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.[1]
Tight–binding description
![](../I/Aubryandre_localizationlenght.svg.png.webp)
Aubry–André model defines a periodic potential over a one-dimensional lattice with hopping between nearest neighbors sites with no interactions. In tight-binding,[2] 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
- ,
where the sum goes over all sites , is a Wannier state on lattice site and the on-site energies are given by
- .
where is the disorder strength, is a phase and is the periodicity of the potential.
The full Hamiltonian can be written as
- ,
where is the hopping constant. This Hamiltonian is self-dual as it retains the same form after a Fourier transformation.[3]
For special values of the system can demonstrate localization. For the case where and (golden ratio), Aubry and André showed that if , then the eigenmodes are exponentially localized, as in the Anderson model. For , the eigenmodes are extended plane waves. This limit between the two behaviors is called the localization transition or the Aubry-André transition.[4] For finite chains, the periodicity of the potential can also be chosen to be the ratio , with primes and larger than number of sites in the chain.[3]
In the Aubry-André model, the energy spectrum as function of , is given by the almost Mathieu equation
- ,
related to Harper equation () 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.[3][4]
Realization
IN 2009, Y. Lahini et al. presented one of the first experimental realizations of the Aubry-André model in photonic lattices.[5]
References
- Aubry, Serge, and Gilles André. "Analyticity breaking and Anderson localization in incommensurate lattices." Ann. Israel Phys. Soc 3.133 (1980): 18.
- Segev, Mordechai; Silberberg, Yaron; Christodoulides, Demetrios N. (2013). "Anderson localization of light". Nature Photonics. 7 (3): 197–204. Bibcode:2013NaPho...7..197S. doi:10.1038/nphoton.2013.30. ISSN 1749-4885. S2CID 35082566.
- Domínguez-Castro, G A; Paredes, R (2019-07-01). "The Aubry–André model as a hobbyhorse for understanding the localization phenomenon". European Journal of Physics. 40 (4): 045403. arXiv:1812.06201. Bibcode:2019EJPh...40d5403D. doi:10.1088/1361-6404/ab1670. ISSN 0143-0807. S2CID 119484117.
- Martínez, Alejandro J.; Porter, Mason A.; Kevrekidis, P. G. (2018-08-28). "Quasiperiodic granular chains and Hofstadter butterflies". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 376 (2127): 20170139. arXiv:1801.09860. Bibcode:2018RSPTA.37670139M. doi:10.1098/rsta.2017.0139. ISSN 1364-503X. PMC 6077862. PMID 30037937.
- Lahini, Y.; Pugatch, R.; Pozzi, F.; Sorel, M.; Morandotti, R.; Davidson, N.; Silberberg, Y. (2009-06-30). "Observation of a Localization Transition in Quasiperiodic Photonic Lattices". Physical Review Letters. 103 (1): 013901. arXiv:0807.2845. Bibcode:2009PhRvL.103a3901L. doi:10.1103/PhysRevLett.103.013901. ISSN 0031-9007. PMID 19659147. S2CID 33770751.