Aubry–André model

File:Aubryandre localizationlenght.svg

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

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

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

:$\backslash epsilon\_n=\backslash lambda\backslash cos(2\backslash pi\; \backslash beta\; n\; +\backslash varphi)$.

Here $\backslash lambda$ is the amplitude of the variation of the onsite energies, $\backslash varphi$ is a relative phase, and $\backslash beta$ 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.{{Cite journal |last1=Domínguez-Castro |first1=G A |last2=Paredes |first2=R |date=2019-07-01 |title=The Aubry–André model as a hobbyhorse for understanding the localization phenomenon |url=https://iopscience.iop.org/article/10.1088/1361-6404/ab1670 |journal=European Journal of Physics |volume=40 |issue=4 |pages=045403 |arxiv=1812.06201 |bibcode=2019EJPh...40d5403D |doi=10.1088/1361-6404/ab1670 |issn=0143-0807 |s2cid=119484117}}

For irrational values of $\backslash beta$, 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 $\backslash lambda$ is varied. For example, for $\backslash beta=(1+\backslash sqrt\{5\})/2$ (the golden ratio) and almost any $\backslash varphi$,{{Cite journal |last=Jitomirskaya |first=Svetlana Ya. |date=1999 |title=Metal-Insulator Transition for the Almost Mathieu Operator |url=https://www.jstor.org/stable/121066 |journal=Annals of Mathematics |volume=150 |issue=3 |pages=1159–1175 |doi=10.2307/121066 |jstor=121066 |issn=0003-486X|arxiv=math/9911265 }} if $\backslash lambda>2J$ the eigenmodes are exponentially localized, while if the eigenmodes are extended plane waves. The Aubry-André metal-insulator transition happens at the critical value of $\backslash lambda$ which separates these two behaviors, $\backslash lambda=2J$.{{Cite journal |last1=Martínez |first1=Alejandro J. |last2=Porter |first2=Mason A. |last3=Kevrekidis |first3=P. G. |date=2018-08-28 |title=Quasiperiodic granular chains and Hofstadter butterflies |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |language=en |volume=376 |issue=2127 |pages=20170139 |doi=10.1098/rsta.2017.0139 |issn=1364-503X |pmc=6077862 |pmid=30037937|arxiv=1801.09860 |bibcode=2018RSPTA.37670139M }}

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 $\backslash lambda$, whereas in one dimension the Anderson transition happens at zero disorder strength.

The energy spectrum $E\_n$ is a function of $\backslash beta$ and is given by the almost Mathieu equation

:$E\_n\backslash psi\_n=-J(\backslash psi\_\{n+1\}+\backslash psi\_\{n-1\})+\backslash epsilon\_n\; \backslash psi\_n$.

At $\backslash lambda=2J$ 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 $\backslash beta$.

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.{{Cite journal |last1=Roati |first1=Giacomo |last2=D’Errico |first2=Chiara |last3=Fallani |first3=Leonardo |last4=Fattori |first4=Marco |last5=Fort |first5=Chiara |last6=Zaccanti |first6=Matteo |last7=Modugno |first7=Giovanni |last8=Modugno |first8=Michele |last9=Inguscio |first9=Massimo |date=June 2008 |title=Anderson localization of a non-interacting Bose–Einstein condensate |url=https://www.nature.com/articles/nature07071 |journal=Nature |language=en |volume=453 |issue=7197 |pages=895–898 |doi=10.1038/nature07071 |pmid=18548066 |issn=1476-4687|arxiv=0804.2609 }}

In 2009, Y. Lahini et al. realized the Aubry–André model in photonic lattices.{{Cite journal |last1=Lahini |first1=Y. |last2=Pugatch |first2=R. |last3=Pozzi |first3=F. |last4=Sorel |first4=M. |last5=Morandotti |first5=R. |last6=Davidson |first6=N. |last7=Silberberg |first7=Y. |date=2009-06-30 |title=Observation of a Localization Transition in Quasiperiodic Photonic Lattices |url=https://link.aps.org/doi/10.1103/PhysRevLett.103.013901 |journal=Physical Review Letters |language=en |volume=103 |issue=1 |pages=013901 |doi=10.1103/PhysRevLett.103.013901 |pmid=19659147 |arxiv=0807.2845 |bibcode=2009PhRvL.103a3901L |s2cid=33770751 |issn=0031-9007}}

• Arnold tongue
• Bose–Hubbard model

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Category:Condensed matter physics