Electrons in deterministic quasicrystalline potentials and hidden conserved quantities

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.

arXiv comment31 pages, 10 figures

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• A Kitaev-type spin liquid on a quasicrystal
• Emergence of quasiperiodic Bloch wave functions in quasicrystals
• Perpendicular space accounting of localized states in a quasicrystal
• Topological superconductivity in Fibonacci quasicrystals
• Properties of the Ammann-Beenker tiling and its square approximants
• Exploiting Anyonic Behavior of Quasicrystals for Topological Quantum Computing
• Stability of quasicrystalline ultracold fermions to dipolar interactions
• The Fibonacci quasicrystal: case study of hidden dimensions and multifractality