Kicked-Harper model vs On-Resonance Double Kicked Rotor Model: From Spectral Difference to Topological Equivalence

TL;DR

This paper compares the spectral and topological properties of the kicked Harper model and on-resonance double kicked rotor model, revealing differences in their spectral bands and dynamical behaviors, but also establishing their topological equivalence under certain extensions.

Contribution

It provides a detailed spectral analysis of the ORDKR, proves the existence of a flat band, and demonstrates topological equivalence between extended versions of KHM and ORDKR.

Findings

ORDKR has a flat band with width decaying as K^{N+2}

KHM lacks flat bands and has linearly scaling band width

Extended models of KHM and ORDKR are topologically equivalent

Abstract

Recent studies have established that, in addition to the well-known kicked Harper model (KHM), an on-resonance double kicked rotor model (ORDKR) also has Hofstadter's butterfly Floquet spectrum, with strong resemblance to the standard Hofstadter's spectrum that is a paradigm in studies of the integer quantum Hall effect. Earlier it was shown that the quasi-energy spectra of these two dynamical models (i) can exactly overlap with each other if an effective Planck constant takes irrational multiples of 2*pi and (ii) will be different if the same parameter takes rational multiples of 2*pi. This work makes some detailed comparisons between these two models, with an effective Planck constant given by 2*pi M/N, where M and N are coprime and odd integers. It is found that the ORDKR spectrum (with two periodic kicking sequences having the same kick strength) has one flat band and $N-1$ non-flat bands whose largest width decays in power law as K to the power of N+2, where K is a kicking strength parameter. The existence of a flat band is strictly proven and the power law scaling, numerically checked for a number of cases, is also analytically proven for a three-band case. By contrast, the KHM does not have any flat band and its band width scales linearly with K. This is shown to result in dramatic differences in dynamical behavior, such as transient (but extremely long) dynamical localization in ORDKR, which is absent in KHM. Finally, we show that despite these differences, there exist simple extensions of KHM and ORDKR (upon introducing an additional periodic phase parameter) such that the resulting extended KHM and ORDKR are actually topologically equivalent, i.e., they yield exactly the same Floquet-band Chern numbers and display topological phase transitions at the same kick strengths. A theoretical derivation of this topological equivalence is provided.