Anderson Localization

Anderson localization is the suppression of wave transport by coherent multiple scattering from static disorder. It can occur without a band gap, a classical barrier, or particle–particle interactions. The disorder randomizes the phases accumulated along different paths, yet the resulting interference is not featureless: repeated scattering can organize amplitudes into spatially localized eigenstates and prevent an initially compact wave packet from diffusing indefinitely.

A short mean free path alone does not establish localization. With weak disorder, elastic scattering first produces an approximately diffusive regime; coherent interference can suppress that diffusion on longer scales. Strong disorder can localize states without a well-separated diffusive interval.

The discussion concerns noninteracting particles in static disorder. The benchmark is the random-site lattice model introduced by Anderson (1958). Inelastic hopping and interacting localization require additional physics.

Required background. Disorder in Quantum Matter supplies disorder ensembles and their spatial correlations. Resolvent Operator supplies the energy-domain Green operator and its spectral boundary prescription. Transfer Matrix Method supplies one-dimensional amplitude matching and matrix composition.

Localized states and suppressed transport

Localization is best established by several compatible statements rather than by one visual impression.

Eigenstate localization

For a normalized one-particle eigenstate centered near $\mathbf r_\alpha$, an exponentially localized envelope obeys

$$\lvert\psi_\alpha(\mathbf r)\rvert \lesssim C_\alpha \exp\!\left( -\frac{\lvert\mathbf r-\mathbf r_\alpha\rvert}{\xi_\alpha} \right)$$

outside a microscopic core. The localization length $\xi_\alpha$ is energy-, disorder-, and symmetry-dependent. Oscillations and rare resonant peaks can sit beneath the envelope, so a straight-line fit to a short semilog tail is not by itself decisive.

Dynamical localization

Launch a state $\lvert\mathbf r_0\rangle$ and define its second spatial moment,

$$m_2(t) = \sum_{\mathbf r} \lvert\mathbf r-\mathbf r_0\rvert^2 \left| \langle\mathbf r| e^{-i\hat Ht/\hbar} |\mathbf r_0\rangle \right|^2.$$

Ordinary diffusion gives

$$\overline{m_2(t)} \simeq 2dDt$$

after microscopic transients. In a dynamically localized regime, the packet may spread initially, but its long-time spatial moments remain bounded under appropriate disorder and energy averaging:

$$\limsup_{t\to\infty} \overline{m_2(t)} < \infty.$$

If only part of the spectrum is localized, first project and normalize the initial packet into that energy range; otherwise its extended component can keep spreading. Exponential eigenstates and bounded dynamics are related statements, but the latter needs control of the whole spectral superposition, not a fit to one eigenstate. Localization does not mean absence of all microscopic motion.

Transport localization

Attach ideal leads and form the dimensionless two-terminal conductance

$$g = \frac{G}{e^2/h} = \operatorname{Tr} \left( t^\dagger t \right),$$

where $t$ is the transmission matrix. Count every spin and valley channel in this trace; multiply by a degeneracy only if those copies were omitted from the matrix. In a long localized sample, the typical conductance

$$g_{\mathrm{typ}} \equiv \exp\!\left( \overline{\ln g} \right)$$

falls exponentially with length. One common quasi-one-dimensional convention is

$$\overline{\ln g} \sim -\frac{2L}{\xi_{\mathrm{tr}}}.$$

Some authors absorb the factor of $2$ into the definition of the transport localization length. A reported $\xi$ is incomplete unless its estimator is stated.

Random on-site energies compete with hopping $t$. The lower panel illustrates the envelope $\lvert\psi(x)\rvert^2\propto e^{-2\lvert x-x_0\rvert/\xi}$: at one amplitude localization length from the center, the probability is reduced by $e^{-2}$. Actual eigenstates fluctuate within such an envelope; this curve is not a numerically computed eigenstate.

