Critical properties in long-range hopping Hamiltonians

Abstract

Some properties of $d$-dimensional disordered models with long-range random hopping amplitudes are investigated numerically at criticality. We concentrate on the correlation dimension $d_2$ (for $d=2$) and the nearest level spacing distribution $P_c(s)$ (for $d=3$) in both the weak ($b^d\gg 1$) and the strong ($b^d\ll 1$) coupling regime, where the parameter $b^{-d}$ plays the role of the coupling constant of the model. It is found that (i) the extrapolated values of $d_2$ are of the form $d_2=c_d{b}^d$ in the strong coupling limit and $d_2=d-a_d/b^d$ in the case of weak coupling, and (ii) $P_{(}s)$ has the asymptotic form $P_c(s)\sim \exp (-A_d{s}^{\alpha })$ for $s\gg$, with the critical exponent $\alpha =2-a_d/b^d$ for $b^d\gg 1$ and $\alpha =1+c_d{b}^d$ for $b^d\ll 1$. In these cases the numerical coefficients $A_d$, $a_d$ and $c_d$ depend only on the dimensionality.

9 pages, 6 .eps figures, contribution to the Festschrift for Michael Schreiber's 50th birthday

References in corpus (39)

• Scaling Theory of the Integer Quantum Hall Effect
• Statistics of energy levels and eigenfunctions in disordered systems
• Transition from localized to extended eigenstates in the ensemble of power-law random banded matrices
• Fluctuations of the inverse participation ratio at the Anderson transition
• Multifractality and critical fluctuations at the Anderson transition
• Mutifractal Analysis of Broadly Distributed Observables at Criticality
• New Class of Random Matrix Ensembles with Multifractal Eigenvectors
• Universal Spectral Correlations at the Mobility Edge
• Relation between the Correlation Dimensions of Multifractal Wavefunctions and Spectral Measures in Integer Quantum Hall Systems
• A generalized ensemble of random matrices
• Asymptotics of Universal Probability of Neighboring Level Spacings at the Anderson Transition
• Multifractality: generic property of eigenstates of 2D disordered metals.
• Impurity States and the Absence of Quasiparticle Localization in Disordered D-Wave Superconductors
• Weak Chaos in a Quantum Kepler Problem
• Fluctuations of the correlation dimension at metal-insulator transitions
• Level spacings at the metal-insulator transition in the Anderson Hamiltonians and multifractal random matrix ensembles
• Critical Hamiltonians with long range hopping
• Critical statistics in a power-law random banded matrix ensemble
• Fluctuation of the Correlation Dimension and the Inverse Participation Number at the Anderson Transition
• Anderson transition in three-dimensional disordered systems with symplectic symmetry
• Strong eigenfunction correlations near the Anderson localization transition
• Shape Analysis of the Level Spacing Distribution around the Metal Insulator Transition in the Three Dimensional Anderson Model
• Spectral Correlations from the Metal to the Mobility Edge
• An exactly solvable many-body problem in one dimension
• Energy level dynamics in systems with weakly multifractal eigenstates: equivalence to 1D correlated fermions
• A class of N-body problems with nearest- and next-to-nearest neighbour interactions
• A new model of quantum chaotic billiards: Spectral Statistics and Wavefunctions in 2D
• $f(\alpha )$ Multifractal spectrum at strong and weak disorder
• Critical spectral statistics in two-dimensional interacting disordered systems
• Coherent propagation of interacting particles in a random potential: the Mechanism of enhancement
• Anomalously large critical regions in power-law random matrix ensembles
• Virial expansion for almost diagonal random matrices
• Multifractality of Hamiltonians with power-law transfer terms
• Metal-insulator transition in two-dimensional disordered systems with power-law transfer terms
• Density of states for almost diagonal random matrices
• Quantum mechanics of chaotic billiards
• Cantori and dynamical localization in the Bunimovich Stadium
• Critical generalized inverse participation ratio distributions
• Critical behaviour at the metal-insulator transition in 3-dimensional disordered systems in a strong magnetic field

Cited by in corpus (7)

• Statistics of the two-point transmission at Anderson localization transitions
• A critical Dyson hierarchical model for the Anderson localization transition
• Two-level correlation function of critical random-matrix ensembles
• Anderson transitions : multifractal or non-multifractal statistics of the transmission as a function of the scattering geometry
• Level number variance and spectral compressibility in a critical two-dimensional random matrix model
• Criticality without self-similarity: a 2D system with random long-range hopping
• Statistical properties of the critical eigenstates in power-law random banded matrices across the band