The Physical Seam · a random product you can operate
The Wave That Could Not Leave
Turn disorder through a one-dimensional quantum chain and measure its localization length from the exact transfer recurrence, then scan the shrinking band-center resonance where the familiar weak-disorder formula fails.
A dirty window dims light because it absorbs. This chain has no absorber. Its Hamiltonian is real and Hermitian, every transfer matrix has determinant one, and the missing transmission is reflected rather than destroyed. What grows below is not probability. It is the norm of a transfer vector, and its exponential growth rate is the inverse length over which a stationary wave amplitude is localized.
Layer 1 · make a length from disorder
One matrix per site
The nearest-neighbor equation is t(ψn+1 + ψn-1) + εnψn = Eψn. Each onsite energy is drawn independently from a box of width W. Move either control. The seed stays fixed, so the change you see comes from the parameter rather than a fresh accident.
[ ψ(n) ] = [ 1 0 ] [ ψ(n-1) ]
measured running γordinary weak-disorder forecast
finite-window γ
calculating
log growth per lattice site
ξamp = 1/γ
calculating
amplitude convention, in sites
box sample
calculating
mean and variance of ε
weak-disorder forecast
calculating
not valid at the center or band edges
After every multiplication the two-vector is divided by its norm. Those discarded logarithms are accumulated, so overflow is prevented without changing the asymptotic exponent. With our convention, typical amplitude decays like exp(-L/ξamp); typical intensity or transmission is often written exp(-2L/ξamp). Boundary definitions and averaging can move that factor of two, so the page never calls either one simply “the localization length.”
At zero disorder the finite product is bounded inside the clean band and the limiting exponent is zero. At nonzero independent onsite disorder the standard one-dimensional Anderson model has exponentially localized states with probability one under the usual regularity assumptions. That statement does not include correlated random dimers, quasiperiodic potentials, off-diagonal disorder, interactions, or dephasing.
Layer 2 · the flagship depth
The sophisticated dismissal
Positive growth in one dimension is elementary. The curve ξ ≈ (96t² - 24E²)/W² is already in the textbook. A long random product that lands near it has shown nothing the approximation did not predict.
Now make the approximation miss
At exactly E = 0, the clean transfer motion has a four-step resonance. Ordinary nondegenerate perturbation theory loses track of the phase distribution. Kappus and Wegner found the anomaly; Derrida and Gardner give its closed-form box-disorder coefficient in Eq. (77a). The naive scaled length tends to computing; the resonant value tends to computing. Their difference, computing, is small enough to hide in a short chain and large enough to survive a careful one.
The horizontal coordinate is Derrida and Gardner's published variable from Eq. (59): x = Et/σ² = 12Et/W² for this box disorder. The crossover is order one in x, but confined to absolute energies of order σ²/t = W²/(12t), vanishingly narrow beside the clean bandwidth 4t. Each dot below is a fresh transfer product at that energy, not a fitted curve and not an eigensolver. Error bars are the standard error across independent pseudorandom streams.
transfer productnaive Thouless curveband-center asymptote
selected x
0.0
E is computed from x
measured W²/(t²γ)
run the scan
standard error across streams
naive prediction
computing
ordinary perturbation theory
work performed
none yet
two-vector transfer steps
A noisy dot is not a refutation. Decreasing W narrows the energy window and also makes γ smaller, so a fixed chain contains fewer localization lengths and becomes noisier. Increase sites or streams and watch whether the central excess remains. The two-disorder repeat is especially unforgiving: it asks for the feature in the scaled coordinate twice, rather than tuning one run until it looks persuasive.
The cyan reference carries the full band factor: W²ξordinary/t² = 96(1 - E²/4t²). It equals 96 only at E = 0. The anomaly has no sharp cutoff. In the weak-disorder scaling limit, its large-|x| behavior is W²ξ/t² ≈ 96[1 + 3/(16x²)], equivalently a 1/E² tail at fixed disorder, as in Derrida and Gardner Eq. (78a).
The red line is not “computed” by the random product. It is an analytic asymptote evaluated live from gamma functions, and it is labeled as such. The amber dots are the measurement. Agreement is evidence that this finite transfer experiment has entered the weak-disorder regime. It is not a proof of the Kappus-Wegner derivation.
Layer 3 · the open edge
The chain closes. Three dimensions do not.
For the standard one-dimensional iid onsite model used here, localization is not the open problem. The nearest open neighbor is the ordinary three-dimensional Anderson transition. One-parameter scaling predicts an unstable critical conductance separating localized and metallic flows. High-precision transfer calculations for the orthogonal class report a weighted critical exponent ν = 1.571 with interval [1.563, 1.579]. That is a numerical finite-size-scaling result from Slevin and Ohtsuki, not a closed form and not a theorem.
, a proof of the extended weak-disorder phase and the localization-delocalization transition for the standard nearest-neighbor three-dimensional lattice with iid box disorder remains missing. Rigorous localization is established in large-disorder and spectral-edge regimes. The open gap is between those theorems and the numerically mapped mobility edge. Interactions, topology, magnetic field, and spin-orbit coupling define different problems and can change the scaling flow.
The check
The browser is recomputing the displayed quantities.
| quantity | how it is obtained | live value | state |
|---|
Every free choice
Every named uncertainty
- Finite length: the reported exponent is a finite-window estimate. The depth error bar measures stream-to-stream scatter, not truncation bias or higher-order corrections in W.
- Error propagation: the depth panel estimates the standard error of the mean exponent with the sample denominator streams - 1, then propagates it to the reciprocal to first order. With one stream it reports the error as unavailable, not zero.
- Pseudorandomness: xorshift32 is deterministic and fast, not a source of physical randomness or a cryptographic generator. Rerolling probes realization dependence.
- Floating point: all live arithmetic uses JavaScript binary64. Per-step renormalization prevents overflow, but this is not interval arithmetic.
- Initial direction: the asymptotic top exponent is independent of a generic starting vector, but a finite product retains a small initial-direction correction. This page performs no discarded burn-in.
- Weak-disorder asymptotics: both reference curves omit higher powers of W. The center coefficient applies as W approaches zero for rectangular diagonal disorder.
- Convention: ξamp = 1/γ. Intensity, typical transmission, mean transmission, Green functions, and participation ratios need not report the same finite-size length.
- Scope: the computation is stationary, coherent, closed, noninteracting, one-dimensional, nearest-neighbor, and diagonal-disorder only. It does not simulate a spreading packet, absorption, dephasing, or interactions.
What this does not license us to say
- “Disorder stops a wave completely” hides transient spreading, exponential tails, finite-sample leakage, and loss of coherence.
- “Localization is absorption” is false here. The ideal Hamiltonian conserves norm.
- “Every one-dimensional disorder model localizes” erases correlated, quasiperiodic, off-diagonal, and interacting exceptions.
- “All two-dimensional systems localize” is only the ordinary noninteracting orthogonal-class scaling conclusion. Other symmetry classes and topology alter it.
- A speckle power-spectrum cutoff can produce an effective perturbative mobility edge over accessible lengths. It is not automatically a true one-dimensional mobility edge.
Source ledger
The arithmetic is reproducible; the historical and experimental record is sourced. “Opened” means the scout or this build exposed the record or text. “Record only” means access stopped at search or publisher metadata, so the page does not pretend to have inspected the full paper.
The independent Node verifier contains more than forty assertions, executes this page’s own engine, and deliberately mutates the equations to prove the checks can fail. Run node research/anderson-localization/verify-anderson-localization.mjs. The full apparatus and source distinctions are in research/anderson-localization/README.md.