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Description
Ultracold atom gases trapped in optical potentials offer a clean and controllable platform to realize quantum models that are difficult to implement in condensed matter systems [1]. Recent theoretical [2] and experimental [3] developments allow to create periodic sub-wavelength potentials that overcome the diffraction limit imposed by the wavelength of the used laser beams. These potentials support the paradigmatic Kronig-Penney-type models which not only describe the behavior of electrons in a one-dimensional crystal but also have been shown to host topologically protected edge states [4]. Developing control strategies for such systems is of fundamental interest in quantum technologies that rely on robust states for computations [5].
In this work we analyze the topological properties of an advanced Kronig-Penney model. The emergent topological behavior is observed under translations and height modulation of the periodic potential in one dimension. The energy bands split into sub-bands displaying Hofstadter's butterfly-like structure under the change of the spatial modulation frequency. This leads to the redistribution of the topological invariants classifying the bands to a set of sub-bands indicating the same charge transport at lower filling. The transport is realized via adiabatic pumping and the spectral function is calculated showing the existence of topologically protected flat edge modes in the many-body case [6,7].
[1] P. Windpassinger et al., Rep. Prog. Phys. 76, 086401, (2013)
[2] M. Lacki et al., Phys. Rev. Lett. 117, 233001, (2016)
[3] Y. Wang et al., Phys. Rev. Lett. 120, 083601, (2018)
[4] I. Reshodko et al., New J. Phys. 21, 013010, (2019)
[5] C. P. Koch et al., EPJ Quantum Technol. 9, 1, (2022)
[6] W.-B. He et al., Phys. Rev. A 111, 013312, (2025)
[7] S. S. Nair et al., Phys. Rev. A 111, 033313, (2025)
