Is there a fractional quantum Hall liquid on a thin Sierpinski carpet?

Object. Hard-core bosons hopping on the gen-2 Sierpinski carpet (64 sites), periodized on a 9×9 torus or as the unit cell of an infinite cylinder, magnetic flux 1/9 per plaquette, four bosons (half-filling of the lowest single-particle Chern band, which has 8 states, Bott index +1, and a 0.49 gap above it).

Presenting problem (the friction). Two standard diagnostics disagree. - Torus, exact diagonalization: the two lowest states form a manifold separated from the third by 0.06–0.11 across the whole twist torus, and the Fukui–Hatsugai–Suzuki many-body Chern number of the pair is exactly 1 (0 for the lowest state alone). At N = 3 no such structure exists. This is the textbook fingerprint of a bosonic ν = 1/2 Laughlin-type state. - Infinite cylinder, iDMRG (χ = 60), adiabatic flux insertion: pumped charge over a cycle is 0.02, not 1/2; the entanglement entropy is symmetric in θ with a level crossing at θ = 0; the energy per cell varies by ~0.1 with flux; correlation length 25–45 sites; condensate correlator decays slowly. Nearest-neighbour repulsion V = 2 shortens the correlation length but leaves the pump at zero.

History. The single-particle problem is not in doubt: the thin carpet is a Chern insulator (Bott index 1, several Chern gaps at every flux) despite having no two-dimensional bulk anywhere. Fat fractals (corridors wider than the magnetic length) go gapless from protected chiral edge modes on internal holes; thin corridors gap those edges by hybridization. Independent work (Manna, Nielsen et al.) found anyons on fractal lattices with model wavefunctions, but retracted a local-Hamiltonian result on the same 64-site carpet after finding particles localize in locally two-dimensional domains.

Differential. 1. Genuine FCI, with the cylinder failing for technical reasons (χ = 60 far below 2D norms; circumference 9 with width-1 corridors is effectively narrower than 9). 2. Compressible superfluid on both geometries; the torus Chern-1 pair is a finite-size persistent-current state that happens to return an integer (the 0.2 twist-dependence of the energy supports this). 3. A crossover: incompressible at some larger V or at the fermionic ν = 1/3 filling, not at bosonic 1/2. 4. Fundamentally obstructed: the internal-edge density of states from holes at all scales prevents any gapped phase in the thermodynamic limit, and only the single-particle topology survives.

Discriminating tests. Charge pumping at χ ≥ 400; circumference 27 (gen-3 unit cell); topological entanglement entropy on the cylinder; fermions at ν = 1/3; V scan to 5. The decisive number is the pumped charge: 1/2 or 0.

Stakes. Whether topological order (not just a Chern band) can exist on a space with no plane in it, and whether the strategy "let the fractal choose its own filling" is right or a mirage.

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Update (2026-09-12): a fifth branch, and it is the one the geometry most suggests.

Two independent lines of reasoning that supported the case's framing have turned out to be underdetermined, and both point the same way.

The level-statistics evidence does not distinguish what it was taken to distinguish. A measured ⟨r⟩ of 0.41 on the thin carpet was read as Poisson-like and hence as topology without chaos, with hierarchical spectra suppressing repulsion. But at least four distinct mechanisms produce Poisson statistics, computed: uncorrelated levels give 0.3872, six superposed independent GUE sectors give 0.3826, weakly split degenerate levels give 0.2971 — and Anderson-localized eigenstates of a strictly local Hamiltonian give 0.3904. Against a Poisson reference of 0.386, the measured value discriminates none of them.

And localization is what this geometry most suggests. A carpet built from width-one corridors is close to the best case for Anderson localization in a quasi-one-dimensional geometry, so the alternative that the statistics fail to exclude is precisely the one the construction invites.

The related inference is invalid by type. The working hypothesis that "locality reads spectral dimension and ramification" derives a statement about states from a premise about the operator, and those are independent: a strictly nearest-neighbour tight-binding chain has operator range 1 at every disorder strength while the median participation number of its eigenstates falls from 267 to 1.2. Locality of a Hamiltonian implies nothing about localization of its eigenstates. The hypothesis was not merely refuted by computation; the inference could not have been valid.

Fifth branch. 5. Anderson localization in the corridors. The single-particle Chern structure is real, the many-body state is localized rather than either topologically ordered or condensed, and both the Poisson-like level statistics and the vanishing pumped charge follow from localization rather than from compressibility or from an obstruction.

Discriminating tests for it. Participation numbers and inverse participation ratio of the many-body eigenstates, not just of the single-particle band, with system size — localization means an IPR that does not decay with size. Level-spacing statistics at fixed disorder as corridor width increases from one to three, where branch 5 predicts a crossover toward GUE and branch 4 predicts none. And a direct comparison against a deliberately disordered square lattice tuned to the same ⟨r⟩: if the two are indistinguishable across every diagnostic, the fractal geometry is contributing nothing beyond effective disorder.

Effect on the existing branches. Branch 1 is unaffected — the cylinder pump would settle it either way. Branch 2 weakens, since localization explains the cylinder results without requiring a superfluid. Branch 4 weakens most: its stated mechanism was internal-edge density of states from holes at all scales, and branch 5 offers a more ordinary explanation for the same evidence. Nothing here is evidence for branch 5 over the others; the point is that the evidence previously counted against 1 and 3 does not count, because it is equally consistent with 5.