Condensed Matter Physics » Disorder & Localization

Disorder & Localization

Disorder in a solid (impurities, vacancies, alloy randomness, rough interfaces) does more than add resistance. Quantum interference between waves scattered by random impurities can trap electrons completely, producing an insulator with no band gap and no interactions. This page covers that phenomenon, Anderson localization, from its weak precursor in good metals (weak localization and antilocalization) through the metal-insulator transition at the mobility edge, the single-parameter scaling theory that organizes the whole subject, transport in the localized phase, and many-body localization, the interacting, finite-temperature extension whose status is still debated. Band theory and Bloch states are covered on the hub; the field-theoretic and Green’s-function methods on Graduate-Level Formalism.

Why disorder matters

In a perfect crystal Bloch’s theorem guarantees extended eigenstates, so a partially filled band is a perfect conductor at $T = 0$. Real metals have finite resistance because electrons scatter from phonons and defects. In the semiclassical (Drude-Boltzmann) picture an electron travels a mean free path $\ell = v_F\tau$ between elastic collisions, and over many collisions its motion becomes diffusive with diffusion constant $D = v_F^2\tau/d$ in $d$ dimensions. The conductivity is

the second form (the Einstein relation) using the density of states per unit volume $\nu(E_F)$. In this picture disorder only limits the conductivity.

In 1958 P. W. Anderson showed that the picture is incomplete. When disorder is strong enough, or in low enough dimension for any disorder, interference between scattered partial waves makes eigenstates exponentially localized:

with localization length $\xi$. A localized electron carries no dc current at $T = 0$, so $\sigma \to 0$ even though the density of states at $E_F$ is finite. The result is an insulator produced by disorder alone. Anderson shared the 1977 Nobel Prize partly for this work, and the effect has since been observed for electrons, light, sound, and ultracold atoms.

Length scales and regimes

Localization physics is organized by comparing a few lengths with each other and with the sample size $L$:

The Ioffe-Regel criterion $k_F\ell \sim 1$ marks where the semiclassical picture must fail: the mean free path cannot be shorter than a wavelength.

A useful dimensionless measure is the Thouless conductance $g = E_{\text{Th}}/\delta$, the ratio of the Thouless energy $E_{\text{Th}} = \hbar D/L^2$ (the inverse time to diffuse across the sample) to the mean level spacing $\delta = 1/(\nu L^d)$. It equals the conductance in units of $e^2/h$ up to a numerical factor. Eigenstates are extended when $g \gg 1$ (levels hybridize across the sample) and localized when $g \ll 1$. How $g$ changes with $L$ is the content of the scaling theory.

Anderson localization

The Anderson model

The canonical model is a tight-binding lattice with random on-site energies,

where $t$ is the nearest-neighbour hopping and the $\varepsilon_i$ are independent random variables, conventionally drawn from a box distribution of width $W$:

The control parameter is the dimensionless disorder strength $W/t$. At $W = 0$ the model is a clean band of width $2zt$ ($z$ the coordination number) with all states extended.

The localization transition

Hopping delocalizes; disorder localizes. An electron on a site of energy $\varepsilon_i$ tunnels efficiently only to a neighbour whose energy lies within about $t$ of its own. When the typical mismatch $W$ greatly exceeds $t$, such resonances are rare and the electron is trapped. Anderson’s analysis of the convergence of the locator (strong-disorder) expansion showed that above a critical disorder every eigenstate is localized:

The outcome depends on dimension:

For $W < W_c$ in three dimensions, band-centre states are extended while the band tails are localized, and the two are separated by a mobility edge (below). Precisely at the transition, eigenstates are neither extended nor localized but multifractal: their moments scale as $\sum_i |\psi(i)|^{2q} \sim L^{-\tau_q}$ with a nonlinear spectrum $\tau_q$.

