Noncommutative space and the low-energy physics of quasicrystals

TL;DR

This paper demonstrates that the low-energy behavior of particles in quasiperiodic potentials can be modeled as nonlinear Schrödinger equations on noncommutative space, linking quasiperiodicity to space noncommutativity.

Contribution

It establishes a theoretical equivalence between quasiperiodic potentials and noncommutative space in the context of the nonlinear Schrödinger equation.

Findings

Quasiperiodic potential effects can be represented by noncommutative geometry.

The effective low-energy dynamics are described by a potential-free nonlinear Schrödinger equation.

This approach provides a new perspective on the physics of quasicrystals.

Abstract

We prove that the effective low-energy, nonlinear Schroedinger equation for a particle in the presence of a quasiperiodic potential is the potential-free, nonlinear Schroedinger equation on noncommutative space. Thus quasiperiodicity of the potential can be traded for space noncommutativity when describing the envelope wave of the initial quasiperiodic wave.