Condensed Matter Physics » Disorder & Localization

Disorder & Localization

Why Disorder Changes Everything

Bloch’s theorem is the foundation of band theory: in a perfect crystal, the eigenstates are extended plane-wave-like states $\psi_{n\mathbf{k}}$ that spread coherently over the entire sample. A perfect crystal with a partially filled band is therefore a perfect conductor — at zero temperature an electron in an extended state propagates forever, and the only thing that gives a real metal finite resistance is scattering off lattice vibrations (phonons) and impurities.

The conventional picture treats that impurity scattering perturbatively. An electron travels ballistically for a mean free path $\ell = v_F \tau$ (with $\tau$ the elastic scattering time and $v_F$ the Fermi velocity), then scatters into a new direction; many such events randomize its momentum and the motion becomes diffusive with diffusion constant $D = v_F^2 \tau / d$ in $d$ dimensions. The Drude/Boltzmann conductivity follows,

where $n$ is the carrier density, $m$ the effective mass, and $\nu(E_F)$ the density of states at the Fermi level. In this semiclassical picture disorder merely limits the conductivity to a finite value; it never destroys metallic behavior outright.

In 1958 P. W. Anderson showed this picture is incomplete. If the disorder is strong enough — or, as we will see, in low enough dimension for any disorder — quantum interference between the many scattered partial waves can cause the electron wavefunction to become exponentially localized:

bound to a region of size $\xi$ (the localization length) around some point $\mathbf{r}_0$. A localized electron cannot carry current at $T = 0$: $\sigma \to 0$ even though the density of states at $E_F$ is finite and nonzero. This is Anderson localization — a genuine insulator made entirely by disorder, with no band gap, no broken symmetry, and no interactions required. It is one of the cleanest examples of a quantum phase transition driven purely by quantum interference.

This page develops the physics in four steps: the Anderson model and the localization transition; the weak-localization quantum correction that survives even in good metals; the mobility edge that separates extended from localized states and drives the metal-insulator transition; the scaling theory that ties it all together; and finally a note on the modern, interacting cousin, many-body localization.

Anderson Localization

The Anderson Tight-Binding Model

The canonical model is a tight-binding lattice with random on-site energies. On a $d$-dimensional lattice with sites $i$,

where $t$ is the hopping amplitude between nearest neighbours and the on-site energies $\varepsilon_i$ are independent random variables, usually drawn from a uniform box distribution of width $W$:

The single dimensionless control parameter is the disorder strength $W/t$. When $W = 0$ the model is a clean tight-binding band of width $2zt$ (with $z$ the coordination number), all states extended. The question is what happens as $W/t$ grows.

The Localization Transition

The physical content is a competition between two effects:

• Hopping $t$ delocalizes: it wants to spread the electron over the lattice into extended Bloch-like states.
• Disorder $W$ localizes: large random energy mismatches between neighbouring sites suppress resonant hopping. An electron on a site of energy $\varepsilon_i$ can only hop efficiently to a neighbour whose energy lies within $\sim t$ of $\varepsilon_i$; when the typical mismatch $\sim W$ greatly exceeds $t$, such resonant neighbours are rare and the electron is trapped.

Anderson’s insight, sharpened by the locator expansion and later by self-consistent and scaling treatments, is that above a critical disorder $W_c$ all eigenstates become localized:

For the 3D cubic-lattice Anderson model with a box distribution the critical value is $(W/t)_c \approx 16.5$. Below threshold, states in the band centre remain extended while states in the band tails are localized — the two are separated by a mobility edge (next section but one).

Inverse Participation Ratio: A Diagnostic

A convenient quantitative measure of how spread out an eigenstate $\psi_n(i)$ is across $N$ sites is the inverse participation ratio (IPR),

• Extended state: $M \sim N$, so $P_2 \sim 1/N \to 0$ as the system grows.
• Localized state: $M \sim \xi^d$, independent of $N$, so $P_2 \to \text{const} > 0$ in the thermodynamic limit.

