Aubry–André model

The Aubry–André model is a toy model of a one-dimensional crystal with periodically varying onsite energies and a metal–insulator transition. 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.^[1]

Hamiltonian of the model

Aubry-André metal-insulator transition, inverse of localization length ${\displaystyle l_{\rm {loc}}}$ as a function of energy ratios ${\displaystyle \lambda /J}$.

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

${\displaystyle H=\sum _{n}{\Bigl (}-J|n\rangle \langle n+1|-J|n+1\rangle \langle n|+\epsilon _{n}|n\rangle \langle n|{\Bigr )}}$,

where the sum goes over all lattice sites ${\displaystyle n}$, ${\displaystyle |n\rangle }$ is a Wannier state on site ${\displaystyle n}$, ${\displaystyle J}$ is the hopping energy, and the on-site energies ${\displaystyle \epsilon _{n}}$ are given by

${\displaystyle \epsilon _{n}=\lambda \cos(2\pi \beta n+\varphi )}$.

Here ${\displaystyle \lambda }$ is the amplitude of the variation of the onsite energies, ${\displaystyle \varphi }$ is a relative phase, and ${\displaystyle \beta }$ 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.^[2]

Localization transition

For irrational values of ${\displaystyle \beta }$, corresponding to a modulation of the onsite energy incommensurate with the underlying lattice, the single particle states of the model go from being delocalized to localized as ${\displaystyle \lambda }$ is varied. For example, for ${\displaystyle \beta =(1+{\sqrt {5}})/2}$ (the golden ratio) and almost any ${\displaystyle \varphi }$,^[3] if ${\displaystyle \lambda >2J}$ the eigenmodes are exponentially localized, while if ${\displaystyle \lambda <2J}$ the eigenmodes are extended. The Aubry-André localization transition happens at the critical value of ${\displaystyle \lambda }$ which separates these two behaviors, ${\displaystyle \lambda =2J}$.^[4]

While this quantum phase transition between a delocalized state and a 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 ${\displaystyle \lambda }$, whereas in one dimension the Anderson transition happens at zero disorder strength.

Energy spectrum

The energy spectrum ${\displaystyle E_{n}}$ is a function of ${\displaystyle \beta }$ and is given by the almost Mathieu equation

${\displaystyle E_{n}\psi _{n}=-J(\psi _{n+1}+\psi _{n-1})+\epsilon _{n}\psi _{n}}$.

At ${\displaystyle \lambda =2J}$ 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.^[2][4] In the Aubry–André model the magnetic field strength maps onto the parameter ${\displaystyle \beta }$.

For general ${\displaystyle \lambda }$ and for any irrational value of ${\displaystyle \beta }$ the spectrum of the almost Mathieu equation is known to be a Cantor set. This results from the solution^[5] of the ten martini problem. For comparison, the energy spectra of periodic lattice models exhibits energy bands, while that of lattice models with uncorrelated disorder (such as the Anderson model) is an everywhere dense set of points.

The many-body system: metal-insulator phase diagram

As proven by mathematicians^[6] the single particle states localize when ${\displaystyle \lambda =2J}$. This suggests the existence of a metal-insulator transition, since extended states can conduct, while localized states can not. However, metals, and insulators, are by definition many-electron systems, and Walter Kohn has argued in a seminal work^[7] that conduction vs. insulation does not necessarily depend on what happens at the single particle level. Instead, it is depend on the collective behavior of all potentially mobile charge carriers. Indeed, in the Aubry-André model, when the single-particle states are filled (i.e. there are more than one particles in the system), the phase diagram takes an unusual form^[8] as a function of particle density. In the thermodynamic limit particle densities which are rational numbers undergo a metal-insulator transition at ${\displaystyle \lambda =2J}$ (where single particle states localize), but particle densities corresponding to certain irrational fillings are always localized (${\displaystyle \lambda =0}$). For example, if ${\displaystyle \beta }$ is the golden ratio, then the fillings at which the system is always insulating are ratios of Fibonacci numbers or sums thereof.

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.^[9]

In 2009, Y. Lahini et al. realized the Aubry–André model in photonic lattices.^[10]

See also

• Arnold tongue
• Bose–Hubbard model