Anderson localization is the phenomenon in which waves stop spreading through a disordered medium, not because they are absorbed but because they interfere with themselves and become trapped. Predicted in 1958 by physicist Philip W. Anderson for electrons in a messy crystal lattice, the idea has since been confirmed for light, sound, and ultracold atoms. What makes it striking is that the trapping has nothing to do with energy loss: the wave simply scatters so many times off random impurities that its many scattered copies cancel each other out everywhere except right where they started. The result is a wave frozen in place inside a material that would otherwise let it pass freely.
How Disorder Traps a Wave
Picture a wave entering a medium sprinkled with randomly placed scatterers. Each time it hits one, it splits into partial waves heading in different directions. Those partial waves hit other scatterers, split again, and so on. In a perfectly ordered material the scattered wavelets add up constructively along certain directions, which is why crystals conduct electricity or transmit light in well-defined ways. In a disordered material, each scattered wavelet picks up a random phase shift depending on the path it travels. When you add up all the wavelets arriving at a distant point, their phases are random enough that, on average, they cancel. The wave’s intensity decays exponentially with distance from its starting point rather than spreading outward.
A particularly clear signature of this interference is coherent backscattering: pairs of scattering paths that trace the same geometric loop in opposite directions accumulate the same total phase and therefore reinforce each other in the backward direction. This constructive interference in the backward direction is a hallmark of localization physics. A subtler companion effect, coherent forward scattering, was predicted more recently and requires strong Anderson localization to appear. Unlike coherent backscattering, coherent forward scattering survives even when certain symmetries of the system are broken, making it a more robust diagnostic of the localized regime.1Nature Communications. Controlling symmetry and localization with an artificial gauge field in a disordered quantum system
Why Dimensionality Changes Everything
One of the most important subtleties of Anderson localization is that it depends heavily on how many spatial dimensions the wave can spread into. In one dimension, any amount of disorder, no matter how small, localizes every state. A wave on a disordered wire has nowhere to go except forward and backward, and the interference that produces localization is overwhelming. In two dimensions the situation is similar in most cases: all states are technically localized, though the localization length can be astronomically large for weak disorder, so in practice some waves look like they are still spreading.
Three dimensions are where things get genuinely interesting. Here, weak disorder is not enough to localize all states. Instead, there is a critical boundary in the energy spectrum called the mobility edge. States on one side of it are extended and can carry current; states on the other side are localized and cannot. The mobility edge marks a true phase transition. Near it, localized states decay exponentially with a characteristic length that grows as the energy approaches the critical value.2PubMed Central. The size of localized States near the mobility edge This transition is sometimes called the Anderson transition, and pinning down its precise properties in three dimensions has been a long-running challenge in theoretical physics. The transition is continuous, with the localization length diverging at the critical energy, and the behavior near it depends on subtle details of the system’s symmetry.
Symmetry Classes and Their Consequences
The symmetry of a disordered system profoundly affects whether localization occurs and how the transition behaves. The three classical symmetry classes are the orthogonal (time-reversal symmetry preserved, no spin-orbit coupling), the unitary (time-reversal symmetry broken, for instance by a magnetic field), and the symplectic (time-reversal symmetry preserved, but with strong spin-orbit coupling). In two dimensions, the orthogonal class localizes all states, which is the standard prediction. The symplectic class, however, can support a band of extended, metallic-like states even in two dimensions, which was a surprising theoretical prediction that has been backed up by numerical work.
One striking demonstration of how delicate these symmetry effects are comes from simulations of two-dimensional electron systems with strong spin-orbit coupling. At zero magnetic field, the energy-level statistics follow the pattern expected for extended states across a wide energy range, suggesting a metal rather than an insulator. Turning on even an incredibly tiny magnetic field is enough to switch the symmetry class from symplectic to unitary and change the statistical behavior of the energy levels, while the states in the relevant energy range remain extended.3Scientific Reports. Absence of localization in disordered two-dimensional electron gas at weak magnetic field and strong spin-orbit coupling Below the mobility edge, the energy levels follow the spacing pattern expected for localized states regardless of symmetry. The take-home point is that whether a two-dimensional disordered system conducts or insulates is not just about how much disorder there is; it also depends on whether the electrons have spin-orbit coupling and whether any magnetic field is present.