The IPR is the workhorse diagnostic in numerical studies of localization. The following code builds a 1D Anderson chain, diagonalizes it, and shows that every eigenstate is localized — the hallmark of one dimension.

import numpy as np

def anderson_1d(N, W, t=1.0, seed=0):
    """Build the 1D Anderson Hamiltonian: random on-site energies + hopping."""
    rng = np.random.default_rng(seed)
    eps = rng.uniform(-W / 2, W / 2, size=N)   # disordered on-site energies
    H = np.diag(eps)
    off = -t * np.ones(N - 1)
    H += np.diag(off, 1) + np.diag(off, -1)    # nearest-neighbour hopping
    return H

def inverse_participation_ratio(psi):
    """IPR of a normalized eigenvector; ~1/N if extended, O(1) if localized."""
    p = np.abs(psi) ** 2
    return np.sum(p ** 2)

# In 1D, ANY nonzero disorder localizes every state.
for N in (500, 1000, 2000):
    H = anderson_1d(N, W=1.0)
    evals, evecs = np.linalg.eigh(H)
    mid = N // 2                       # a band-centre state
    ipr = inverse_participation_ratio(evecs[:, mid])
    # IPR stays ~constant as N grows -> localized (a 1/N trend would mean extended)
    print(f"N={N:5d}  IPR(mid)={ipr:.4e}  ->  effective extent ~ {1/ipr:.1f} sites")

Running this shows the IPR (and hence the localization length read off as $1/\text{IPR}$) saturating to an $N$-independent value: the state occupies a fixed number of sites no matter how long the chain. Contrast this with a clean chain ($W = 0$), where the IPR would fall off as $1/N$.

Weak Localization & Quantum Corrections to Conductivity

Anderson localization in its strong form needs large disorder. But quantum interference leaves a fingerprint even in good metals with $k_F \ell \gg 1$, in the form of a small, temperature- and field-dependent correction to the Drude conductivity called weak localization. It is the precursor of full localization and one of the most experimentally accessible interference effects in solids.

The Coherent Backscattering Picture

Consider the quantum amplitude for an electron to return to its starting point after diffusing through the disordered medium. The return probability is a sum over all closed diffusive paths. For any such loop there is a time-reversed partner that traverses exactly the same impurities in the opposite order. In the absence of magnetic fields or magnetic impurities, the two amplitudes $A_+$ and $A_- = A_+$ are identical (time-reversal symmetry), so when we square the total amplitude the cross term survives:

Magnitude and Dimensionality

Summing over diffusive loops of all durations from the elastic time $\tau$ up to the phase-coherence time $\tau_\phi$ (beyond which inelastic scattering randomizes the phase and kills the interference) gives the weak-localization correction. In two dimensions it is logarithmic:

where $L_\phi = \sqrt{D \tau_\phi}$ is the phase-coherence length. The general dimensional dependence of the correction (per unit volume in 3D, per square in 2D, per length in 1D) is

The crucial observation is that in $d \le 2$ the correction diverges as $\tau_\phi \to \infty$ (i.e. as $T \to 0$, since $\tau_\phi$ grows on cooling). The metallic state in one and two dimensions is therefore unstable: lowering the temperature drives $\sigma$ down without bound, and the system flows toward an insulator. This is the perturbative signature of the scaling result that there is no true metal in 1D or 2D for noninteracting electrons with time-reversal symmetry.

Magnetoresistance: The Experimental Smoking Gun

Weak localization is most cleanly identified through its response to a magnetic field. A perpendicular field $B$ threads the time-reversed loops with opposite Aharonov-Bohm phases, breaking the equality $A_+ = A_-$ once the flux through a typical coherent loop reaches a flux quantum. The field thus destroys the interference correction and restores the Drude conductivity:

A characteristic negative magnetoresistance at low fields and low temperatures, with a magnitude of order $e^2/h$ per square and a width set by $B_\phi \sim \hbar/(e L_\phi^2)$, is the experimental signature. Measuring its field and temperature dependence is the standard way to extract the phase-coherence length $L_\phi(T)$ in mesoscopic metals and 2D electron gases.

Spin-Orbit Coupling and Antilocalization

The sign of the effect depends on the symmetry class. Strong spin-orbit scattering rotates the electron spin around the loop in a way that flips the relative sign of the time-reversed amplitudes, so the interference becomes destructive and enhances the conductivity. This is weak antilocalization, which produces a positive low-field magnetoresistance instead — a hallmark of materials with strong spin-orbit coupling (heavy-element metals, topological-insulator surface states). The three canonical behaviours map onto the Wigner-Dyson symmetry classes: orthogonal (weak localization), symplectic (weak antilocalization), and unitary (no correction, time-reversal already broken by $B$).

The Mobility Edge & Metal-Insulator Transition

Mott’s Mobility Edge

In three dimensions, weak and moderate disorder does not localize everything. Instead, as N. F. Mott emphasized, localized and extended states coexist in energy but never at the same energy — they are separated by a sharp boundary called the mobility edge $E_c$:

• States with energy $

  E

  > E_c$ (in the band tails, where the density of states is low) are localized.