Diagnostics: participation ratio and level statistics

The inverse participation ratio (IPR) of a normalized eigenstate on $N$ sites,

measures the number of sites it occupies, $1/P_2$. An extended state has $P_2 \sim 1/N \to 0$; a localized state has $P_2 \to \text{const}$ independent of $N$. The code below diagonalizes the 1D Anderson chain and compares the disorder-averaged IPR of band-centre states with the clean chain.

import numpy as np

def anderson_1d(N, W, t=1.0, rng=None):
    """1D Anderson Hamiltonian: box-distributed on-site energies + NN hopping."""
    eps = rng.uniform(-W / 2, W / 2, size=N)
    return np.diag(eps) - t * (np.eye(N, k=1) + np.eye(N, k=-1))

def mean_ipr(N, W, window=0.5, samples=20, seed=0):
    """Average IPR of eigenstates with |E| < window over disorder realizations."""
    rng = np.random.default_rng(seed)
    iprs = []
    for _ in range(samples):
        evals, evecs = np.linalg.eigh(anderson_1d(N, W, rng=rng))
        sel = np.abs(evals) < window
        iprs.extend(np.sum(np.abs(evecs[:, sel]) ** 4, axis=0))
    return np.mean(iprs)

for W in (0.0, 2.0):
    for N in (250, 500, 1000, 2000):
        p2 = mean_ipr(N, W, samples=1 if W == 0 else 20)
        print(f"W={W:3.1f}  N={N:5d}  <IPR>={p2:.3e}  N*<IPR>={N * p2:7.1f}")

Typical output:

W=0.0  N=  250  <IPR>=5.976e-03  N*<IPR>=    1.5
W=0.0  N= 2000  <IPR>=7.496e-04  N*<IPR>=    1.5
W=2.0  N=  250  <IPR>=4.511e-02  N*<IPR>=   11.3
W=2.0  N= 2000  <IPR>=3.798e-02  N*<IPR>=   76.0

In the clean chain $N \cdot P_2$ is constant (it equals $3/2$ for standing waves), so the IPR falls as $1/N$. With $W = 2t$ the IPR saturates at about $0.04$: each state occupies a few tens of sites however long the chain. (For localization lengths comparable to $N$ the IPR still drifts, which is why numerical work uses finite-size scaling or the transfer-matrix method, where $\xi$ is the inverse of the smallest Lyapunov exponent of a product of random $2\times 2$ transfer matrices.)

A second diagnostic uses the spectrum alone. Localized states at nearby energies sit in different places and do not repel, so their level spacings follow Poisson statistics; extended states mix and show random-matrix (Wigner-Dyson) level repulsion. The mean ratio of consecutive spacings, $\langle r\rangle = \langle\min(s_n, s_{n+1})/\max(s_n, s_{n+1})\rangle$, needs no unfolding of the density of states and takes the values $2\ln 2 - 1 \approx 0.386$ (Poisson), $\approx 0.531$ (GOE), $\approx 0.600$ (GUE), and $\approx 0.674$ (GSE).

Weak localization and antilocalization

Strong localization needs strong disorder, but interference leaves a measurable fingerprint even in good metals with $k_F\ell \gg 1$: a small, temperature- and field-dependent correction to the Drude conductivity called weak localization. It is the perturbative precursor of Anderson localization.

Coherent backscattering

The probability for a diffusing electron to return to its starting point is a sum over closed paths. Every closed path has a time-reversed partner that visits the same impurities in the opposite order. With time-reversal symmetry the two amplitudes are equal, $A_+ = A_-$, so

twice the classical value $2|A_+|^2$. Enhanced return probability means reduced forward transport, so the conductivity falls below the Drude value. In diagrammatic language this is the sum of maximally crossed diagrams (the Cooperon). The same effect appears in optics as the coherent-backscattering cone of light reflected from a random medium.

Size of the correction

Loops contribute from the elastic time $\tau$ up to the phase-coherence time $\tau_\phi$, beyond which inelastic scattering (electron-electron, electron-phonon) randomizes the phase. In two dimensions the correction per square is logarithmic:

with $L_e = \sqrt{D\tau}$ (spin degeneracy included). In general dimension, up to positive numerical prefactors,

The correction is negative in every dimension. In $d = 3$ it stays finite as $L_\phi \to \infty$, but in $d \le 2$ it diverges. Because $\tau_\phi$ grows on cooling (typically $\tau_\phi \propto T^{-p}$ with $p$ between 1 and 3), the conductivity of a 1D or 2D metal decreases without bound as $T \to 0$. This is the perturbative signature of the scaling-theory result that non-interacting, time-reversal-symmetric electrons have no true metal in $d \le 2$.