Interference and the Absence of Diffusion

A propagator through a disordered region can be organized schematically as a sum over multiple-scattering paths $\mathcal P$,

$$K(\mathbf r_f,\mathbf r_i;t) = \sum_{\mathcal P} A_{\mathcal P} e^{i\phi_{\mathcal P}}.$$

Its probability contains diagonal and interference terms:

$$\begin{aligned} \lvert K\rvert^2&=P_{\mathrm{diag}}+P_{\mathrm{int}},\\ P_{\mathrm{diag}}&=\sum_{\mathcal P}\lvert A_{\mathcal P}\rvert^2,\\ P_{\mathrm{int}}&=\sum_{\mathcal P\ne\mathcal P'} B_{\mathcal P}B_{\mathcal P'}^\ast. \end{aligned}$$

Here $B_{\mathcal P}=A_{\mathcal P}e^{i\phi_{\mathcal P}}$ includes the path phase. A classical random walk retains only the first sum. A disorder average suppresses many cross terms, but not all of them. Time-reversed loops, repeated returns, and families of paths constrained by symmetry retain correlated phases. Their cumulative effect changes transport at scales larger than the elastic mean free path.

When the weak-disorder scales are well separated, increasing the observation length can reveal

$$\text{ballistic} \;\longrightarrow\; \text{diffusive} \;\longrightarrow\; \text{localized},$$

with the last crossover possible only while phase information survives. This sequence is a useful weak-disorder picture, not a required intermediate stage for every localized system. If the phase-coherence length $L_\phi$ is shorter than the localization length, dephasing truncates the interference buildup and the sample can display only a precursor. Magnetic flux, spin–orbit coupling, and internal symmetries also change which path pairs interfere, so dimensionality alone never specifies the localization class.

Weak localization is the perturbative correction produced by a restricted set of coherent return paths while $g\gg1$. Anderson localization is the nonperturbative regime in which transmission itself becomes exponentially small. The two are connected, but fitting a small magnetoconductance cusp does not establish strong localization.

The Anderson Model

The standard tight-binding benchmark is

$$\hat H = \sum_i \epsilon_i \lvert i\rangle\langle i\rvert - \sum_{\langle i,j\rangle} \left( t_{ij}\lvert i\rangle\langle j\rvert + t_{ij}^\ast\lvert j\rangle\langle i\rvert \right).$$

The simplest diagonal-disorder Anderson model uses a regular lattice, nearest-neighbor hopping $t_{ij}=t$, and independent site energies drawn from

$$P(\epsilon) = \begin{cases} 1/W, & -W/2\le\epsilon\le W/2,\\ 0, & \text{otherwise}. \end{cases}$$

Then

$$\overline{\epsilon_i}=0, \qquad \overline{\epsilon_i\epsilon_j} = \frac{W^2}{12}\delta_{ij}.$$

Only dimensionless combinations such as $W/t$, $E/t$, and $L/a$ can be compared across calculations. Replacing the box distribution by a Gaussian, correlating nearby $\epsilon_i$, randomizing the hoppings, adding magnetic phases, or adding spin–orbit matrices defines a different model. Universal critical behavior may survive some changes, but mobility edges and microscopic localization lengths generally do not.

For a clean hypercubic lattice,

$$E(\mathbf k) = -2t \sum_{\mu=1}^{d} \cos(k_\mu a).$$

The eigenstates are extended Bloch waves. In the opposite atomic limit $t=0$, each nondegenerate site orbital is an exact localized eigenstate. Anderson localization describes the interference-controlled connection between these limits; it is not simply the statement that the atomic limit is localized.

Localized Wavefunctions

A Green-function definition

The retarded resolvent

$$G^R_{ij}(E) = \left\langle i\left| \frac{1}{E+i0^+-\hat H} \right|j\right\rangle$$

connects spatial decay to spectral information. For energies in the disordered spectrum, a typical decay length can be extracted from

$$\frac{1}{\xi(E)} = - \lim_{r_{ij}\to\infty} \frac{1}{r_{ij}} \overline{ \ln\lvert tG^R_{ij}(E)\rvert }.$$

Here $t>0$ supplies an energy unit so the logarithm is dimensionless; any fixed reference energy gives the same asymptotic length. The $i0^+$ prescription is essential: finite artificial broadening also attenuates propagation. Check the decay as that broadening tends to zero rather than identifying a fixed-broadening attenuation length with $\xi$. A clean resolvent decays inside a spectral gap too, so exponential decay alone is not a disorder diagnosis.

The logarithm is essential. Localized systems have broad, often strongly skewed distributions; the arithmetic mean can be dominated by rare resonant samples. The Resolvent Operator develops the underlying operator identities.

