research35 min studyLesson 557 in Physics

Anderson localization and disorder

Distinguish interference-induced localization from an interaction gap and test spatial participation without overinterpreting a finite sample.

Before you begin

On this page

The question for today

What you’ll understand

Define a disordered noninteracting lattice model and localization length.
Use inverse participation ratios with normalization checks.
Separate single-particle localization, transport and many-body localization claims.

An electronic system can fail to conduct even without a repulsive charge gap. Static disorder scatters coherent waves repeatedly, and interference can make its eigenstates spatially localized. Anderson localization is therefore distinct from the atomic Mott mechanism, although real materials can contain both disorder and interactions. The distinction must be made using spatial and transport diagnostics, not merely the observation that a sample has high resistance.

Specify the disordered Hamiltonian and coherence

Start with one orbital per lattice site and a real nearest-neighbor hopping, but assign static site energies from a declared random distribution. For this noninteracting problem each one-particle eigenstate solves a random linear equation. Disorder breaks translation symmetry, so a single crystal momentum no longer labels an exact eigenstate. The average density of states can remain nonzero where eigenstates are localized; absence of a spectral gap does not guarantee diffusion.

Localization arises from interference among repeated scattering paths, not from particles being classically unable to climb every local energy barrier. Phase coherence and dimensionality matter. In the conventional noninteracting one-dimensional model with generic independent short-range disorder, states are localized in the thermodynamic limit, while three-dimensional models can have mobility edges separating localized and extended energy regions. Symmetry classes, correlated disorder and additional couplings can change these statements, so the distribution and model must be specified before declaring a universal disorder threshold.

H=-t\sum_{\langle ij\rangle}(|i\rangle\langle j|+|j\rangle\langle i|)+\sum_i\epsilon_i|i\rangle\langle i|,\qquad \epsilon_i\in[-W/2,W/2]\ \text{independently}

t and disorder width W are energies. This example uses a uniform independent on-site distribution and once-counted bonds; it is a one-particle Anderson model, not an interacting Hubbard model.↔ Scroll wide equations horizontally.

Measure spatial concentration with normalized states

A localized eigenstate typically has an exponentially decreasing envelope around a localization center, with a characteristic length that can depend strongly on energy and disorder. A state spread over only part of a finite sample can resemble localization even when its eventual thermodynamic behavior differs. Therefore compare sample lengths with the inferred localization length and distinguish boundaries or a confining potential from disorder-induced localization.

The inverse participation ratio sums the fourth powers of normalized site amplitudes. It equals the inverse number of equally populated sites for a uniform distribution on that support. An extended state on an increasing sequence of similar samples has a ratio that tends to zero roughly as inverse volume; an exponentially localized state can approach a nonzero value once the sample is much larger than its localization region. A single finite-state ratio measures concentration but does not prove the mechanism or its infinite-size limit.

\sum_i|\psi_i|^2=1,\qquad P_2=\sum_i|\psi_i|^4,\qquad N_{\rm eff}=P_2^{-1},\qquad |\psi(x)|\sim e^{-|x-x_0|/\xi}

P2 and Neff are dimensionless, while localization length ξ has length units. Neff is a participation diagnostic, not always an integer number of occupied sites; the exponential specifies an amplitude envelope, whose probability decays twice as fast.↔ Scroll wide equations horizontally.

Separate a wave diagnostic from conduction and interactions

Coherent transmission through a localized region is often exponentially small in sample length. Its logarithm and disorder distribution are useful because rare resonant samples can dominate an arithmetic mean. A measured finite-temperature resistance also includes contacts, phonon-assisted hopping, dephasing and interactions; it is not determined by one eigenstate participation ratio alone. Temperature-dependent hopping conduction can coexist with localized noninteracting states and requires an additional inelastic mechanism.

Adding interactions raises different questions about charge correlations, thermalization and many-body dynamics. A one-particle Anderson result does not establish many-body localization, and the stability of proposed interacting localized phases needs explicit model and limit qualifications. Likewise a Mott gap does not prove all one-electron eigenfunctions are Anderson localized. The wave lab can show coherent cancellation algebra but has no random landscape, repeated paths, finite-size scaling or transport channels, so it cannot demonstrate Anderson localization.

T_{\rm typ}(L)\propto e^{-2L/\xi}\quad\text{for an amplitude localization-length convention}

This illustrative coherent-transmission scaling uses a convention where amplitude decays on ξ. Detailed prefactors and the definition of a typical transmission depend on geometry and disorder; a single finite sample can show resonances.↔ Scroll wide equations horizontally.

Work it through

A participation-ratio comparison

Compare two normalized four-site wavefunctions. State A has equal probability $1/4$ on every site. State B has probabilities $(1/2,1/2,0,0)$ . Find their inverse participation ratios and effective participation numbers.

The ratio sums squared probabilities, because each probability is already an amplitude squared. For A, $P_2=4(1/4)^2=1/4$ . Its participation number is four.
For B, $P_2=(1/2)^2+(1/2)^2=1/2$ . Its participation number is two, indicating a more concentrated state in this finite basis.
Both probability lists sum to one, so their ratios can be compared directly. The result does not determine whether B arose from random interference, a clean potential well or an intentionally prepared superposition; that conclusion needs Hamiltonian and size dependence.

The result

State A has P2 = 1/4 and Neff = 4; state B has P2 = 1/2 and Neff = 2, without by itself proving Anderson localization.

The living notebook

Explore the model

Interactive lab

Interactive lab · waves

Interference is necessary context, not a localization simulation

Reset and set BOTH Phase difference = 0° and Amplitude ratio = 1; predict amplitude 2. Reset and set Phase difference = 180° and Amplitude ratio = 1; predict zero. The two equal-frequency waves show phase-sensitive cancellation only. There is no disorder control, ensemble sampling, spatial localization, repeated scattering network or conductivity readout.

SumFirst componentSecond component

Sum at x = 0: 0.866
Resultant amplitude: 1.732
Phase difference: 60°

Phase difference60°Amplitude ratio1

t = 0 s

Two waves share the same frequency and wave number. Playback advances t in sin(2x − t) + a sin(2x − t + phase). Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forInterference can suppress a signal, but localization requires a specified disordered spatial problem and scaling evidence.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

Localized single-particle states can contribute finite density of states while failing to carry diffusive current. Spectral availability and spatial transport are distinct diagnostics.

Your turn

Try it yourself

A normalized state is uniform over N sites. Find P2 and Neff for N = 20 and N = 80. Compare this scaling with a state remaining uniform over the same four central sites as the surrounding sample is enlarged.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

For uniform support, each probability is 1/N; compare the support size rather than the total number of available basis states.

Compare with the solution

A uniform twenty-site state has $P_2=20(1/20)^2=0.05$ and $N_{\rm eff}=20$ . At eighty sites these become $0.0125$ and 80. The state confined to the same four central sites retains $P_2=1/4$ and $N_{\rm eff}=4$ despite the larger sample. This contrasting scaling is consistent with extended versus fixed-support behavior. It still does not identify Anderson interference as the cause, because a clean confining potential could produce the latter pattern as well.

Keep the idea

Take it with you

Localization can suppress transport without a hard density-of-states gap.
Participation ratios require normalized states and finite-size interpretation.
Single-particle disorder localization does not prove an interacting many-body localized phase.

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