Magnetoresistance as the fingerprint

A perpendicular magnetic field gives time-reversed loops opposite Aharonov-Bohm phases, destroying the interference once the flux through a typical coherent loop reaches about one flux quantum. The field therefore restores conductance, giving a negative magnetoresistance of order $e^2/h$ per square at low field. For a 2D film, Hikami, Larkin, and Nagaoka (HLN) derived

where $\psi$ is the digamma function. The bracket is positive and grows with $B$. In the simplest limits $\alpha = 1$ for weak localization (conductance rises with field) and $\alpha = -1/2$ for weak antilocalization (conductance falls). Fitting this form to low-field data is the standard way to extract $L_\phi(T)$ in thin metal films, semiconductor heterostructures, graphene, and topological-insulator surfaces.

Symmetry classes and antilocalization

Strong spin-orbit scattering rotates the spin along a loop so that the time-reversed amplitudes acquire a relative phase of $\pi$ (a Berry phase from the $2\pi$ spin rotation), turning constructive interference into destructive. This weak antilocalization raises the conductivity and produces a positive low-field magnetoresistance. The three behaviours correspond to the Wigner-Dyson symmetry classes:

The index $\beta$ is the Dyson index of the corresponding random-matrix ensemble. The Altland-Zirnbauer extension to ten classes (adding particle-hole and chiral symmetries) underlies the classification of topological insulators and superconductors: topological surface states are exactly those that cannot be localized by disorder respecting the protecting symmetry.

Interaction corrections

Electron-electron interactions in a disordered metal add a separate correction, the Altshuler-Aronov effect: diffusive motion makes the Coulomb interaction more effective, producing a $\ln T$ correction to $\sigma$ in 2D and a zero-bias anomaly (a dip in the tunneling density of states at $E_F$). It is distinguished experimentally from weak localization because it is insensitive to weak perpendicular fields. Interactions also supply the dephasing that sets $\tau_\phi$.

The mobility edge and the metal-insulator transition

Mott’s mobility edge

In three dimensions moderate disorder localizes only part of the spectrum. As N. F. Mott emphasized, localized and extended states are separated in energy by a sharp mobility edge $E_c$: states in the band tails, where the density of states is low, are localized; states in the band centre are extended. The two cannot coexist at the same energy, because a localized state degenerate with an extended one would hybridize with it and delocalize.

Driving the transition

Whether a sample is a metal at $T = 0$ depends on where the Fermi level sits relative to $E_c$. There are two ways to cross the transition:

1. Tune the disorder. Increasing $W$ moves both mobility edges inward. At $W_c$ they meet at the band centre and all states localize.
2. Tune the carrier density. In doped semiconductors, the textbook case being phosphorus-doped silicon (Si:P), changing the dopant concentration sweeps $E_F$ through $E_c$. Below a critical density $n_c$ the material is insulating; above it, metallic. Mott’s criterion $n_c^{1/3}a_B^\ast \approx 0.26$, with $a_B^\ast$ the effective Bohr radius, holds across many semiconductors.

This Anderson transition is a $T = 0$ quantum phase transition. It is distinct from the Mott transition, driven by electron-electron repulsion ($U/t$ in the Hubbard model) rather than disorder. Real doped semiconductors involve both, and the combined Anderson-Mott problem remains incompletely understood.

Critical behaviour

Near the transition the localization length on the insulating side and the dc conductivity on the metallic side follow power laws:

Wegner’s scaling relation ties the exponents together,

so $s = \nu$ in three dimensions and $s \to 0$ as $d \to 2$, another sign that two is the lower critical dimension. Transfer-matrix and finite-size-scaling numerics for the 3D orthogonal class give $\nu \approx 1.57$ (Slevin and Ohtsuki), and the $2 + \epsilon$ expansion gives $\nu \approx 1/\epsilon$ near two dimensions. Experiments on doped semiconductors have long reported smaller conductivity exponents in some systems (early Si:P data gave $s \approx 1/2$), a discrepancy usually attributed to Coulomb interactions placing real materials in a different universality class.

Transport in the localized phase

An Anderson insulator still conducts at $T > 0$, by phonon-assisted tunneling (“hopping”) between localized states. Mott argued that at low temperature the optimal hop balances distance against energy mismatch, giving variable-range hopping:

Long-range Coulomb interactions open a soft Coulomb gap in the single-particle density of states at $E_F$, $\nu(E) \propto |E - E_F|^{d-1}$, which changes the law to the Efros-Shklovskii form

in every dimension. A crossover from Mott to Efros-Shklovskii behaviour on cooling is commonly observed.