Participation ratios

On a lattice, define generalized inverse participation ratios

$$P_q^{(\alpha)} = \sum_i \lvert\psi_\alpha(i)\rvert^{2q}, \qquad q>1.$$

For $q=2$,

$$P_2 = \sum_i \lvert\psi_i\rvert^4, \qquad N_{\mathrm{part}} \equiv \frac{1}{P_2}.$$

For the following scaling laws, measure $L$ and $\xi$ in lattice spacings. An extended state occupying $L^d$ sites has

$$P_q \propto L^{-d(q-1)}.$$

A localized state with $L\gg\xi$ instead approaches an $L$-independent value set by its localization volume:

$$P_q \propto \xi^{-d(q-1)}.$$

At an Anderson transition, critical wavefunctions are multifractal,

$$P_q \propto L^{-\tau_q}, \qquad \tau_q\ne d(q-1),$$

and a distributional finite-size analysis is required. A large $P_2$ in one finite sample can also come from a surface state, defect bound state, flat band, or ordinary confinement.

Local density of states

The local density of states is

$$\rho_i(E) = -\frac{1}{\pi} \operatorname{Im} G^R_{ii}(E).$$

Two averages answer different questions (use a fixed density-of-states unit when taking a logarithm):

$$\rho_{\mathrm{avg}}(E) = \overline{\rho_i(E)}.$$ $$\rho_{\mathrm{typ}}(E) = \exp\!\left[ \overline{\ln\rho_i(E)} \right].$$

A localized band may retain a nonzero arithmetic density of states while the typical local spectral weight becomes extremely small when the infinite-volume limit is taken before the spectral broadening tends to zero. Reversing those limits makes even a finite clean spectrum consist of isolated peaks. Thus a density of available eigenvalues is not a density of current-carrying states.

Transfer Matrices in One Dimension

For a chain with uniform hopping $t>0$,

$$-t(\psi_{n+1}+\psi_{n-1}) + \epsilon_n\psi_n = E\psi_n.$$

Rearranging gives

$$\begin{pmatrix} \psi_{n+1}\\ \psi_n \end{pmatrix} = M_n(E) \begin{pmatrix} \psi_n\\ \psi_{n-1} \end{pmatrix}.$$

The site transfer matrix is

$$M_n(E) = \begin{pmatrix} (\epsilon_n-E)/t & -1\\ 1 & 0 \end{pmatrix}.$$

Because

$$\det M_n=1,$$

the two Lyapunov exponents of a long matrix product occur with opposite signs. Define

$$\mathcal M_N(E) = M_N(E)\cdots M_2(E)M_1(E)$$

and

$$\gamma(E) = \lim_{N\to\infty} \frac{1}{Na} \overline{ \ln\lVert\mathcal M_N(E)\rVert }.$$

The amplitude localization length is $\xi(E)=1/\gamma(E)$.

For independent weak diagonal disorder with variance $\sigma^2$, the leading result away from the clean band edges and special commensurate energies is

$$\frac{\xi(E)}{a} \simeq \frac{8t^2\sin^2(ka)}{\sigma^2}.$$

Here $E=-2t\cos(ka)$ is the clean dispersion. For box disorder, $\sigma^2=W^2/12$, so

$$\frac{\xi(E)}{a} \simeq 96 \frac{t^2}{W^2} \sin^2(ka).$$

This formula exposes two robust lessons: arbitrarily weak generic disorder gives a finite one-dimensional localization length, and weaker disorder can make that length far larger than any available sample. It is not valid at the clean band edges, and the band center has the Kappus–Wegner anomaly, where naive perturbation theory misses a finite correction. The Transfer-Matrix Method gives the general scattering construction.

Dimension and Symmetry

The conventional noninteracting picture can be summarized as follows:

Orthogonal symmetry means time reversal with spin-rotation symmetry (or a spinless model with time reversal squaring to $+1$). Breaking time reversal gives the unitary class. Spinful time reversal with spin–orbit coupling and no spin-rotation symmetry gives the symplectic class. The classification and its exceptions are reviewed by Evers and Mirlin (2008).