Seeing Anderson Localization in the Lab
For decades after Anderson’s original prediction, direct experimental proof was elusive. Electrons in real solids interact with each other and with lattice vibrations, both of which can mask or destroy localization. The cleanest observations have come from systems where those complications can be removed or minimized.
Photonic systems provided some of the first convincing demonstrations. In 2007, researchers observed transverse Anderson localization of light in a two-dimensional photonic lattice with controlled random fluctuations. Light injected into the lattice remained confined transversely instead of spreading, exactly as the theory predicts.4Nature. Transport and Anderson localization in disordered two-dimensional photonic lattices Because photons do not interact with each other under normal conditions, these experiments sidestep the complications that plague electronic systems.
Ultracold atoms have been another major platform. By cooling fermionic atoms to near absolute zero and releasing them into a disordered optical potential created by laser speckle, experimenters observed three-dimensional Anderson localization directly. The expanding atomic cloud split into two components: a mobile fraction that kept spreading and a localized fraction that stayed put, exactly what the theory predicts when some energies lie above and some below the mobility edge.5PubMed. Three-dimensional Anderson localization of ultracold matter Follow-up experiments varied the disorder’s spatial correlation length and mapped out how the localization properties changed, confirming theoretical expectations about the role of disorder scale.6PubMed. Three-dimensional Anderson localization in variable scale disorder
More recently, three-dimensional Anderson localization has been reported for ultrasound. By sending ultrasonic pulses through a random medium and measuring the transmitted intensity over time and space, researchers observed the hallmark signature: the transverse width of the transmitted beam saturated instead of continuing to grow, indicating confinement. Analysis of the frequency dependence of the localization length revealed two mobility edges, boundaries in frequency space separating localized from extended acoustic modes.7Nature. Frozen sound That result was particularly significant because three-dimensional localization of classical waves had been debated for years, with earlier claims for light contested on the grounds that absorption might mimic localization. Ultrasound, being a scalar wave in a medium where absorption can be carefully controlled, offered a cleaner test.
Disorder-Based Optical Fibers
One of the more unexpected practical outgrowths of Anderson localization is in optical fiber design. Conventional optical fibers guide light through a carefully engineered core-cladding structure. Fibers exploiting transverse Anderson localization work on a completely different principle: the fiber’s cross-section is intentionally disordered, with random variations in refractive index, while the fiber is uniform along its length. Strong transverse scattering from the disorder confines light transversely, while allowing it to propagate freely along the fiber’s axis.8PubMed Central. Transverse Anderson Localization in Disordered Glass Optical Fibers: A Review
The key advantage is that any point across the fiber’s face can serve as an independent transport channel. This makes Anderson-localizing fibers natural candidates for image transport. Researchers have demonstrated high-quality optical image transmission through these disordered fibers, with the image resolution determined by the localization radius of the transverse modes. Stronger disorder and larger refractive-index fluctuations produce tighter localization and therefore better resolution. The self-averaging behavior of the localized modes also ensures that image quality is roughly uniform across the entire fiber cross-section, rather than varying from spot to spot.9Nature Communications. Image transport through a disordered optical fibre mediated by transverse Anderson localization The approach is still far from replacing conventional imaging fibers in medical endoscopes or industrial inspection tools, but it is a compelling proof of concept that disorder can be a design feature rather than a defect.
Many-Body Localization
Anderson’s original theory was about waves or non-interacting particles. A natural question is what happens when the particles interact. In a metal, electrons repel each other through the Coulomb force. Does that interaction help waves escape localization, or can localization survive?
