cm}^{-1}$7, scale-invariant edge fractions $\epsilon_m=0\,\mathrm{cm}^{-1}$8, and a minimal excitation gap that decays as $\epsilon_m=0\,\mathrm{cm}^{-1}$9 with $A$00. Chiral transport survives in a non-quantized form, and conventional bulk-edge separation is effectively lost. By contrast, inhomogeneous fractals such as the Sierpiński carpet can recover gapped Bott-nontrivial phases (Agarwala et al., 2018).

Other literatures use the phrase in broader topological senses. In polymer point-clouds, reducing the fractal dimension of a self-avoiding walk by loop deletion lowers the Betti-growth exponent from $A$01 to $A$02, while multifractal contact clouds raise it to $A$03; here fractality shapes persistent-homology complexity rather than band topology (Jia et al., 2020). In scale-free networks, maximal nearest-neighbor disassortativity does not generate network fractality; the decisive ingredient is long-range repulsive correlation among similar degrees (Fujiki et al., 2017). In combinatorial graph theory, one obtains a discrete topology/fractality dictionary through

$A$04

with Sierpiński gasket graphs furnishing the explicit example $A$05 and $A$06 (Skums et al., 2019). At an even more formal level, a fractal family of topological spaces is defined by nested inclusions $A$07 and subspace topologies

$A$08

so that increasing the number of nested spaces yields strictly finer topologies (Porchon, 2012).

Extensions also reach beyond single-particle band theory. In anisotropic 3D topological order with fractal subsystem symmetries, rigid Sierpiński-type membranes in the $A$09-plane and Wilson lines along $A$10 generate mixed anomalies, size-dependent ground-state degeneracy, and a low-energy-but-not-long-distance BF-like effective field theory. Here fractality enters not as a lattice decoration but as the symmetry algebra itself (Casasola et al., 2024). Conversely, on the surface of a 3D topological insulator, a time-reversal-symmetric Sierpiński-carpet potential produces monofractal Dirac states with dimension $A$11 while leaving the surface sector gapless; fractality reshapes the support of topological surface states without changing the underlying $A$12 class (Pullasseri et al., 2023).

A coherent view therefore requires a three-part distinction. First, fractal geometry can directly induce topological boundary states through exact reduction to effective higher-order models. Second, fractal lattices can host topological phases whose phenomenology is qualitatively altered by nested boundaries and the loss of translation symmetry. Third, fractal metrics can diagnose topology or topology-changing criticality without being the microscopic cause. Much of the current research frontier lies in clarifying which of these meanings applies in a given system, and in determining when self-similarity is a generator, an organizer, or merely a fingerprint of topology.