Electrons in deterministic quasicrystalline potentials and hidden conserved quantities

Abstract

Abstract. We propose an ansatz for the wave function of a non-interacting quantum particle in a deterministic quasicrystalline potential. It is applicable to both continuous and discrete models and includes Sutherland's hierarchical wave function as a special case. The ansatz is parameterized by a first cohomology class of the hull of the structure. The structure of the ansatz and the values of its parameters are preserved by the time evolution. Numerical results suggest that the ground states of the standard vertex models on Ammann-Beenker and Penrose tilings belong to this class of functions. This property remains valid for the models perturbed within their MLD class, e.g. by adding links along diagonals of rhombi. The convergence of the numerical simulations in a finite patch of the tiling critically depends on the boundary conditions, and can be significantly improved when the choice of the latter respects the structure of the ansatz.

Critical eigenstates and their properties in one- and two-dimensional quasicrystals

“…One can define environment-specific distributions, N (t) µ (h), where µ denotes the local environment. There are three different local environments on the Fibonacci chain which we define according to the nature of the arrow immediately following the site, namely, µ = r, l, u, appearing with the frequencies (15). The N (t) µ (h) give the number of times height h is found on the local environment µ in region R t .…”

“…Although this Hamiltonian is simple, no solutions were known for any of its eigenstates, apart from trivial confined eigenstates at the middle of the spectrum [17]. This situation changed recently, when Kalugin and Katz [15], building on the work of Sutherland [35], deduced the form of the ground state of the Hamiltonian (1) on the Penrose and Ammann-Beenker tilings.…”

“…Taking their cue from periodic crystals, where Bloch states are given by ψ k (r) = u k (r)e ik.r with u a periodic function, Kalugin and Katz [15] proposed that the ground state of the model (1) be given by a product of two factors, namely ψ(i) = C(i)e κh(i) (2) In this expression, κ is a real constant (note that in [15] the notation λ = exp(2κ) is used). The pre-exponential factor C(i) of the ansatz (2) is a quasiperiodic function, a site-dependent amplitude which depends only on the arrangement of the atoms around the site i.…”

Landau levels in quasicrystals

“…The scaling factor κ is related to λ = e 2κ defined in Ref. [29], where it was numerically found that λ = 1.07500(1) for the ground-state wave function of the Penrose tiling. When plugged in (D2) and (D3), this gives β = 0.9991897(2), in agreement with the above numerical result.…”

Perpendicular space accounting of localized states in a quasicrystal

“…The perpendicular space images of eigenstates in the one-dimensional Fibonacci chain [20] and spin models on quasicrystals [21] have been investigated. Perpendicular space images have been calculated for particular eigenstates satisfying an ansatz [22] for two-dimensional tight-binding models [23].…”