Dimension and class	Large-system behavior
1D, generic independent disorder	Arbitrarily weak disorder localizes; $\xi$ may exceed the sample
2D, orthogonal	Flow toward localization; weak disorder can give enormous $\xi$
2D, symplectic	A metallic regime and an Anderson transition are possible
2D, quantum Hall	Localized states separated by critical extended energies
3D, orthogonal	Extended and localized regions separated by mobility edges

These statements concern noninteracting models with short-range hopping and generic disorder. Chiral or other extra symmetries, protected channels, correlated disorder and quasiperiodic potentials need separate analysis. In particular, the ordinary one-dimensional random-disorder conclusion is not the phase diagram of a quasiperiodic chain.

For the simple-cubic box-disorder model at $E=t$, the finite-size study of Slevin and Ohtsuki (2014) gives a benchmark of approximately

$$\frac{W_c}{t} \simeq 16.5.$$

This is not a universal constant of disordered matter. It belongs to a specific Hamiltonian, disorder normalization, energy, and symmetry class. Mobility Edges develops the phase diagram and critical scaling.

Mobility Edges

A mobility edge $E_c$ separates localized and extended states in energy even when the density of states is nonzero on both sides. In three dimensions, changing $E$ at fixed disorder can cross from a localized spectral tail into an extended core and possibly back into localized states; changing $W/t$ at fixed $E$ can drive the same transition.

Finite temperature complicates the electronic interpretation. Inelastic processes impose $L_\phi(T)$, occupations sample an energy window of order $k_{\mathrm B}T$, interactions modify both states and transport, and phonon-assisted hopping can produce nonzero conductivity among localized orbitals. The single-particle mobility edge is therefore an input to a finite-temperature transport model, not a complete prediction of the measured resistance.

The Integer Quantum Hall Effect provides an important variant: localized states broaden Landau levels and support plateaus, while critical extended states mediate plateau transitions. Those mobility edges carry topological structure absent from the plain zero-field orthogonal model.

Diagnostic Toolkit

Level statistics

After unfolding within one irreducible symmetry sector, spatially separated localized states have weak mutual overlap and approximately Poisson statistics. Extended states in a coherent diffusive sample approach the Wigner–Dyson class selected by antiunitary symmetry, while Anderson-critical spectra are scale invariant and intermediate.

Random Matrix Theory in Quantum Matter explains unfolding and symmetry-resolved comparisons. Poisson behavior also occurs in integrable systems or mixed symmetry blocks; pair level statistics with spatial or transport scaling.

Boundary-condition sensitivity

Thread a twist $\theta$ through a periodic sample,

$$\psi(\mathbf r+L\hat{\mathbf e}_x) = e^{i\theta}\psi(\mathbf r),$$

and measure a typical level shift $\delta E$ relative to the mean spacing $\Delta$. The Thouless number

$$g_{\mathrm{Th}} \sim \frac{\delta E}{\Delta}$$

measures sensitivity to going around the periodic sample. It becomes exponentially small for deeply localized states. Extended diffusive states have a response related to conductance, but an extended state need not give $g_{\mathrm{Th}}\gg1$ in every geometry. This diagnostic links spectral sensitivity to transport without assuming a relaxation time.

The conductance distribution supplies complementary information: rare resonant samples can make $\overline g$ much larger than $g_{\mathrm{typ}}$. Keep the distribution or quantiles as well as the mean when comparing system sizes.

Experimental Signatures

No single experimental curve is universally sufficient. The most persuasive evidence combines controlled disorder, a size or time axis, an interference-sensitive observable, and tests against classical trapping, absorption, dephasing, and interactions.

Probe	What must be distinguished
Exponential conductance versus length	Localization, depletion, tunneling barriers and contacts
Confined spatial modes	Random interference, defect binding and surface states
Arrested matter-wave expansion	Localization, classical trapping and a short observation time
Small wave transmission	Coherent localization, absorption and leakage

For example, Billy et al. (2008) studied the expansion of matter waves in controlled one-dimensional disorder. The initial energy distribution, disorder correlations and residual interactions determine how a measured cloud profile can be compared with localized single-particle states. A finite field of view or a stationary trapped fraction cannot establish localization of every populated energy.

