What does the disorder parameter W mean?

Each site energy ε_i is uniform on [−W/2, W/2]. W = 0 is a perfect chain with bandwidth 4t. Increasing W broadens the spectrum and tends to localize states in the band center in 1D; W is always quoted in the same units as the hopping t (here t = 1 by default).

How should I read the IPR?

With normalized ψ, IPR = Σ|ψ_i|⁴. For a state spread over all N sites with comparable amplitude, IPR ~ 1/N. For a state confined to a few sites, IPR approaches O(1). The dashed reference line marks 1/N on the IPR–energy plot.

Why do pink bars appear on |ψ|²?

They mark states with IPR > 5/N, a simple heuristic for “more localized than a uniformly extended state” on this finite chain. It is a visualization aid, not a rigorous critical criterion.

Is there a mobility edge in 1D?

For the 1-D Anderson model with site disorder, essentially all eigenstates are localized for any W > 0 in the infinite-system limit. Mobility edges appear in higher dimensions; this page is intentionally 1D to show eigenvectors and IPR directly.

Anderson Localization (1D Tight-Binding)

Anderson localization explains why electronic (or wave) eigenstates in a disordered crystal can remain spatially confined even when the underlying lattice supports propagation. This simulator implements the standard 1-D tight-binding Anderson Hamiltonian on an open chain: H = −t Σ_i (|i⟩⟨i+1| + h.c.) + Σ_i ε_i |i⟩⟨i| with hopping amplitude t and independent random onsite energies ε_i drawn uniformly from [−W/2, W/2]. The disorder strength W is measured in units of t. The full N×N matrix is diagonalized numerically to obtain eigenenergies E_n and eigenvectors ψ_n. For each state you can inspect the probability density |ψ_i|² along the chain, the onsite disorder landscape ε_i, and the inverse participation ratio IPR = Σ_i |ψ_i|⁴ — small (of order 1/N) for delocalized band states and large for localized states. A scatter plot of IPR versus E highlights how states near the band center localize as W increases, while the clean chain (W = 0) recovers extended Bloch-like eigenstates. Localization length is estimated two ways: the RMS spatial extent of |ψ|² and an exponential fit of ln|ψ| to distance from the peak amplitude. Finite chain length and open boundaries are important caveats: edge reflections and level repulsion modify the spectrum compared with an infinite system, but the IPR diagnostic remains the standard classroom measure of localization.

Who it's for: Graduate or advanced undergraduate solid-state physics students studying disordered systems, after band structure and before scaling/localization length concepts in 2D/3D.

Key terms

Anderson localization
Tight-binding model
Disorder potential
Inverse participation ratio
Localization length
Eigenstate
Density of states

How it works

Anderson localization in a 1-D tight-binding chain with random onsite disorder: eigenstates, |ψ|² profiles, inverse participation ratio, and localization length estimates — the standard condensed-matter disorder demo.

Frequently asked questions

What does the disorder parameter W mean?: Each site energy ε_i is uniform on [−W/2, W/2]. W = 0 is a perfect chain with bandwidth 4t. Increasing W broadens the spectrum and tends to localize states in the band center in 1D; W is always quoted in the same units as the hopping t (here t = 1 by default).
How should I read the IPR?: With normalized ψ, IPR = Σ|ψ_i|⁴. For a state spread over all N sites with comparable amplitude, IPR ~ 1/N. For a state confined to a few sites, IPR approaches O(1). The dashed reference line marks 1/N on the IPR–energy plot.
Why do pink bars appear on |ψ|²?: They mark states with IPR > 5/N, a simple heuristic for “more localized than a uniformly extended state” on this finite chain. It is a visualization aid, not a rigorous critical criterion.
Is there a mobility edge in 1D?: For the 1-D Anderson model with site disorder, essentially all eigenstates are localized for any W > 0 in the infinite-system limit. Mobility edges appear in higher dimensions; this page is intentionally 1D to show eigenvectors and IPR directly.

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