Scaling theory of localization

Single-parameter scaling

The unifying framework is the scaling theory of Abrahams, Anderson, Licciardello, and Ramakrishnan (the “Gang of Four”, 1979). Its central hypothesis is that the dimensionless conductance of a block of size $L$,

is the only parameter that matters: when blocks are combined into larger blocks, the new conductance depends only on the old one. Its flow is captured by a single beta function

If $\beta > 0$, $g$ grows with size and the system is a metal; if $\beta < 0$, $g$ shrinks toward zero and the system is an insulator.

Limits fix the shape of the beta function

• Good metal ($g \gg 1$). Ohm’s law gives $g \propto L^{d-2}$, so $\beta(g) = d - 2 - a/g + \ldots$ The $-a/g$ term ($a > 0$ in the orthogonal class) is exactly the weak-localization correction computed above.
• Strong insulator ($g \ll 1$). Conductance is set by tunneling through the sample, $g \sim e^{-L/\xi}$, so $\beta(g) = \ln g + \text{const} \to -\infty$.

Assuming $\beta$ is continuous and monotonic between these limits gives the standard scaling diagram:

What the diagram says

• $d = 1$. $\beta < 0$ for all $g$: every state is localized for any disorder, confirming the numerics above.
• $d = 2$. The classical term $d - 2$ vanishes and the weak-localization correction makes $\beta$ slightly negative for all $g$. Conductance flows logarithmically slowly to zero, so two dimensions is the lower critical dimension and non-interacting 2D electrons with time-reversal and spin-rotation symmetry are always localized, though $\xi$ can be astronomically large. In the symplectic class $a < 0$, $\beta$ is positive at large $g$, and a genuine 2D metal-insulator transition exists.
• $d = 3$. $\beta$ crosses zero at an unstable fixed point $g_c$. Samples starting above $g_c$ flow to a metal, those below to an insulator; the fixed point is the mobility edge. Linearizing, $\beta(g) \approx (g - g_c)/(\nu g_c)$, so the slope at the crossing gives the localization-length exponent $\nu = 1/[g_c\,\beta’(g_c)]$.

One curve therefore explains why disorder always wins in low dimensions, why three dimensions has a true metal-insulator transition, and how weak localization and strong localization are two ends of the same flow. The field-theoretic formulation is the nonlinear sigma model of Wegner and Efetov, in which $\beta(g)$ is the RG beta function of the coupling $1/g$.

Where single-parameter scaling is modified

• Quantum Hall systems. In a strong field (unitary class) almost all states in a disordered Landau level are localized, which produces the plateaus of the quantum Hall effect, but one critical energy per level carries extended states. The plateau transition is a two-parameter flow in $(\sigma_{xx}, \sigma_{xy})$; numerical studies of the Chalker-Coddington network model give $\nu \approx 2.6$, and the critical theory is still not established.
• Interacting 2D systems. Clean, low-density silicon MOSFETs and other 2D carrier systems show an apparent metal-insulator transition (Kravchenko and co-workers, 1994 onward), contrary to the non-interacting prediction. Strong interactions ($r_s \gg 1$) are believed responsible; the nature of the metallic phase remains debated.
• Topological protection. Surface states of 3D topological insulators and edge states of quantum spin Hall insulators are not localized by time-reversal-symmetric disorder. The quantum spin Hall edge, for example, cannot backscatter without breaking time-reversal symmetry.

Experimental realizations

Anderson localization is a wave phenomenon and has been observed well beyond electrons in solids:

Many-body localization

Everything above concerns non-interacting particles, with interactions entering only as a source of dephasing. Many-body localization (MBL) asks whether localization survives in an isolated, interacting system at nonzero energy density. If it does, the system never reaches thermal equilibrium on its own.

Thermalization and its failure

An isolated quantum many-body system is normally its own heat bath: under unitary evolution, small subsystems approach a thermal state. The eigenstate thermalization hypothesis (ETH) explains this by positing that individual highly excited eigenstates already look thermal to local observables. MBL is the proposed failure of ETH. Basko, Aleiner, and Altshuler (2006) argued perturbatively that weakly interacting electrons with localized single-particle states remain insulating up to a finite temperature, and numerical studies of spin chains such as the random-field Heisenberg model

found an apparent transition from ergodic to localized behaviour near $W/J \approx 3.5$-$4$.