In electronic solids, the dimensionless disorder marker

$$k_{\mathrm F}\ell_{\mathrm{tr}} \sim 1$$

is an Ioffe–Regel warning scale for a simple transport picture, not a definition of Anderson localization. For smooth disorder, $\ell_{\mathrm{tr}}$ and the quantum lifetime can differ greatly; a large transport mean free path alone does not establish a narrow quasiparticle linewidth. A rising residual resistivity or a negative temperature coefficient of resistance can have several nonlocalization causes.

Comparing finite samples

Use several lengths at a fixed energy and disorder distribution. A state with $L<\xi$ can look extended even when the infinite system localizes. In a long bar, vary its transverse width too: its quasi-one-dimensional localization length is not by itself the bulk localization length.

For transfer products, use periodic QR reorthogonalization to avoid losing the contracting direction. For Green functions, vary the spectral broadening. For eigenstates, resolve symmetry sectors and energy windows before pooling statistics. The finite-size scaling guide explains how to test corrections and uncertainties.

Two distinctions are especially useful. A density of states measures available energies, not their ability to carry current; localized states can fill a mobility gap. And the single-particle mechanism discussed here does not establish many-body localization, which concerns interacting dynamics and needs additional tests.

Exercises

1. Clean-chain transfer eigenvalues

Set $\epsilon_n=0$ in the one-dimensional transfer matrix. Show that its eigenvalues lie on the unit circle inside the clean band and recover the dispersion.

Solution

The clean matrix is

$$M_0 = \begin{pmatrix} -E/t & -1\\ 1 & 0 \end{pmatrix}.$$

Its characteristic equation is

$$\lambda^2 + \frac{E}{t}\lambda + 1 = 0.$$

Because $\det M_0=1$, write $\lambda_\pm=e^{\pm ika}$ when $\lvert E/t\rvert\le2$. Then

$$\lambda_++\lambda_- = 2\cos(ka) = -\frac{E}{t},$$

so

$$E=-2t\cos(ka).$$

The eigenvalues have unit modulus, and repeated transfer does not generate exponential growth. Outside the clean band, $k$ becomes complex and evanescence is ordinary band-edge decay, not disorder-induced Anderson localization.

2. Normalize an exponential state

On an infinite chain, let

$$p_n = Aq^{\lvert n\rvert}, \qquad q=e^{-2a/\xi}.$$

Normalize $p_n$, compute $P_2=\sum_n p_n^2$, and find its large-$\xi/a$ limit.

Solution

Normalization gives

$$1 = A \left( 1+2\sum_{n=1}^{\infty}q^n \right) = A\frac{1+q}{1-q},$$

so

$$A = \frac{1-q}{1+q}.$$

Then

$$\begin{aligned} P_2 &= A^2 \left( 1+2\sum_{n=1}^{\infty}q^{2n} \right)\\ &= \frac{(1-q)(1+q^2)}{(1+q)^3}. \end{aligned}$$

For $\xi\gg a$,

$$q \simeq 1-\frac{2a}{\xi},$$

and therefore

$$P_2 \simeq \frac{a}{2\xi}.$$

Thus the participation number is of order $2\xi/a$, as expected for a state spread over one localization length on either side of its center.

3. Reciprocal Lyapunov exponents

Show that both the eigenvalues and the singular values of a finite transfer product $\mathcal M_N$ occur in reciprocal pairs. Use the singular values to explain the pairing of its Lyapunov exponents.

Solution

Each factor has unit determinant, so

$$\det\mathcal M_N = \prod_{n=1}^{N}\det M_n = 1.$$

If $\Lambda_+$ and $\Lambda_-$ are the two eigenvalues,

$$\Lambda_+\Lambda_-=1.$$

The positive singular values $s_+\geq s_-$ instead measure stretching and contraction. Since their product is the absolute determinant,

$$s_+s_- = \lvert\det\mathcal M_N\rvert = 1.$$

For a stationary ergodic transfer sequence satisfying the usual logarithmic integrability condition, their long-product limits exist almost surely:

$$\begin{gathered} \gamma_\pm = \lim_{N\to\infty} \frac{\ln s_\pm(\mathcal M_N)}{Na} = \pm\gamma, \\ \gamma\geq0. \end{gathered}$$

This agrees with the matrix-norm definition above. Reciprocal eigenvalues alone do not measure finite-product stretching: the matrix $\left(\begin{smallmatrix}1&u\\0&1\end{smallmatrix}\right)$ has both eigenvalues equal to one but unequal singular values when $u\ne0$.