The answer, at least under certain conditions, is that a form of localization can persist even with interactions. This phenomenon is called many-body localization, or MBL. In a many-body localized system, the particles are not just localized in real space; they are localized in the abstract space of all possible configurations of the system. The practical consequence is dramatic: a many-body localized system fails to reach thermal equilibrium. If you prepare it in some non-equilibrium initial state, it retains a memory of that state indefinitely instead of relaxing to a featureless thermal state the way normal matter does.
This was observed experimentally with interacting fermions in a one-dimensional quasirandom optical lattice. Researchers prepared the atoms in a charge density wave pattern (atoms alternating between occupied and empty sites) and watched how the pattern evolved over time. For weak disorder, the pattern washed out quickly as the system thermalized. Above a critical disorder strength, a substantial fraction of the initial pattern persisted, signaling the breakdown of thermalization and the onset of many-body localization.10PubMed. Observation of many-body localization of interacting fermions in a quasirandom optical lattice
On the theoretical side, detailed studies of spin chains with bond disorder have revealed some of the fingerprints of the MBL phase: the distribution of entanglement entropy in the energy eigenstates becomes multimodal rather than sharply peaked, and the energy-level spacing statistics shift to a pattern characteristic of localized states.11PubMed. Phenomenology of Many-Body Localization in Bond-Disordered Spin Chains Whether true MBL survives in the thermodynamic limit, meaning for an infinitely large system rather than a finite one, remains one of the most actively debated questions in condensed matter physics. Some researchers have argued that very slow thermalization processes might eventually destroy MBL at extremely long timescales, but the question is far from settled.
Interactions can also affect single-particle Anderson localization directly. In disordered wires, weak electron-electron interactions introduce dephasing, which at intermediate temperatures replaces the sharp exponential localization with a power-law decay in conductivity. At even lower temperatures, localization in the abstract configuration space of the interacting system kicks in and drives the conductivity to zero below a critical temperature.12PubMed. Interacting electrons in disordered wires: Anderson localization and low-T transport
The Topological Anderson Insulator
Perhaps the most counterintuitive twist in the Anderson localization story is the discovery that disorder can create topological order rather than simply destroying metallic behavior. In a topological insulator, the bulk of the material is insulating, but the edges or surfaces carry protected conducting states that are immune to backscattering. These edge states arise from the material’s band structure and are usually thought of as a property of clean, well-ordered crystals.
The topological Anderson insulator flips this expectation on its head. Computer simulations of HgTe quantum wells predicted that introducing random impurities into an ordinary (topologically trivial) band insulator could invert the band gap and drive the system into a topological phase with quantized conductance at the edges.13PubMed. Topological Anderson insulator Disorder, normally the enemy of clean electronic transport, is in this case fundamentally responsible for creating the extended edge states. An effective medium theory subsequently explained the mechanism: in a material described by a Dirac-like equation with certain corrections, a random potential can shift the effective mass term and flip the sign of the band gap, turning a trivial insulator into a topological one.14PubMed. Theory of the Topological Anderson Insulator
The phenomenon turns out not to be limited to on-site disorder (randomness in the potential energy at each lattice site). Disorder in the hopping between lattice sites can also drive the transition. Studies of HgTe/CdTe quantum wells found that randomness in the connections between identical atomic orbitals on neighboring sites is enough to induce the topological phase at an appropriate disorder strength.15Journal of Applied Physics. Topological Anderson insulator induced by inter-cell hopping disorder The topological Anderson insulator is still primarily a theoretical and computational prediction, but it has reshaped how physicists think about the relationship between disorder and order. It shows that randomness in a material is not always degrading; under the right conditions, it can build structure that pure crystals lack.
When Nonlinearity Breaks the Localization
Anderson localization is a consequence of linear wave physics: the wave equation is linear, so scattered wavelets superpose without affecting each other. In many real systems, though, nonlinear effects are present. The intensity of a light wave can change the refractive index of the medium it travels through. Atoms in a Bose-Einstein condensate interact, introducing a nonlinear term in their wave equation. What does nonlinearity do to a localized wave?