Phenomenology of the MBL regime

Deep in the localized regime the Hamiltonian can be written in terms of an extensive set of quasi-local conserved quantities $\tau_i^z$, the l-bits, which are dressed versions of the local degrees of freedom:

with couplings decaying exponentially with distance. Because each $\tau_i^z$ is conserved, nothing is transported, yet the exponentially small couplings $J_{ij}$ dephase distant l-bits over times $t \sim e^{|i-j|/\xi}$. That is why entanglement grows as $S(t) \sim \ln t$ even though energy and particles do not move, the sharpest difference between MBL and Anderson localization.

Experiments

Signatures of MBL have been reported in several quantum simulators: persistent density imbalance of fermions in a 1D quasi-periodic optical lattice (Munich, 2015) and of bosons in 2D (2016), and localization with slow entanglement growth in trapped-ion chains and superconducting-qubit processors. Related disorder-free forms exist: Stark MBL in a linear potential, and quasi-periodic localization (Aubry-André), which has a sharp transition even in 1D. Many-body localization was also the first proposed mechanism for stabilizing discrete time crystals, observed in trapped ions and NV-centre ensembles in 2017 and on a superconducting processor in 2021-22.

The stability debate

Whether MBL is a true phase of matter in the thermodynamic limit, rather than a very long-lived transient, has been the central question since about 2020.

• Finite-size drifts. Šuntajs, Bonča, Prosen, and Vidmar (2020) pointed out that the apparent critical disorder in exact-diagonalization studies drifts upward with system size, so the thermodynamic-limit transition could lie at much stronger disorder, or not exist.
• Avalanches. De Roeck and Huveneers argued that rare, locally ergodic regions can act as baths that thermalize their surroundings, with the thermal region growing like an avalanche. This destabilizes MBL in $d > 1$ and with sufficiently slowly decaying interactions, and in 1D pushes the critical disorder well above the early estimates.
• Prethermal MBL. The emerging picture distinguishes a prethermal MBL regime, in which dynamics are frozen on all experimentally and numerically accessible time scales, from asymptotic MBL at infinite times. The existence of the latter in 1D at very strong disorder is still unresolved, and in two or more dimensions it is widely thought not to exist.

For experiments and quantum devices, which always operate at finite times, the prethermal regime already delivers the practical payoff: slow relaxation that protects local quantum information and non-equilibrium order. Sierant, Lewenstein, Scardicchio, Vidmar, and Zakrzewski, “Many-body localization in the age of classical computing” (Rep. Prog. Phys., 2025), review the current state of the problem.

References

1. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492 (1958).
2. E. Abrahams, P. W. Anderson, D. C. Licciardello, T. V. Ramakrishnan, “Scaling theory of localization,” Phys. Rev. Lett. 42, 673 (1979).
3. P. A. Lee and T. V. Ramakrishnan, “Disordered electronic systems,” Rev. Mod. Phys. 57, 287 (1985).
4. F. Evers and A. D. Mirlin, “Anderson transitions,” Rev. Mod. Phys. 80, 1355 (2008).
5. E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (2007).
6. B. I. Shklovskii and A. L. Efros, Electronic Properties of Doped Semiconductors (1984).
7. D. A. Abanin, E. Altman, I. Bloch, M. Serbyn, “Colloquium: Many-body localization, thermalization, and entanglement,” Rev. Mod. Phys. 91, 021001 (2019).

See Also

• Condensed Matter Physics (Hub) — crystal structure, band theory, and the Bloch states that disorder localizes.
• Metals & Magnetism — Drude and Fermi-liquid theory of the clean metal.
• Superconductivity, Quantum Hall & Topological Phases — quantum Hall plateaus rely on localized states between extended ones.
• Graduate-Level Formalism — Green’s functions, the tenfold way, and scaling theory of quantum phase transitions.
• Experimental Techniques — magnetotransport and the measurement of $L_\phi$.
• Quantum Field Theory — field-theoretic methods, including the nonlinear sigma model.
• Computational Physics — exact diagonalization and transfer-matrix methods used to map localization transitions.