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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]

A Mott insulator is insulating because of electron interactions even where a noninteracting electronic band structure would predict conduction. This mechanism differs from disorder-driven Anderson localization.

A Mott transition is a change between a conducting state and a Mott insulator, driven by the competition between electron motion and electron repulsion. A partially filled energy band need not conduct if repulsion makes moving an electron onto an already occupied site too costly.

The half-filled Hubbard model illustrates the mechanism: hopping lowers the kinetic energy, whereas on-site repulsion $U$ penalizes double occupation. Increasing $U$ relative to the energy-band width can suppress charge motion and open a gap for charge excitations. Changing pressure, composition or carrier concentration can alter that competition, although actual materials may also change their crystal structure or magnetic order. This interaction mechanism differs from a band insulator with filled bands and from disorder-driven Anderson localization.

Multiple scattering occurs when a disturbance encounters more than one scatterer, or revisits scatterers after earlier deflections. Each object is driven by both the original incident wave and fields produced by other objects. Solving the coupled responses therefore differs from adding independent single-scattering intensities.

When wave phases remain correlated, alternative paths must be added as complex amplitudes before squaring, retaining their interference. If suitable randomization makes those cross terms negligible, transport may instead be approximated by a sequence of random deflections; the relevant mean free path compares the typical distance between encounters with sample size. A thick cloudy sample can transmit diffuse light even when its direct beam is weak. Under appropriate coherent conditions, repeated wave scattering also produces weak localization or Anderson localization, which cannot be inferred from independent path probabilities alone.

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.

In quasi-one-dimensional quantum transport, transverse confinement makes discrete subbands while motion along a quantum wire remains continuous. Only occupied propagating subbands conduct. For elastic scattering between equilibrated reservoirs, the Landauer formula isThe sum explicitly counts spin. A perfectly transmitting spin-degenerate subband gives $2e^2/h$ at low temperature; a quantum point contact shows steps as a gate opens additional subbands. The one-dimensional density of states cancels the carrier velocity in the current, giving a conductance quantum rather than a material-dependent velocity factor. Finite two-terminal contact resistance remains even without internal scattering.

Transport is ballistic transport for length $L\ll \ell$, where $\ell$ is the mean free path. In a multichannel wire with $\ell \ll L$ below its localization length, scattering reduces transmission and gives approximately Ohmic behaviour. If the phase coherence length exceeds the sample length, amplitudes interfere; weak localization and sample-dependent universal conductance fluctuations can appear. At greater lengths, generic coherent disordered wires undergo Anderson localization, with localization length of order the channel number times $\ell$, up to symmetry-dependent factors. Thermal averaging and dephasing smear the quantum features.

Strong interactions require more than independent-electron transmission. A clean single-channel interacting wire is described at low energy by a Luttinger liquid, with collective modes and possible spin-charge separation; impurities can generate temperature-dependent power laws. Ideal contacts to noninteracting leads can still fix the two-terminal conductance of a clean wire, so it must not automatically be multiplied by a bulk Luttinger parameter.

Philip W. Anderson showed that disorder can prevent quantum transport in a lattice. His 1958 model assigns random energies to sites connected by hopping, demonstrating regimes in which diffusion fails even though motion between neighboring sites is allowed. The phenomenon now called Anderson localization depends on coherent wave propagation through a random system. It changed how an insulating response could be understood, connecting transport with the spatial character of quantum states.^[ref]

A physicist.^[ref]

The degeneracy of a Landau level is $eBA/h$ per resolved internal branch in area $A$. The filling factor $f=n_{s}h/(eB)$ varies continuously. Additional localized eigenstates need not carry bulk transport, allowing an integer plateau over an interval of $f$; a mobility gap need not be an empty spectral gap.^[ref]

The plot is an original phenomenological construction, not measured data or a calculation of Anderson localization. It stipulates dimensionless transverse sheet conductivity $k(f)=1+\frac{1}{2}{\sum }_{m=1}^5[1+\tanh ((f-m-1/2)/w)]$ and longitudinal sheet conductivity $s(f)=a{\sum }_{m=1}^5{\mathrm{sech}}^2((f-m-1/2)/w)$, with chosen $w=0.06$, $a=0.08$. Both are measured in units $e^2/h$. The quantum Hall conductivity tensor inversion produces $|{\rho }_{yx}|/R_K=k/(s^2+k^2)$ and ${\rho }_{xx}/R_K=s/(s^2+k^2)$. Regions near $k=N$, $s=0$ yield plateaus near $R_K/N$; the model alone does not explain their microscopic robustness. Each step represents one resolved branch, without assuming unresolved spin or valley multiplicities.

Robin Kaiser studies collective scattering of light and coherent transport in clouds of cold chemical atoms. His ANDLICA project asks whether Anderson localization of light can be observed in three-dimensional cold ytterbium samples. The project distinguishes localization from effects such as absorption and inelastic scattering of light, which can imitate some transport signatures. It connects controlled atomic samples and time-resolved measurements with the physical question of whether disorder can prevent extended wave transport.^[ref]

A physicist.^[ref]

Wave propagation in a random medium studies how fluctuations of properties such as refractive index, mass density or sound speed change an incident wave. Different paths acquire different wave phases, and spatial inhomogeneities generate wave scattering. One realization can therefore show a complicated transmitted pattern even under simple illumination.

The coherent field $⟨u⟩$ is the ensemble average of the field, while the mean wave intensity in a scalar normalization is $⟨|u{|}^2⟩$. They obeyThus decay of the coherent field need not mean loss of total wave energy: randomized wave phases can transfer weight to fluctuating contributions. Comparing a phase-sensitive average with an average of detector intensities tests this distinction. Weak fluctuations can permit perturbation methods, whereas repeated wave scattering with retained interference can lead to Anderson localization.

Interference of time-reversed multiple-scattering paths can enhance return probability and reduce conductance in a coherent disordered conductor with the appropriate symmetries. Magnetic fields, spin-orbit coupling and dephasing alter or suppress the correction; some symmetry classes exhibit antilocalization. This small-correction regime must be distinguished from strong Anderson localization.