Anderson model

Anderson model. The Anderson model is a fundamental theoretical construct in condensed matter physics that describes the behavior of non-interacting electrons in a disordered potential. Introduced in a seminal 1958 paper by Philip W. Anderson, it provides the cornerstone for understanding the phenomenon of Anderson localization, where disorder can cause the electronic wavefunctions to become spatially localized, leading to an insulating state. The model's profound implications have shaped the study of disordered systems and metal–insulator transitions across multiple fields of physics.

Introduction and definition

The model was formulated by Philip W. Anderson to explain how the presence of disorder can inhibit the quantum-mechanical diffusion of electrons. It is defined on a lattice model (physics) where each site has a randomly chosen on-site energy, representing the disorder introduced by impurities or defects in a material. The central prediction of the model is that beyond a critical disorder strength, all electronic states become localized, a phase transition now known as Anderson transition. This work was a key contribution for which Anderson was awarded the Nobel Prize in Physics in 1977, shared with Nevill Mott and John Hasbrouck Van Vleck.

Physical interpretation and significance

Physically, the model captures the competition between the kinetic energy of an electron, which favors delocalization via quantum tunneling between lattice sites, and the disorder potential, which tends to trap the electron. The localization transition is not driven by interactions but by the purely quantum effect of wave interference in a random medium, analogous to phenomena in wave propagation. Its significance extends far beyond electronic structure, influencing the study of classical wave localization in photonic crystals and acoustic waves, and forming a basis for understanding the mobility edge in the energy spectrum of disordered systems.

Mathematical formulation

The Hamiltonian for the single-particle Anderson model on a Bravais lattice is typically written in second quantization as: \( H = \sum_i \epsilon_i c_i^\dagger c_i - t \sum_{\langle i,j \rangle} (c_i^\dagger c_j + \text{h.c.}) \). Here, \( \epsilon_i \) are independent random variables, often drawn from a uniform distribution of width \( W \), representing the disorder. The parameter \( t \) is the nearest-neighbor hopping integral on the lattice, such as a cubic lattice. The model's properties are studied through the Green's function (many-body theory) and the analysis of the Lyapunov exponent, which characterizes the exponential decay of wavefunctions. Key mathematical techniques applied include the transfer-matrix method and scaling theory of localization.

Extensions and related models

Numerous important extensions have been developed to incorporate additional physical effects. The Anderson–Hubbard model combines the disorder of the Anderson model with electron-electron interactions described by the Hubbard model, crucial for studying Mott insulators and disordered strongly correlated materials. The Anderson model of localization has been generalized to different symmetry classes, leading to the study of Anderson localization in periodic potentials and the tenfold Altland–Zirnbauer classification. Other related frameworks include models for many-body localization, which explore localization in the presence of interactions, and the random matrix theory approach to spectral statistics in disordered systems.

Experimental realizations

The predictions of the Anderson model have been verified in diverse experimental platforms. In solid-state physics, early evidence came from studies of doped semiconductors like silicon and germanium, and measurements of the temperature dependence of conductivity in amorphous solids. More direct observations use ultracold atoms in optical lattices created with lasers, where disorder can be engineered using speckle patterns or incommensurate lattices, allowing direct imaging of localized states. In photonics, localization has been observed in disordered dielectric materials and waveguide arrays. Furthermore, experiments with microwaves in chaotic cavities and ultrasound in elastic media have demonstrated the universality of Anderson localization for classical waves.

Category:Condensed matter physics Category:Quantum models Category:Disordered systems