Aubry–André Model

The Aubry–André model is a statistical toy model to study thermodynamic properties in condensed matter. The model is usually employed to study quasicrystals and the transition metal-insulator in disordered systems predicted by Anderson localization. It was first developed by Serge Aubry and Gilles André in 1980.

Tight–binding description

Aubry–André model defines a periodic potential over a one-dimensional lattice with hopping between nearest neighbors sites with no interactions. In tight-binding, the on-site energies of Aubry-André potential have a periodicity that is incommensurate with the periodicity of the lattice. The potential can be written as
:$V=\sum_ \epsilon_n , n\rangle\langle n,$,
where the sum goes over all sites $n$, $, n\rangle$ is a Wannier state on lattice site $n$ and the on-site energies are given by
:$\epsilon_n=\lambda\cos(2\pi \beta n +\varphi)$.
where $\lambda$ is the disorder strength, $\varphi$ is a phase and $\beta$ is the periodicity of the potential.

The full Hamiltonian can be written as

:$H=-J\sum_( , n\rangle\langle n+1, +, n+1\rangle\langle n, )+ V$,

where $J$ is the hopping constant. This Hamiltonian is self-dual as it retains the same form after a Fourier transformation.

For special values of $\beta$ the system can demonstrate localization. For the case where $\varphi=0$ and $\beta=(1+\sqrt)/2$ (golden ratio), Aubry and André showed that if $\lambda>2J$, then the eigenmodes are exponentially localized, as in the Anderson model. For $\lambda<2J$, the eigenmodes are extended plane waves. This limit between the two behaviors $\lambda=2J$ is called the localization transition or the Aubry-André transition. For finite chains, the periodicity of the potential can also be chosen to be the ratio $\beta=P/Q$, with primes $P$ and $Q$ larger than number of sites in the chain.

In the Aubry-André model, the energy spectrum $E_n$ as function of $\beta$, is given by the almost Mathieu equation

:$E_n\psi_n=-J(\psi_+\psi_)+\epsilon_n \psi_n$,

related to Harper equation ($\lambda=2J$) that leads to a fractal spectrum known as the Hofstadter's butterfly, which describes the motion of an electron in a two-dimensional lattice under a magnetic field.

Realization

IN 2009, Y. Lahini et al. presented one of the first experimental realizations of the Aubry-André model in photonic lattices.
Condensed matter physics

References

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