Hubbard Models for Quasicrystalline Potentials

TL;DR

This paper develops a numerical method to construct Hubbard models for 2D quasicrystals without relying on Bloch's theorem, enabling analysis of their many-body physics and topological properties.

Contribution

It introduces a novel approach to derive Hubbard Hamiltonians for non-periodic potentials and provides a configuration-space framework for large-scale many-body simulations.

Findings

Constructed Hubbard models for 2D optical quasicrystals.

Identified conditions for Mott insulator phases at unit filling.

Provided a smooth parameterization of Hamiltonian elements across the quasicrystal.

Abstract

Quasicrystals are long-range ordered, yet not periodic, and thereby present a fascinating challenge for condensed matter physics, as one cannot resort to the usual toolbox based on Bloch's theorem. Here, we present a numerical method for constructing the Hubbard Hamiltonian of non-periodic potentials without making use of Bloch's theorem and apply it to the case of an eightfold rotationally symmetric 2D optical quasicrystal that was recently realized using cold atoms. We construct maximally localised Wannier functions and use them to extract on-site energies, tunneling amplitudes, and interaction energies. In addition, we introduce a configuration-space representation, where sites are ordered in terms of shape and local environment, that leads to a compact description of the infinite-size quasicrystal in which all Hamiltonian parameters can be expressed as smooth functions. This configuration-space picture allows one to construct arbitrarily large tight-binding graphs for numerical many-body calculations and enables new analytic arguments on the topological structure and many-body physics of these models, for instance the conclusion that this quasicrystal will host unit-filling Mott insulators in the thermodynamic limit.