A semiclassical theory of the Anderson transition

Abstract

We study analytically the metal-insulator transition in a disordered conductor by combining the self-consistent theory of localization with the one parameter scaling theory. We provide explicit expressions of the critical exponents and the critical disorder as a function of the spatial dimensionality, $d$. The critical exponent $\nu$ controlling the divergence of the localization length at the transition is found to be $\nu =\frac{1}{2}+\frac{1}{d-2}$. This result confirms that the upper critical dimension is infinity. Level statistics are investigated in detail. We show that the two level correlation function decays exponentially and the number variance is linear with a slope which is an increasing function of the spatial dimensionality.

arXiv comment4 pages, journal version

References in corpus (19)

• Statistics of energy levels and eigenfunctions in disordered systems
• Transition from localized to extended eigenstates in the ensemble of power-law random banded matrices
• Anderson Localization of Expanding Bose-Einstein Condensates in Random Potentials
• Fluctuations of the inverse participation ratio at the Anderson transition
• Matterwave localization in disordered cold atom lattices
• New Class of Random Matrix Ensembles with Multifractal Eigenvectors
• Spectral Statistics Beyond Random Matrix Theory
• Universal Spectral Correlations at the Mobility Edge
• Relation between the Correlation Dimensions of Multifractal Wavefunctions and Spectral Measures in Integer Quantum Hall Systems
• Spectral Rigidity and Eigenfunction Correlations at the Anderson Transition
• Reconciling Conductance Fluctuations and the Scaling Theory of Localization
• Anomalous diffusion at the Anderson transitions
• SPECTRAL CORRELATIONS IN DISORDERED ELECTRONIC SYSTEMS: CROSSOVER FROM METAL TO INSULATOR REGIME
• Level spacings at the metal-insulator transition in the Anderson Hamiltonians and multifractal random matrix ensembles
• Dimensional dependence of the metal-insulator transition
• Spectral Compressibility at the Metal-Insulator Transition of the Quantum Hall Effect
• Critical level statistics at the Anderson transition in four-dimensional disordered systems
• Multifractality of Hamiltonians with power-law transfer terms
• Classical intermittency and quantum Anderson transition

Cited by in corpus (33)

• Critical exponent for the Anderson transition in the three dimensional orthogonal universality class
• Critical parameters from generalised multifractal analysis at the Anderson transition
• Anomalous Thouless energy and critical statistics on the metallic side of the many-body localization transition
• Observation of the Anderson Metal-Insulator Transition with Atomic Matter Waves: Theory and Experiment
• Experimental Test of Universality of the Anderson Transition
• Experimental Observation of Two-Dimensional Anderson Localization with the Atomic Kicked Rotor
• Critical properties of the Anderson localization transition and the high dimensional limit
• Self-Consistent Theory of Anderson Localization: General Formalism and Applications
• Drawing Phase Diagrams of Random Quantum Systems by Deep Learning the Wave Functions
• Complexity of 2D random laser modes at the transition from weak scattering to Anderson localization
• Anderson transition in a three dimensional kicked rotor
• Universality of the Anderson transition with the quasiperiodic kicked rotor
• Dimensional Dependence of Critical Exponent of the Anderson Transition in the Orthogonal Universality Class
• Phase diagram of the Anderson transition with atomic matter waves
• Metallic phase of the quantum Hall effect in four-dimensional space
• Transverse confinement of waves in 3D random media
• Multifractal finite-size scaling at the Anderson transition in the unitary symmetry class
• Numerical simulation of the localization of elastic waves in two- and three-dimensional heterogeneous media
• Resolution of the exponent puzzle for the Anderson transition in doped semiconductors
• Strong-disorder renormalization group study of the Anderson localization transition in three and higher dimensions
• Testing the universality of the many-body metal-insulator transition by time evolution of a disordered one-dimensional ultracold fermionic gas
• Boltzmann transport theory for many body localization
• Character of eigenstates of the 3D disordered Anderson Hamiltonian
• Renormalization group for Anderson localization on high-dimensional lattices
• Marginal and Irrelevant Disorder in Einstein-Maxwell backgrounds
• Critical Phenomena of Dynamical Delocalization in Quantum Maps:standard map and Anderson map
• Critical Phenomena of Dynamical Delocalization in Quantum Anderson Map
• Interpretation of high-dimensional numerical results for Anderson transition
• Presence and Absence of Delocalization-localization Transition in Coherently Perturbed Disordered Lattices
• Level statistics for quantum $k$-core percolation
• Multifractality and quantum diffusion from self-consistent theory of localization
• Critical behavior of Anderson transitions in higher dimensional Bogoliubov-de Gennes symmetry classes
• Disorder and Interactions in Graphene and other Quantum Systems