• States with energy $

  E

  < E_c$ (near the band centre, high density of states) are extended.

The mobility edge cannot lie at an energy where extended and localized states overlap, because an extended and a localized state at the same energy would hybridize and the localized one would delocalize. The transition at $E_c$ is therefore sharp.

Driving the Metal-Insulator Transition

Whether the material is a metal or an insulator at $T = 0$ depends on where the Fermi level sits relative to the mobility edge:

There are two experimental knobs:

1. Tune the disorder $W$. Increasing disorder pushes the mobility edge inward toward the band centre. At the critical disorder the two mobility edges (one in each tail) merge at the band centre, $E_c \to 0$, and all states localize — the Anderson transition described earlier.
2. Tune the carrier density / Fermi level. In doped semiconductors (the textbook example is phosphorus-doped silicon, Si:P) one sweeps $E_F$ through a fixed $E_c$ by changing the dopant concentration. Below a critical density $n_c$ the system is a disorder-driven insulator; above it, a metal.

This Anderson metal-insulator transition is a genuine $T = 0$ quantum critical point. It is distinct from the Mott metal-insulator transition, which is driven by electron-electron interaction ($U/t$ in the Hubbard model) rather than disorder; real materials such as Si:P near the transition involve both effects, and disentangling them (the “Anderson-Mott” problem) remains an active research question.

Critical Behaviour

For the 3D orthogonal class numerics give a critical exponent $\nu \approx 1.57$. Wegner’s scaling relation ties the conductivity exponent to the localization-length exponent,

so in three dimensions $s = \nu$. The vanishing of $s$ at $d = 2$ is yet another statement that two dimensions is the lower critical dimension for the Anderson transition.

Scaling Theory of Localization

The Single-Parameter Scaling Hypothesis

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

controls everything: when blocks are combined into larger blocks, the only quantity needed to predict the new conductance is the old conductance. The way $g$ changes with system size is captured by a single beta function,

which depends on $g$ alone (not separately on microscopic details). The sign of $\beta$ decides the fate of the system as $L \to \infty$:

• $\beta(g) > 0$: $g$ grows with size $\to$ flows to large $g$ $\to$ metal (extended states).
• $\beta(g) < 0$: $g$ shrinks with size $\to$ flows to $g \to 0$ $\to$ insulator (localized states).

Limits Fix the Shape of $\beta$

Two limits pin down the beta function:

• Good metal (large $g$): Ohm’s law holds, $g = \sigma L^{d-2}$, so \(\beta(g) = d - 2 + O(1/g).\) The $O(1/g)$ correction is exactly the weak-localization term computed above, and it is negative, pushing $\beta$ below its classical value $d-2$.

• Strong insulator (small $g$): conductance is set by exponentially small tunnelling, $g \sim e^{-L/\xi}$, so \(\beta(g) = \ln g + \text{const} \;\xrightarrow{g \to 0}\; -\infty.\)

Interpolating smoothly between these limits gives the celebrated scaling diagram:

   beta(g) = d ln g / d ln L
        ^
        |                              ___________  d = 3  (beta = +1 at large g)
   +1 --|--------------------_________/
        |               ____/        UNSTABLE FIXED POINT g_c
        |          ____/  *............... d = 2  (beta -> 0 from below)
    0 --|------*--/-------------------------------------------> ln g
        |     /:
        |    / :
        |   /  :                      d = 1  (always below: beta < 0)
        |  /   :
        | /    : metal  --->  if beta>0, g grows  ->  EXTENDED
        |/     : insulator -> if beta<0, g shrinks -> LOCALIZED
   -inf *      :
              g_c (mobility edge, d=3 only)

What the Diagram Tells Us

Reading off the three dimensionalities:

• $d = 1$: $\beta < 0$ for all $g$. Every state is localized for any disorder. A one-dimensional disordered wire is always an insulator at $T = 0$, no matter how weak the disorder — confirming the exact-diagonalization result above.

• $d = 2$: $\beta(g) \to 0^-$ from below as $g \to \infty$. The classical term $d - 2 = 0$ vanishes, and the weak-localization correction tips $\beta$ slightly negative for all $g$. So $g$ always flows (logarithmically slowly) to zero: two dimensions is the marginal, lower critical dimension — noninteracting electrons in 2D with time-reversal symmetry are always localized, though the localization length can be astronomically large. (Strong spin-orbit coupling changes the symmetry class and can rescue a 2D metal — the antilocalization story.)

