Self-similar spectrum in effective time independent Hamiltonians for kicked systems

TL;DR

This paper investigates the multifractal and self-similar spectral properties of effective time-independent Hamiltonians derived from delta-kicked quantum systems, revealing generic fractal spectra and their statistical characteristics.

Contribution

It introduces a perturbative method to derive effective Hamiltonians for kicked systems and demonstrates their self-similar spectra using specific models like the double kicked SU(2) and kicked Harper.

Findings

Effective Hamiltonians exhibit self-similar, fractal spectra.

The spectra's fractal dimensions are quantitatively characterized.

Generic SU(2) Hamiltonians show multifractal spectral properties.

Abstract

We study multifractal properties in the spectrum of effective time-independent Hamiltonians obtained using a perturbative method for a class of delta-kicked systems. The evolution operator in the time-dependent problem is factorized into an initial kick, an evolution dictated by a time-independent Hamiltonian, and a final kick. We have used the double kicked $SU(2)$ system and the kicked Harper model to study butterfly spectrum in the corresponding effective Hamiltonians. We have obtained a generic class of $SU(2)$ Hamiltonians showing self-similar spectrum. The statistics of the generalized fractal dimension is studied for a quantitative characterization of the spectra.