The short answer is that it slowly destroys localization. An initially localized wave packet in a disordered nonlinear medium does not stay put forever. Instead, it spreads subdiffusively, meaning much more slowly than ordinary diffusion but without any ultimate bound. The second moment of the packet (a measure of its width) grows as a power law in time, with an exponent around one-third, which is dramatically slower than the linear-in-time growth of normal diffusion.16PubMed. Universal spreading of wave packets in disordered nonlinear systems This spreading arises because the nonlinearity shifts the frequencies of the localized modes, occasionally bringing neighboring modes into resonance and allowing energy to leak from one to the next. The process is driven by weak chaos inside the packet, which slowly heats up the surrounding cold lattice.
There appears to be a critical nonlinearity strength above which this spreading sets in. Below it, the packet remains localized for all practical purposes. Above it, subdiffusion takes over and the packet expands indefinitely, though very slowly.17PubMed. Destruction of Anderson localization by a weak nonlinearity Self-consistent theoretical treatments support this picture, predicting that nonlinearity replaces the exponential localization with algebraic (power-law) tails on the wave packet while leaving ordinary diffusion unaffected at higher nonlinearity strengths.18PubMed Central. A self-consistent theory of localization in nonlinear random media For anyone hoping to use Anderson localization in a practical device, this is a key constraint: if the waves involved can interact with each other or with the medium in a nonlinear way, the localization will eventually leak.
Quasiperiodic Systems and the Aubry-André Model
Anderson localization is typically discussed in the context of random disorder, but closely related physics appears in quasiperiodic systems, where the potential is not random but also not strictly periodic. The textbook example is the Aubry-André model, a one-dimensional lattice with a potential whose spatial period is incommensurate with the lattice spacing. Unlike truly random one-dimensional systems, where any disorder localizes all states, the Aubry-André model has a sharp transition at a critical disorder strength. Below the critical value, all states are extended; above it, all states are localized.
Numerical studies of many-body versions of this model with interacting particles have revealed additional structure. At many particle densities the localization transition occurs at or near the single-particle critical disorder strength. However, at certain special densities the transition occurs at weaker disorder, producing narrow “spikes” in the phase diagram. These special densities approach irrational numbers expressible as Fibonacci ratios as the system size increases, reflecting the deep connection between quasiperiodicity and the golden ratio that underlies the Aubry-André model’s mathematics.19PubMed Central. A numerical study of the localization transition of Aubry-André type models Quasiperiodic potentials have become a popular experimental tool for studying localization in ultracold atom setups precisely because they offer a clean, tunable way to drive the localization transition without genuine randomness.
Anderson Localization Beyond Condensed Matter
Although Anderson localization began as a theory about electrons in dirty metals, its underlying mechanism is pure wave interference and therefore applies to any wave in any disordered medium. This universality has pushed the concept well beyond its original home. In seismology, the scattering of seismic waves by heterogeneous rock structures has been analyzed through the lens of localization theory. In acoustics, as the recent ultrasound experiments demonstrate, three-dimensional localization of sound waves has moved from theoretical possibility to laboratory fact. In wireless communications, the multiple scattering of radio waves in dense urban environments shares some of the same interference physics, though practical localization is rare because absorption and finite system size usually intervene first.
The breadth of settings where Anderson localization shows up, or nearly shows up, is a testament to how fundamental destructive interference is as a physical process. Any time a wave encounters enough random scattering, the possibility of localization is on the table. Whether it actually occurs depends on the dimensionality, the symmetry of the scattering, the strength of the disorder, and whether nonlinear or many-body effects are present. Those qualifiers matter enormously in practice, but the core idea remains one of the cleanest examples in physics of how randomness can produce order: a wave, confronted with chaos, does the most orderly thing imaginable and stops moving entirely.