• $d = 3$: $\beta$ crosses zero at an unstable fixed point $g = g_c$. This sign change is the mobility edge. If a sample starts with $g > g_c$ it flows to a metal; if $g < g_c$ it flows to an insulator. The fixed point is a genuine quantum critical point, and linearizing $\beta$ about $g_c$ gives the localization-length exponent $\nu = 1/\beta’(g_c)$, recovering the critical scaling $\xi \sim

  E - E_c

  ^{-\nu}$ of the previous section.

Single-parameter scaling thus explains, from one curve, why disorder always wins in low dimensions, why three dimensions has a true metal-insulator transition, and how the weak-localization correction and the strong-localization insulator are two ends of the same flow.

A Note on Many-Body Localization

Everything above concerns noninteracting electrons (or, in the weak-localization story, interactions treated only as a source of dephasing). A profound modern question is what survives when strong interactions are present at $T > 0$. The answer is many-body localization (MBL) — a phase in which an interacting, isolated quantum system fails to reach thermal equilibrium because of disorder.

Localization Meets Thermalization

An isolated quantum many-body system is normally expected to act as its own heat bath: under unitary evolution local subsystems reach a thermal (Gibbs) state, a statement formalized by the eigenstate thermalization hypothesis (ETH). MBL is the dramatic failure of ETH. In a strongly disordered, interacting system the highly excited many-body eigenstates remain non-thermal: the system retains memory of its initial conditions forever, transport is absent at finite energy density, and the usual machinery of statistical mechanics simply does not apply.

Whereas Anderson localization concerns single-particle wavefunctions at $T = 0$, MBL is a property of all (or many) eigenstates across the entire many-body spectrum, surviving at nonzero energy density (effectively $T > 0$) even with interactions switched on.

Signatures of the MBL Phase

The MBL phase is characterized by a cluster of related properties:

• Emergent integrability / local integrals of motion (l-bits). Deep in the MBL phase the Hamiltonian can be written in terms of an extensive set of quasi-local conserved quantities $\tau_i^z$ (“l-bits”), \(H = \sum_i h_i\, \tau_i^z + \sum_{ij} J_{ij}\, \tau_i^z \tau_j^z + \sum_{ijk} K_{ijk}\, \tau_i^z \tau_j^z \tau_k^z + \cdots,\) with couplings $J_{ij}$ that decay exponentially with distance. These conserved l-bits are dressed versions of the bare single-particle localized states and are the structural reason equilibration fails.
• Logarithmic entanglement growth. Starting from a product state, the entanglement entropy grows only logarithmically in time, $S(t) \sim \ln t$, in stark contrast to the ballistic (linear-in-$t$) growth of a thermalizing system. Crucially, this slow growth happens even though transport of energy and charge is frozen — interactions dephase the l-bits without moving them.
• Poisson level statistics. Energy levels in the MBL phase are uncorrelated and obey Poisson statistics (no level repulsion), whereas a thermalizing system shows Wigner-Dyson (random-matrix) level repulsion. The crossover in the level-spacing ratio is a standard numerical diagnostic of the MBL transition.
• Area-law excited states. Highly excited MBL eigenstates obey an area law for entanglement entropy, like ground states, rather than the volume law expected of thermal states.

Why It Matters and Open Questions

MBL is significant because it is one of very few known mechanisms that protects quantum coherence at finite temperature in a generic interacting system — it can shelter quantum order and even forbidden phases (such as time crystals and certain symmetry-protected topological orders) in highly excited states where thermal fluctuations would normally destroy them.

The field remains unsettled. The existence and stability of a true MBL phase in the thermodynamic limit, especially in dimensions above one and in the presence of rare thermal “avalanche” inclusions, is actively debated; much of the firm evidence comes from one-dimensional numerics and from cold-atom and trapped-ion experiments that realize isolated, tunable disordered chains. MBL sits at the intersection of localization physics, quantum thermodynamics, and quantum information, and is treated further among the non-equilibrium frontiers in Graduate-Level Formalism & Experiment.

See Also

• Condensed Matter Physics (Hub) — crystal structure, band theory, and the Bloch states that disorder localizes.
• Superconductivity, Quantum Hall & Topological Phases — quantum Hall plateaus rely on disorder-localized states between extended edge channels.
• Graduate-Level Formalism & Experiment — Green’s functions, ARPES/transport probes, and the many-body-localization frontier.
• Quantum Field Theory — field-theoretic (nonlinear sigma model) treatments of the localization classes.
• Computational Physics — exact diagonalization and transfer-matrix methods used to map localization transitions.