Numerical Models in Cond Mat Physics

AAH Model

The Aubry-Andre-Harper Model is a quassi-periodic in space.
It is known to Manifest Anderson Localisation

The Hamiltonian of the model is given by

H= -t \sum_r c^{\dagger}_rc_r - 2\lambda \sum_r \cos(2 \pi Q r + \delta)c^{\dagger}_rc_r

where c are fermionic operators and Q is incommensurate.

It is known that for incommensurate Q all the single particle states are

\text{localised}\quad \text{if} \quad |t| < |\lambda|\\ \text{delocalised}\quad \text{if} \quad |t| > |\lambda|\\

for arbitrary offset \lambda. At the critical point V=t, the states exhibit fractal properties.

The localisation length at the critical point diverges with exponent \nu =1

\xi = \frac{1}{\log|V/t|}\sim |V/t|^{-1}

The Chandran Laumann Version

AAH Duality:

This duality corresponds to a pi/2 rotational symmetry of the associated 2D Model.

(Think of particles hopping on an anisotropic square lattice with flux Q per plaquette and hopping strength t,V in x,y direction.

At t=V this is the hoffstader model whose fractal character is well known.)

References

It is known that for incommensurate Q all the single particle states area

\text{localised}\quad \text{if} \quad |t| < |\lambda|\\ \text{delocalised}\quad \text{if} \quad |t| > |\lambda|\\

for arbitrary offset \lambda. At the critical point V=t, the states exhibit fractal properties.

The localisation length at the critical point diverges with exponent \nu =1

\xi = \frac{1}{\log|V/t|}\sim |V/t|^{-1}