Aubry–André model

The Aubry–André model is a toy model of a one-dimensional crystal with periodically varying onsite energies. The model is employed to study both quasicrystals and the Anderson localization metal-insulator transition in disordered systems. It was first developed by Serge Aubry and Gilles André in 1980.

Hamiltonian of the model
The Aubry–André model describes a one-dimensional lattice with hopping between nearest-neighbor sites and periodically varying onsite energies. It is a tight-binding (single-band) model with no interactions. The full Hamiltonian can be written as
 * $$H=\sum_{n}\Bigl(-J |n\rangle\langle n+1| -J|n+1\rangle\langle n| + \epsilon_n |n\rangle\langle n|\Bigr)$$,

where where the sum goes over all lattice sites $$n$$, $$|n\rangle$$ is a Wannier state on site $$n$$, $$J$$ is the hopping energy, and the on-site energies $$\epsilon_n$$ are given by
 * $$\epsilon_n=\lambda\cos(2\pi \beta n +\varphi)$$.

Here $$\lambda$$ is the amplitude of the variation of the onsite energies, $$\varphi$$ is a relative phase, and $$\beta$$ is the period of the onsite potential modulation in units of the lattice constant. This Hamiltonian is self-dual as it retains the same form after a Fourier transformation interchanging the roles of position and momentum.

Metal-insulator phase transition
For irrational values of $$\beta$$, corresponding to a modulation of the onsite energy incommensurate with the underlying lattice, the model exhibits a quantum phase transition between a metallic phase and an insulating phase as $$\lambda$$ is varied. For example, for $$\beta=(1+\sqrt{5})/2$$ (the golden ratio) and almost any $$\varphi$$, if $$\lambda>2J$$ the eigenmodes are exponentially localized, while if $$\lambda<2J$$ the eigenmodes are extended plane waves. The Aubry-André metal-insulator transition happens at the critical value of $$\lambda$$ which separates these two behaviors, $$\lambda=2J$$.

While this quantum phase transition between a metallic delocalized state and an insulating localized state resembles the disorder-driven Anderson localization transition, there are some key differences between the two phenomena. In particular the Aubry–André model has no actual disorder, only incommensurate modulation of onsite energies. This is why the Aubry-André transition happens at a finite value of the pseudo-disorder strength $$\lambda$$, whereas in one dimension the Anderson transition happens at zero disorder strength.

Energy spectrum
The energy spectrum $$E_n$$ is a function of $$\beta$$ and is given by the almost Mathieu equation


 * $$E_n\psi_n=-J(\psi_{n+1}+\psi_{n-1})+\epsilon_n \psi_n$$.

At $$\lambda=2J$$ this is equivalent to the famous fractal energy spectrum known as the Hofstadter's butterfly, which describes the motion of an electron in a two-dimensional lattice under a magnetic field. In the Aubry–André model the magnetic field strength maps onto the parameter $$\beta$$.

Realization
Iin 2008, G. Roati et al experimentally realized the Aubry-André localization phase transition using a gas of ultracold atoms in an incommensurate optical lattice.

In 2009, Y. Lahini et al. realized the Aubry–André model in photonic lattices.