In the localized regime with $\gamma>0$, the positive exponent sets the inverse localization length, $\xi^{-1}=\gamma$. If $\gamma=0$, both exponents vanish and this diagnostic gives no finite exponential localization length; it does not by itself establish ballistic transport. Numerically, direct multiplication overflows and loses the contracting direction, so stable transfer calculations periodically use QR or related reorthogonalization.

4. Weak-disorder localization length

For box disorder with $W/t=0.5$ at $E=-t$, estimate $\xi/a$ using the leading weak-disorder formula. Would a chain of $L=100a$ reliably show asymptotic localization?

Solution

At $E=-t$,

$$\cos(ka) = \frac12, \qquad \sin^2(ka) = \frac34.$$

Therefore

$$\frac{\xi}{a} \simeq 96 \frac{t^2}{W^2} \frac34 = 96(4)\frac34 = 288.$$

The available ratio is

$$\frac{L}{\xi} \simeq 0.35.$$

The chain is shorter than one localization length and can look extended. A reliable claim needs longer systems or finite-size scaling, not merely an IPR snapshot at $L=100a$.

5. Diagnose an arrested cloud

An atomic cloud stops expanding and has an exponential tail. A colleague concludes that every populated state is Anderson localized. Explain two independent reasons why that conclusion is premature, and propose a measurement for each.

Solution

First, the cloud contains an energy distribution. A trapped or localized low-energy fraction may coexist with extended particles that have left the imaging region. Resolve or vary the initial energies and measure the expanding fraction as well as the stationary tail.

Second, a finite observation interval cannot distinguish saturation from very slow expansion. Extend the time range and field of view, while checking loss and imaging sensitivity. Varying the disorder and comparing with a coherent calculation helps distinguish interference from classical trapping; residual interactions must also be small enough for a single-particle comparison.

Continue with localization physics

• Weak Localization derives the small coherent-return correction on the diffusive side.
• Scaling Theory of Localization follows conductance as the observation length changes.
• Anderson Insulators explains finite-temperature hopping between localized orbitals.
• Mobility Edges studies the energy boundary between localized and extended states.

References

• P. W. Anderson, “Absence of Diffusion in Certain Random Lattices,” Physical Review 109, 1492–1505 (1958), doi:10.1103/PhysRev.109.1492. The original random-site model and localization argument.
• J. Billy, V. Josse, Z. Zuo, et al., “Direct Observation of Anderson Localization of Matter Waves in a Controlled Disorder,” Nature 453, 891–894 (2008), doi:10.1038/nature07000. Observes arrested expansion and exponential matter-wave localization.
• F. Evers and A. D. Mirlin, “Anderson Transitions,” Reviews of Modern Physics 80, 1355–1417 (2008), doi:10.1103/RevModPhys.80.1355. Modern review of universality classes, multifractality, and critical statistics.
• M. Kappus and F. Wegner, “Anomaly in the Band Centre of the One-Dimensional Anderson Model,” Zeitschrift für Physik B 45, 15–21 (1981), doi:10.1007/BF01294272. Identifies the weak-disorder band-center anomaly.
• B. Kramer and A. MacKinnon, “Localization: Theory and Experiment,” Reports on Progress in Physics 56, 1469–1564 (1993), doi:10.1088/0034-4885/56/12/001. Broad account of transfer methods, scaling, experiments, and dimensionality.
• A. MacKinnon and B. Kramer, “One-Parameter Scaling of Localization Length and Conductance in Disordered Systems,” Physical Review Letters 47, 1546–1549 (1981), doi:10.1103/PhysRevLett.47.1546. Establishes the recursive transfer-matrix finite-size program.
• K. Slevin and T. Ohtsuki, “Critical Exponent for the Anderson Transition in the Three-Dimensional Orthogonal Universality Class,” New Journal of Physics 16, 015012 (2014), doi:10.1088/1367-2630/16/1/015012. High-precision finite-size scaling across several disorder distributions.
• D. J. Thouless, “Electrons in Disordered Systems and the Theory of Localization,” Physics Reports 13, 93–142 (1974), doi:10.1016/0370-1573(74)90029-5. Classic review of spectral, transport, and localization distinctions.