Wang Hailong, Ho Derek Y H, Lawton Wayne, Wang Jiao, Gong Jiangbin
Department of Physics and Center for Computational Science and Engineering, National University of Singapore, Singapore 117542, Singapore.
School of Mathematics and Statistics, University of Western Australia, Crawley, Western Australia 6009, Perth, Australia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Nov;88(5):052920. doi: 10.1103/PhysRevE.88.052920. Epub 2013 Nov 27.
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.
最近的研究表明，除了著名的受驱哈珀模型（KHM）之外，共振双驱转子（ORDKR）模型也具有霍夫施塔特蝴蝶弗洛凯谱，与作为整数量子霍尔效应研究范例的标准霍夫施塔特谱极为相似。早些时候有研究表明，这两个动力学模型的准能谱：（i）如果有效普朗克常数取2π的无理数倍，则它们能精确重叠；（ii）如果相同参数取2π的有理数倍，则它们会有所不同。这项工作对这两个模型进行了详细比较，有效普朗克常数由2πM/N给出，其中M和N是互质的奇数整数。研究发现，ORDKR谱（两个周期驱动序列具有相同的驱动强度）有一个平带和N - 1个非平带，最大带宽按幂律衰减，约为~K(N + 2)，其中K是驱动强度参数。严格证明了平带的存在，并且通过数值验证了在多种情况下的幂律标度关系，还针对三带情况进行了解析证明。相比之下，KHM没有任何平带，其带宽与K呈线性标度关系。这导致了动力学行为上的显著差异，例如ORDKR中存在瞬态（但极其持久）的动力学局域化，而KHM中则不存在。最后，我们表明，尽管存在这些差异，但KHM和ORDKR模型存在简单扩展（通过引入一个额外的周期相位参数），使得扩展后的KHM和ORDKR模型实际上在拓扑上是等价的，即它们产生完全相同的弗洛凯带陈数，并在相同的驱动强度下显示拓扑相变。给出了这种拓扑等价性的理论推导。考虑到经过简单的正则变换后，KHM和ORDKR模型具有完全相同的经典极限，这些结果对于我们当前对量子 - 经典对应关系的理解也具有重要意义。
