Statistics of the two-point transmission at Anderson localization transitions

Abstract

At Anderson critical points, the statistics of the two-point transmission $T_L$ for disordered samples of linear size $L$ is expected to be multifractal with the following properties [Janssen {\it et al} PRB 59, 15836 (1999)] : (i) the probability to have $T_L\sim 1/L^{\kappa }$ behaves as $L^{\Phi (\kappa )}$, where the multifractal spectrum $\Phi (\kappa )$ terminates at $\kappa =0$ as a consequence of the physical bound $T_L\le 1$; (ii) the exponents $X(q)$ that govern the moments $\overline{T_L^q}\sim 1/L^{X(q)}$ become frozen above some threshold: $X(q\ge q_{sat})=-\Phi (\kappa =0)$, i.e. all moments of order $q\ge q_{sat}$ are governed by the measure of the rare samples having a finite transmission ($\kappa =0$). In the present paper, we test numerically these predictions for the ensemble of $L\times L$ power-law random banded matrices, where the random hopping $H_{i,j}$ decays as a power-law $(b/|i-j|{)}^a$. This model is known to present an Anderson transition at $a=1$ between localized ($a>1$) and extended ($a<1$) states, with critical properties that depend continuously on the parameter $b$. Our numerical results for the multifractal spectra ${\Phi }_b(\kappa )$ for various $b$ are in agreement with the relation $\Phi (\kappa \ge 0)=2[f(\alpha =d+\frac{\kappa }{2})-d]$ in terms of the singularity spectrum $f(\alpha )$ of individual critical eigenfunctions, in particular the typical exponents are related via the relation ${\kappa }_{typ}(b)=2({\alpha }_{typ}(b)-d)$. We also discuss the statistics of the two-point transmission in the delocalized phase and in the localized phase.

arXiv commentv2=final version with two new appendices with respect to v1; 12 pages, 10 figures

References in corpus (26)

• Anderson Transitions
• Transition from localized to extended eigenstates in the ensemble of power-law random banded matrices
• Statistics and Scaling in Disordered Mesoscopic Electron Systems
• Fluctuations of the inverse participation ratio at the Anderson transition
• Multifractality and critical fluctuations at the Anderson transition
• Exact relations between multifractal exponents at the Anderson transition
• Multifractal analysis with the probability density function at the three-dimensional Anderson transition
• Multifractal analysis of the metal-insulator transition in the 3D Anderson model I: Symmetry relation under typical averaging
• Point-Contact Conductances at the Quantum Hall Transition
• Fluctuations of the correlation dimension at metal-insulator transitions
• Weak Chaos in a Quantum Kepler Problem
• Multifractal analysis of the metal-insulator transition in the 3D Anderson model II: Symmetry relation under ensemble averaging
• Critical statistics in a power-law random banded matrix ensemble
• Critical Hamiltonians with long range hopping
• Fluctuation of the Correlation Dimension and the Inverse Participation Number at the Anderson Transition
• $f(\alpha )$ Multifractal spectrum at strong and weak disorder
• Probing the eigenfunction fractality with a stop watch
• Anomalously large critical regions in power-law random matrix ensembles
• Boundary multifractality in critical 1D systems with long-range hopping
• Scattering at the Anderson transition: Power--law banded random matrix model
• Level compressibility in a critical random matrix ensemble: The second virial coefficient
• Metal-insulator transition in two-dimensional disordered systems with power-law transfer terms
• Power spectrum for critical statistics: A novel spectral characterization of the Anderson transition
• Critical wave-packet dynamics in the power-law bond disordered Anderson Model
• Two-level correlation function of critical random-matrix ensembles
• Critical properties in long-range hopping Hamiltonians

Cited by in corpus (14)

• Many-body localization transition in a lattice model of interacting fermions: statistics of renormalized hoppings in configuration space
• Exotic Multifractal Conductance Fluctuations in Graphene
• Statistics of renormalized on-site energies and renormalized hoppings for Anderson localization models in dimensions d=2 and d=3
• Statistical properties of the Green function in finite size for Anderson Localization models with multifractal eigenvectors
• Anderson localization transition with long-ranged hoppings : analysis of the strong multifractality regime in terms of weighted Levy sums
• A critical Dyson hierarchical model for the Anderson localization transition
• Random elastic networks : strong disorder renormalization approach
• The Fock-space landscape of many-body localisation
• Extreme Value Statistics Distributions in Spin Glasses
• Scattering and transport statistics at criticality
• Anderson transitions : multifractal or non-multifractal statistics of the transmission as a function of the scattering geometry
• Criticality without self-similarity: a 2D system with random long-range hopping
• Directional Localization in Disordered 2D Tight-Binding Systems: Insights from Single Particle Entanglement Measures
• Statistical properties of two-particle transmission at Anderson transition