Anderson localization

Anderson localization is the confinement of Electron wave functions, or other wave modes, by interference between repeated wave scattering paths in static disorder. A localized eigenstate occupies a limited region even in a much larger sample. Disorder can therefore suppress diffusion without absorbing the wave. In 1958, Philip Warren Anderson studied random site energies and intersite coupling, finding a regime without transport. His paper develops a theoretical model; it does not report an Anderson laboratory measurement.^[ref]

The distinction from ordinary multiple scattering is the retention of wave phases: alternative paths contribute probability amplitudes before their squared modulus is taken. Adding path probabilities alone loses the cross terms responsible for interference. This makes Anderson localization useful as a warning about transport models: repeated deflections need not produce unlimited diffusion. It also connects Electron transport with wave propagation in a random medium, although the detailed equations and experimental diagnostics differ between Electrons, light and acoustic waves.

Philip Warren Anderson's 1977 lecture distinguishes localized magnetic moments from disorder-localized states. It credits Nevill Francis Mott with recognizing the significance of the mobility edge. A Mott insulator instead requires Electron interactions and can exist without disorder.^[ref]

John Hasbrouck Van Vleck developed quantum mechanics explanations of magnetic susceptibility, including its temperature dependence; this is a distinct contribution from Anderson localization.^[ref]

An energy interval with no extended bulk energy eigenstates available for transport. It can contain localized eigenstates and have a nonzero density of states; a mobility gap need not be a spectral gap.

A localized eigenstate is an energy eigenstate whose spatial weight remains concentrated near some position as the system grows. An exponentially localized eigenstate has a decaying envelope characterized by a localization length. A small participation ratio measures concentration in a chosen site basis; it does not by itself establish an exponential tail or a transport law.

An energy eigenstate changes only by an overall wave phase under time evolution, whether localized or extended. Its stationary probability distribution alone therefore does not distinguish transport mechanisms. The important distinction concerns spatial extent and how superpositions of energy eigenstates carry or retain a disturbance. A wave packet assembled from localized eigenstates with nearby centres can remain confined, whereas extended modes can support propagation.

The localization length $\xi$ measures the decay scale of a localized eigenstate's wave amplitude envelope. Here $a$ is the site spacing and $n_0$ its centre. With this amplitude convention, probability decays on half that scale:Individual random tails fluctuate; this notation describes their typical envelope, not exact equality at every site. A localization length much greater than a finite sample's length makes a state look extended inside that sample. The participation ratio below is an effective site count, not an independent fitted value of $\xi$.

For an eigenvector normalized by ${\sum }_{n=1}^N|{\psi }_n{|}^2=1$, the participation ratio in its site basis isIt estimates how many sites carry appreciable probability. Equal weight on $K$ sites and zero elsewhere gives $P=K$, irrespective of their wave phases. A single occupied site gives $P=1$; uniform probability on all sites gives $P=N$. These are exact constructed examples, not claims about particular Hamiltonian operators.

To prove the bounds, write $p_n=|{\psi }_n{|}^2$. Since $0\le p_n\le 1$, ${\sum }_n{p}_n^2\le {\sum }_n{p}_n=1$, giving $P\ge 1$. The Cauchy-Schwarz inequality gives $1=({\sum }_n{p}_n{)}^2\le N{\sum }_n{p}_n^2$, giving $P\le N$. Thus the measure has useful limits without assuming an exponential shape. Comparing $P$ as $N$ increases helps distinguish a state filling the sample from one occupying a fixed region, provided energy selection and disorder statistics are controlled.

A mobility edge separates localized from extended energy eigenstates on the energy axis. Three-dimensional disordered models can have both. In the infinite one-dimensional tight-binding model with independent continuous random site energies, all states are localized for any nonzero disorder, with probability one. This statement excludes specially correlated disorder.^[ref]

A mobility edge is not a band gap: states can exist on both sides of $E_c$, but differ in spatial extent. For Electrons, the location of the Fermi energy relative to a mobility edge matters for low-temperature electrical conductivity. A three-dimensional sample need not have an extended interval at every disorder strength. In particular, the finite one-dimensional calculations below neither locate a three-dimensional mobility edge nor establish a nonzero universal critical disorder.

