Kicked-Harper model versus on-resonance double-kicked rotor model: From spectral difference to topological equivalence

Abstract

Recent studies have established that, in addition to the well-known kicked-Harper model (KHM), an on-resonance double-kicked rotor (ORDKR) model also has Hofstadter's butterfly Floquet spectrum, with strong resemblance to the standard Hofstadter spectrum that is a paradigm in studies of the integer quantum Hall effect. Earlier it was shown that the quasienergy spectra of these two dynamical models (i) can exactly overlap with each other if an effective Planck constant takes irrational multiples of 2π and (ii) will be different if the same parameter takes rational multiples of 2π. This work makes detailed comparisons between these two models, with an effective Planck constant given by 2πM/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 nonflat bands with the largest bandwidth decaying in a power law as ~K(N+2), where K is a kick 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 bandwidths scale 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 the KHM. Finally, we show that despite these differences, there exist simple extensions of the KHM and ORDKR model (upon introducing an additional periodic phase parameter) such that the resulting extended KHM and ORDKR model 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. These results are also of interest to our current understanding of quantum-classical correspondence considering that the KHM and ORDKR model have exactly the same classical limit after a simple canonical transformation.

Topological effects in chiral symmetric driven systems

“…Within each single k y subspace, this is indeed the familiar form of the 1D KHM Floquet operator, with k y playing the role of a phase shift parameter as introduced in our early studies 47,63,66 . By choosing b = 2πM/N , where M, N ∈ Z, we again obtain an N -band quasienergy spectrum just like we did for the DKL Floquet operator.…”

Floquet Second-Order Topological Phases in Momentum Space

“…Furthermore, under the quantum resonance condition that has been considered experimentally [ 113 , 114 , 115 , 116 , 117 , 118 , 119 ], we obtain the two-dimensional extension of on-resonance double kicked rotor (or lattice) model, whose Floquet operator reads It is then clear that, once , with p and q being coprime integers, the Floquet operator will have translational symmetries in both and with the common period q , i.e., a periodic crystal structure in the momentum space of the two-dimensional on-resonance double-kicked lattice. In one-dimensional (1D) descendant models of Equation ( 7 ), rich first-order Floquet topological phases have been discovered, which are characterized by large Chern (winding numbers), multiple chiral (dispersionless) edge modes and topologically quantized acceleration in momentum space [ 107 , 120 , 121 , 122 , 123 ]. These discoveries further motivate us to explore HOTPs in the 2D on-resonance double-kicked lattice model.…”

Topological flat bands in time-periodically driven uniaxial strained graphene nanoribbons

“…This kind of kicking layout can be hard to be reached in experimental conditions, therefore, we discuss the experimental feasibility of our model in a special section (see section VII), therein, we also study a more realistic kind of driving: harmonic driving. However it is worth mentioning that many theoretical papers consider a quite similar kind of kicking 41,[47][48][49][50][51][52] . From here, we will study the t 1 → T limit, then the driving protocol can be written as…”