Fibonacci Tight-Binding

random-matrix-quantum · 37 views
random-matrix-quantumquasiperiodic

What It Is

1D tight-binding chain with hopping (off-diagonal) amplitudes t_n in {t_A, t_B} (t_A/t_B=2) modulated by Fibonacci substitution rule A->AB, B->A --- Cantor-set spectrum, multifractal critical eigenstate at the chiral-protected band center E=0 (Kohmoto-Kadanoff-Tang 1983; deterministic multifractality from substitution-aperiodic mechanism)

Interpretation

Standard analysis sees: heavy-tailed; right-skewed; few distinct values; nonstationary / drifting. The atlas additionally detects discrete-map sensitive dependence, Attractor Reconstruction:lyap_entropy, Higher-Order Statistics:c3_energy.

What standard analysis sees
tail heaviness0.99
asymmetry0.99
occupancy0.02
short-range corr0.20
long-range memory0.70
spectral colour0.47
periodicity0.39
complexity0.45
time-irreversibility0.68
volatility clustering0.16
multifractality0.79
dimensionality0.21
nonstationarity0.98
What the atlas adds
discrete-map sensitive dependence+9.4z
deterministic chaos (positive λ_max, sensitive dependence)
names a discrete-map-scoped estimate, NOT chaos in general — continuous-flow chaos (Lorenz, Rössler) reads weak/neutral here despite being genuinely chaotic; spiky arithmetic sources can false-positive on the finite-time estimate
Attractor Reconstruction:lyap_entropy+4.5z
Kolmogorov-Sinai entropy production rate (sum of positive λ)
Higher-Order Statistics:c3_energy+2.9z
third-cumulant burst energy (SOC irreducibility signature)
Atlas-extreme metrics the standard bank can’t predict for this source
Ulam Spiral (Square):polynomial_concentration+5.9zbank-miss 1.6σ
H3 Icosahedral:temporal_coherence+4.3zbank-miss 2.5σ
2-adic:mean_distance-3.9zbank-miss 1.0σ
Ulam Spiral (Square):radial_dispersion+3.7zbank-miss 1.9σ
Hodge–Laplacian:source_fraction+3.3zbank-miss 1.5σ
Nonstationarity:variance_trend-3.2zbank-miss 1.9σ
H3 Icosahedral:nn_enrichment+3.1zbank-miss 2.8σ
Wavelet Cascade:coarse_scale_kurtosis_slope-2.6zbank-miss 1.5σ

Composition

dtypefloat64
range[8.793e-15, 0.02254]
unique values16384 / 16384
mean ± std6.1e-05 ± 0.000413

Render Gallery

Native > phase_plot
Native > phase_plotraw
Native > timeseries
Native > timeseriesraw
Native > triple_lattice
Native > triple_latticeraw
Boltzmann > lag_map
Boltzmann > lag_map
Boltzmann > spin_grid
Boltzmann > spin_grid
Cayley > default
Cayley > default
Chladni > free_plate_nodal
Chladni > free_plate_nodal
Chladni > plate_bottom
Chladni > plate_bottom
Chladni > plate_top
Chladni > plate_top
Heisenberg (Nil) (centered) > signed_log_z
Heisenberg (Nil) (centered) > signed_log_z
Heisenberg (Nil) (centered) > xy_path
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
Hodge–Laplacian > default
Inflation (Substitution) > barcode
Inflation (Substitution) > barcode
Inflation (Substitution) > d_curve
Inflation (Substitution) > d_curve
Information Theory > default
Information Theory > default
Julia Set > escape_histogram
Julia Set > escape_histogram
Klein Bottle > default
Klein Bottle > default
Laplacian > default
Laplacian > default
Logarithmic Spiral > default
Logarithmic Spiral > default
Möbius-S³ > mobius_cylinder
Möbius-S³ > mobius_cylinder
Möbius-S³ > s2_base_map
Möbius-S³ > s2_base_map
Ordinal Partition > default
Ordinal Partition > default
Penrose (Quasicrystal) > phi_spectrum
Penrose (Quasicrystal) > phi_spectrum
Persistent Homology > barcode
Persistent Homology > barcode
Persistent Homology > h1_diagram
Persistent Homology > h1_diagram
Predictability > default
Predictability > default
Recurrence Quantification > default
Recurrence Quantification > default
Spectral Analysis > default
Spectral Analysis > default
Spectral Graph > default
Spectral Graph > default
Spirograph > mode_spectrum
Spirograph > mode_spectrum
Spirograph > phasor_trajectory
Spirograph > phasor_trajectory
Symplectic > flux_grid
Symplectic > flux_grid
Symplectic > phase_space
Symplectic > phase_space
Ulam Spiral (Sacks) > default
Ulam Spiral (Sacks) > default
Ulam Spiral (Square) > default
Ulam Spiral (Square) > default
Visibility Graph > default
Visibility Graph > default
Wavelet Cascade > default
Wavelet Cascade > default

