The Physical Seam · topology under pressure
The Integer Hiding in the Gap
Split a lattice band with magnetic flux, compute a gap Chern number, then add seeded disorder and watch localized states enter without immediately changing the bulk marker. The page keeps quantization, plateau width, and finite-size uncertainty visibly separate.
A Hall plateau contains two facts that are easy to blur. A clean isolated band already carries an integer Hall conductance. Moderate disorder can then widen the interval over which transport keeps that integer, because many of the new bulk states are localized. Disorder is not being asked to invent the integer below. It is being asked a narrower question: which states can change it?
The operable base
Put flux through one square
Drag the reduced flux. The denominator is not a label pasted onto the picture: it is the dimension of the magnetic Bloch Hamiltonian being diagonalized at every sampled momentum.
All reduced proper fractions with denominators 2 through 31 are available.
Click an empty horizontal strip to select that gap. A visual touching is not counted as a gap unless the sampled upper minimum exceeds the sampled lower maximum.
Magnetic bands
3
matrix dimension q
Selected gap
r = 1
sampling
Gap label
1 = 0·3 − (−1)·1
r = s q − C p
Predicted Hall integer
C = −1
sign follows this page's gauge convention
This is the square-lattice Harper model, not a claim that every ordinary Hall bar has a butterfly spectrum. The lattice makes the topology unusually visible: for coprime p,q, the magnetic unit cell has q sites and therefore q magnetic bands, with possible touchings.
The flagship depth layer
First the integer, then the plateau
A butterfly plus the word topology is not enough. Run the occupied-subspace invariant, then put the same flux into an open, disordered lattice and move the Fermi energy through its states.
The phase of each normalized determinant overlap joins two neighboring occupied subspaces. Four links make one plaquette. The sum of all plaquette phases is divided by 2π.
FHS sum
not run
predicted |Rxy| waits for resolved C
Minimum direct gap
not run
same k, bands r and r+1
Smallest link determinant
not run
overlap conditioning
Refinement verdict
waiting
15 by 15 versus 21 by 21
Each dot is an eigenstate. Height is the participation fraction 1/(N·IPR). A state on one site would be 1/N; a perfectly uniform state would be 1. Dot color follows the inverse participation ratio.
Occupied states
building
out of N = 144
Gaussian DOS at EF
building
η = 0.12t, states per t
Nearest state
building
IPR and participation
Central local Chern marker
building
central 6 by 6, never rounded
What the clean calculation shows
When the occupied subspace remains isolated, its link phases add to an integer. In this noninteracting model, σxy = C e²/h. No disorder was needed to produce C.
What the disordered calculation can show
States can enter while the central real-space marker moves much less than the occupied count. Low participation supports a localized finite-sample interpretation. Near a delocalized region, participation and the marker can change together. This 12 by 12 example illustrates the mechanism. It is not a thermodynamic-limit proof.
The finite open sample also has boundaries. Its marker cancels when averaged over the whole lattice, so this page averages only the central 6 by 6 sites and says so. Chiral edge transport is the boundary consequence of the bulk invariant, but a low-participation bulk state in this plot is not an edge state. Position at the boundary and localization in the bulk are different diagnostics.
The check
These values are rebuilt by the shipped page engine. The independent Node verifier derives its own references, executes this page's inline engine in a sandbox, and then mutates inputs and logic to prove the checks can fail.
Exact quotient versus realization. This page computes h/e² from h = 6.62607015 × 10⁻³⁴ J s and e = 1.602176634 × 10⁻¹⁹ C, both exact in the SI since 20 May 2019. The quotient is exact as a definition. The displayed decimal is rounded, and a laboratory realization still has experimental uncertainty. Under ideal integer-plateau conditions, |Rxy| = RK/|C|, not always RK.
Free choices. The hopping is t = 1. Layer 1 uses a 16 by 16 midpoint mesh and reports sampled band extrema, not certified global extrema. Layer 2 uses 21 by 21 and 15 by 15 meshes, calls the invariant unresolved below a direct gap of 10⁻⁶t or a link determinant of 10⁻⁸, and requires the two meshes to agree within 0.2. The disorder is uniform on [−W/2,W/2) from a named 32-bit seed. The open lattice is 12 by 12, the central average is 6 by 6, and the density-of-states Gaussian width is η = 0.12t. The local marker convention is −4π Im diag(PXPYP).
Uncertainties and truncations. All eigensystems use double-precision cyclic complex Jacobi rotations with tolerance 10⁻¹¹ and at most 40 sweeps. The Chern sum is a discretized gauge-invariant computation, not symbolic integration. The local marker is approximate because the lattice is small, open, disordered, and evaluated at one seed. Its deviation from the nearest integer is printed instead of hidden. Finite-size participation cannot by itself prove Anderson localization, which is an infinite-size statement. Strong disorder can close the mobility gap and destroy the plateau.
What the verifier tests
More than forty independent assertions cover exact SI arithmetic, rational reduction, Diophantine gap labels, Hermiticity, traces, spectra at solvable points, eigenvector orthogonality, residuals, mesh Chern values, determinant conditioning, deterministic disorder, onsite bounds, eigenvalue sum rules, IPR bounds, projector idempotence, marker finiteness, page coupling, required markup, and mutation sensitivity. Run node research/quantum-hall-chern/verify-quantum-hall-chern.mjs.
The open edge
The transition still lacks one settled exponent story
As of 1 August 2026, the existence of integer quantization is not the open problem. The critical theory connecting adjacent plateaux, especially how interactions, screening, dephasing, and disorder range select measured exponents, remains unsettled. A high-precision noninteracting Chalker-Coddington calculation reported a localization-length exponent ν = 2.593 with interval [2.587, 2.598] in 2009. A 2024 trilayer-graphene experiment instead reported γ = 2.4 ± 0.2 and κ = 0.41 ± 0.02 across integer and fractional transitions, while documenting a much wider historical experimental range for κ. An accepted 27 July 2026 experiment reports that a nearby screening layer changes the dynamical critical behavior, direct evidence that electron interaction range matters. The live 12 by 12 marker above cannot decide this thermodynamic, interacting scaling dispute.