On the generalized dimensions of multifractal eigenstates

Abstract

Recently, based on heuristic arguments, it was conjectured that an intimate relation exists between any multifractal dimensions, Dq and D q ′ , of the eigenstates of critical random matrix ensembles:Here, we verify this relation by extensive numerical calculations on critical random matrix ensembles and extend its applicability to q < 1/2 but also to deterministic models producing multifractal eigenstates and to generic multifractal structures. We also demonstrate, for the scattering version of the power-law banded random matrix model at criticality, that the scaling exponents σq of the inverse moments of Wigner delay times, τ

Effect of Hilbert space truncation on Anderson localization

“…It would be interesting to further explore the connection between the PRBM ensemble and the truncated Anderson models to shed more light on the role of randomness and symmetry-breaking in driving a metal-insulator transition in 1 dimension, in a similar spirit to previous works [98,99].…”

On the multifractal dimensions and statistical properties of critical ensembles characterized by the three classical Wigner-Dyson symmetry classes

“…These relations have been proved to correctly describe the multifractal behavior of critical states for the PBRM model in the presence (β = 1) [21] and absence (β = 2) [22] of time reversal invariance, in a relatively wide range of the model parameters. Furthermore, it also accounts for the multifractal properties of other models showing critical behavior [21,22]. However, the PBRM model corresponding to the third of the three classical Wigner-Dyson ensembles, i.e, the symplectic case, has been left out.…”

“…More recently [21,22], this model has been used to verify the validity of existing heuristic relations, established between the multifractal properties of eigenstates and their spectra at criticality. These relations have been proved to correctly describe the multifractal behavior of critical states for the PBRM model in the presence (β = 1) [21] and absence (β = 2) [22] of time reversal invariance, in a relatively wide range of the model parameters. Furthermore, it also accounts for the multifractal properties of other models showing critical behavior [21,22].…”

The Multifractal Property of Heterogeneous Microstructure in Cement Paste

“…It is usually found that the fractal dimension as a single variable is not sufficient to capture the underlying dynamics of real-world systems. Therefore, the multifractal theory has been developed which adopts a continuous spectrum, i.e., the generalized dimension or singularity spectrum depending on the scale or region of measure [43][44][45].…”