Atlas Position

Nearest neighborDistance
Aubry-André Critical4.50
Anderson 1D Localized5.15
Geomagnetic ap Index5.31cross-origin

Open in Atlas →

Which Geometries Light Up

Attractor Reconstruction › Attractor Reconstruction:lyap_entropyrank 1/3065.0000
Attractor Reconstruction › Attractor Reconstruction:lyapunov_maxrank 1/3060.8698
AutoRegressive › AutoRegressive:ar_coef_2rank 1/3061.5382
Cantor Set › Cantor Set:bit_plane_autocorrelationrank 304/3060.0007
Catch24 › Catch24:DN_HistogramMode_10rank 2/3062.5806
Catch24 › Catch24:DN_HistogramMode_5rank 2/3065.3090
Catch24 › Catch24:CO_Embed2_Dist_tau_d_expfit_meandiffrank 303/3060.0094
Catch24 › Catch24:DN_Meanrank 304/3060.0027
Catch24 › Catch24:DN_Spread_Stdrank 304/3060.0183
Chladni › Chladni:temporal_burstinessrank 4/3060.6034
Continued Fraction › Continued Fraction:gauss_kuzmin_distancerank 2/3060.8713
H3 Icosahedral › H3 Icosahedral:temporal_coherencerank 2/3060.4179
Higher-Order Statistics › Higher-Order Statistics:skew_meanrank 2/3063.0869
Higher-Order Statistics › Higher-Order Statistics:kurt_maxrank 4/3067.1664
Hodge–Laplacian › Hodge–Laplacian:source_fractionrank 1/3060.7595
Klein Bottle › Klein Bottle:rank_deficitrank 4/3060.9521
Laplacian › Laplacian:poisson_recovery_errorrank 2/30615.6161
Multi-Scale Wasserstein › Multi-Scale Wasserstein:w_stdrank 302/3060.0003
Nonstationarity › Nonstationarity:variance_trendrank 304/306-0.3267
SL(2,ℝ) (Thurston) › SL(2,ℝ) (Thurston):cocycle_growth_raterank 4/3061.9804
Spectral Analysis › Spectral Analysis:spectral_bandwidthrank 4/3060.2088
Spherical S² › Spherical S²:concentrationrank 4/3060.9986
Spherical S² › Spherical S²:angular_spreadrank 302/3060.0646
Symplectic › Symplectic:phase_reflection_symmetryrank 2/3060.9994
Symplectic › Symplectic:windowed_area_cvrank 3/3069.0044
Symplectic › Symplectic:recurrence_raterank 4/3060.9751
S² × ℝ (Thurston) › S² × ℝ (Thurston):sphere_concentrationrank 3/3060.9989
Ulam Spiral (Square) › Ulam Spiral (Square):polynomial_concentrationrank 3/3060.1706
Ulam Spiral (Square) › Ulam Spiral (Square):radial_dispersionrank 3/3060.7148
Visibility Graph › Visibility Graph:degree_r_squaredrank 3/3060.9745
Wasserstein › Wasserstein:dist_from_uniformrank 2/3060.4831
Wasserstein › Wasserstein:concentrationrank 4/30631.4375
Wasserstein › Wasserstein:entropyrank 304/3060.1580
Wavelet Cascade › Wavelet Cascade:cascade_couplingrank 4/3060.8097
Zariski › Zariski:nonsep_fractionrank 4/3060.9527
Zipf–Mandelbrot (16-bit) › Zipf–Mandelbrot (16-bit):gini_coefficientrank 5/3060.9419
Zipf–Mandelbrot (8-bit) › Zipf–Mandelbrot (8-bit):gini_coefficientrank 5/3060.9411
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Structure Atlas — Open Atlas · GitHub