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

Abstract

We study an extended Aubry-André-Harper model with simultaneous modulation of hopping, on-site potential, and p-wave superconducting pairing. For the case of commensurate modulation of β = 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.

Topological properties of the long-range Kitaev chain with Aubry-André-Harper modulation

“…We then study the scaling of this quantity as a function of the size of the system. In particular, we study the scaling in terms of the index n of the Fibonacci number that corresponds to the size of the system, similar to [26,28]. This is depicted in Fig.…”

“…For a nonzero AAH chemical potential and both ξ, α 1, we recover the short-range Kitaev model with an AAH modulation studied in Refs. [24,25,28,30]. For ∆ = 0 and ξ 1, we retrieve the AAH model with long-range hopping amplitudes characterized in Refs.…”

“…For small values of µ t, the totality of the states of the system are ergodic, while for large values of µ t, all the eigenvectors are localized. In between, in contrast to the conventional Aubry-André-Harper model without pairing, which features a direct transition between ergodic and localized states, this system also has a multifractal region for µ t ∈ [|∆ − t|, |∆ + t|] [24,25,28,30]. In this Section, we characterize the phase diagram of the Hamiltonian H ξ,α 1 with the help of three global observables.…”

“…The mean inverse participation ratio (MIPR) is one of the most used quantities to characterize the localization of the eigenstates. It is defined as [28]…”

“…In addition, on can also study the effect of interactions. In particular, density-density interactions [22] or Hamiltonians with superconducting pairing term [23][24][25][26][27][28][29][30] have been considered, and lead to an intermediate extended multifractal regime, which exists for a finite range of chemical potential strengths. Other works have considered coupling of more than one Aubry-André-Harper chains [31,32] or two-dimensional Aubry-André-Harper models [33], which leads to hybridization of ergodic and localized bands.…”