Topological Equivalence between the Fibonacci Quasicrystal and the Harper Model

One-dimensional quasiperiodic systems, such as the Harper model and the Fibonacci quasicrystal, have long been the focus of extensive theoretical and experimental research. Recently, the Harper model was found to be topologically nontrivial. Here, we derive a general model that embodies a continuous deformation between these seemingly unrelated models. We show that this deformation does not close any bulk gaps, and thus prove that these models are in fact topologically equivalent. Remarkably, they are equivalent regardless of whether the quasiperiodicity appears as an on-site or hopping modulation. This proves that these different models share the same boundary phenomena and explains past measurements. We generalize this equivalence to any Fibonacci-like quasicrystal, i.e., a cut and project in any irrational angle.

Figure 1

Graphical presentations of 2D Hamiltonians. The electrons hop on a rectangular lattice in the presence of a perpendicular magnetic field with $b$ flux quantum per unit cell. A hopping amplitude from one vertex to another is denoted by an arrow. (a) The Hamiltonian ${\mathcal{H}}_{\mathrm{d}}^{\mathrm{H}}$, the “ancestor” of the diagonal Harper model. The hopping is to nearest neighbors, and each rectangular plaquette (marked in cyan) is pierced by $b$ flux quantum. (b) The 2D ancestor Hamiltonian of the off-diagonal Harper model, ${\mathcal{H}}_{\mathrm{od}}^{\mathrm{H}}$. The hopping is to nearest neighbors in the $n$ direction, and also to next-nearest neighbors. Here, there are two types of plaquettes, a triangle (dark green) and a parallelogram (cyan), both pierced by $b$ flux quantum. The magnetic translation group is the same in both Hamiltonians, implying that they are topologically equivalent.

Figure 2

(a) The smooth continuation between the Harper and the Fibonacci modulations $V_n^{\mathrm{S}}$ [cf. Eq. (4)] for $b=2/(1+\sqrt{5})$, $k=0$, and several values of the smoothing parameter $\beta$. For $\beta \to 0$, $V^{\mathrm{S}}$ resembles the Harper modulation $V^{\mathrm{H}}$, whereas for $\beta \to \infty$ it approaches the Fibonacci modulation $V^{\mathrm{F}}$. Varying $\beta$ has no influence on the distribution of the Chern numbers, meaning that the Harper model and the Fibonacci QC are topologically equivalent. (b) Illustration of the space of quasiperiodic systems spanned by the smoothing parameter $\beta$ and the diagonal–off-diagonal ratio ${\lambda }^{\mathrm{od}}/{\lambda }^{\mathrm{d}}$, when substituted in the general Hamiltonian $H^{\mathrm{S}}$ [cf. Eq. (5)]. All the systems in this space are topologically equivalent, with the same number of gaps and the same Chern numbers.