Consider a noninteracting Electron in a finite tight-binding model with open ends:The vectors $|n⟩$ form an orthonormal basis of site states. The hopping energy is $t$; it is not the time coordinate. Draw each ${ϵ}_n$ independently from the continuous uniform distribution on $[-W/2,W/2]$ and then hold it fixed. The disorder strength $W/t$ is dimensionless. Open ends mean no hopping bond connects sites $N$ and $1$; equivalently set ${\psi }_0={\psi }_{N+1}=0$. There are no Electron interactions, phonons or wave absorption in this model.

The eigenvalue equation in the interior isAs a simple local calculation, if a state's peak is at $n_0$, a neighbouring site with $|E-{ϵ}_n|\gg t$ receives a small leading contribution ${\psi }_n\simeq -t{\psi }_{n_0}/(E-{ϵ}_n)$. Several successive large mismatches multiply small factors. This explains why large disorder can restrict spreading, but is not a proof of Anderson localization: near-resonant denominators and alternative interference paths prevent treating this expansion as uniformly valid.

For the first original calculation, take $N=80$, $t=1$, $W/t=6$ and the fixed random seed $20261015$. Diagonalizing the resulting real symmetric matrix with numpy.linalg.eigh and selecting the eigenvalue closest to zero gives $E/t\simeq -0.05962$, a probability peak at site $29$, and $P\simeq 6.0031$. The selected eigenvector has total probability one. This is one chosen realization, not a disorder average or a measured specimen.

The clean comparison needs care. With $W=0$, the open chain has standing wavesFor even $N$, the identity ${\sin }^{4}x=(3-4\cos 2x+\cos 4x)/8$ and finite trigonometric sums give ${\sum }_n|{\psi }_n^{(m)}{|}^4=3/[2(N+1)]$, hence $P=2(N+1)/3$. Thus the clean $80$-site comparison has $P=54$, although it extends across the chain. Extended does not mean uniform probability. For odd $N$, the special central mode $m=(N+1)/2$ has zeros on alternating sites and instead $P=(N+1)/2$; the even-size assumption avoids that exception.

The second original calculation compares $N=40$ and $80$ over $W/t=0,0.5,1,2,3,4,6,8$. For each size, draw $24$ independent site patterns using seed $20261016$, with the size included in the random-seed construction. Scale each fixed pattern across disorder strengths. At every point, average $P$ over the four energy eigenstates closest to $E=0$. The solid line is the median across these sample averages; shading spans their $10$th–$90$th percentiles. The selected states can change as $W/t$ varies: lines connect one sample's selections rather than tracking a single mode.

At $W/t=0$ the two clean values are $27\frac{1}{3}$ and $54$. At $W/t=8$, the computed median sample averages are approximately $3.17$ and $3.01$. In the latter regime, doubling $N$ hardly changes the typical occupied site count, so $P/N$ falls. At weak disorder the larger chain permits more spatial extent. Neither a single curve nor the overlap of finite-size bands proves the infinite-system statement under mobility edge; a long localization length can hide confinement in either size.

Nevill Francis Mott's 1977 lecture discusses silicon deposited in a glow discharge. Injected Electrons provided a drift-mobility readout; electrical conductivity and temperature dependence tested localized-state transport and excitation toward a mobility edge. Varying temperature distinguishes transport regimes. The lecture gives no contact geometry or timing circuit here, so none is inferred. Its Figure 2 is schematic, not measured data.^[ref]

This evidence motivates a transport question beyond the closed-chain eigenvalue problem: how does a prepared charge disturbance move under the conditions of a real sample? The numerical figures above supply neither drift-mobility measurements nor a model of the deposition process. Finite-size concentration alone cannot reproduce an experimental electrical conductivity curve.

Finite temperature permits phonon-assisted hopping between localized eigenstates or thermal excitation toward a mobility edge; localized states therefore need not imply zero measured electrical conductivity at every nonzero temperature.^[ref]

Real dephasing destroys the relative wave phases needed for long-path interference. Observing coherent confinement requires the distance over which those wave phases survive to be long enough to explore the localized region, and the sample to be large enough to resolve it. Temperature alone is not an automatic switch: the actual inelastic processes set the relevant scales. Static elastic wave scattering preserves coherence within each realization. Loss of the ensemble-averaged coherent field through random wave phases is therefore not, by itself, physical dephasing of an individual sample.