Generalized Aubry-André-Harper model with modulated hopping and $p$-wave pairing

Abstract

We study an extended Aubry-Andr{é}-Harper model with simultaneous modulation of hopping, on-site potential, and $p$-wave superconducting pairing. For the case of commensurate modulation of $\beta =1/2$ it is shown that the model hosts four different types of topological states: adiabatic cycles can be defined which pump particles, two types of Majorana fermions, or Cooper pairs. In the incommensurate case we calculate the phase diagram of the model in several regions. We characterize the phases by calculating the mean inverse participation ratio and perform multi-fractal analysis. In addition, we characterize whether the phases found are topologically trivial or not. We find an interesting critical extended phase when incommensurate hopping modulation is present. The rise between the inverse participation ratio in regions separating localized and extended states is gradual, rather than sharp. When, in addition, the on-site potential modulation is incommensurate, we find several sharp rises and falls in the inverse participation ratio. In these two cases all different phases exhibit topological edge states. For the commensurate case we calculate the evolution of the Hofstadter butterfly and the band Chern numbers upon variation of the pairing parameter for zero and finite on-site potential. For zero on-site potential the butterflies are triangular-like near zero pairing, when gap-closure occurs, they are square-like, and hexagonal-like for larger pairing, but with the Chern numbers switched compared to the triangular case. For the finite case gaps at quarter and three-quarters filling close and lead to a switch in Chern numbers.

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• Dynamics of a quantum phase transition in the Aubry-André-Harper model with $p$-wave superconductivity
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• Critical and Topological Phases of Dimerized Kitaev Chain in Presence of Quasiperiodic Potential
• Emergent strength-dependent scale-free mobility edge in a non-reciprocal long-range Aubry-André-Harper model
• Adatom engineering magnetic order in superconductors: Applications to altermagnetic superconductivity
• Colloquium: Synthetic quantum matter in non-standard geometries
• Topological superconductivity in Fibonacci quasicrystals
• Su-Schrieffer-Heeger quasicrystal: Topology, localization, and mobility edge
• Majorana zero modes, unconventional real-complex transition and mobility edges in a one-dimensional non-Hermitian quasi-periodic lattice
• Generalized phase-space description of non-linear Hamiltonian systems and the Harper-like dynamics
• Topological Materials for Near-Field Radiative Heat Transfer
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• Fate of Majorana zero modes by a modified real-space-Pfaffian method and mobility edges in a one-dimensional quasiperiodic lattice