Time propagation of a 3d wavepacket

Localization can be studied by propagating in time an initially well localized wavepacket (with energy approximately 0) and observe how it spreads in time. The propagation is here performed using an expansion of the initial state on the eigenbasis (computed by exact diagonalization).

time_propagation_anderson_model_3d.py L W nr t_max nsteps

L: system size
W: disorder strength
nr: number of disorder realizations
t_max: propagate from time 0 to t_max
nsteps: number of intermediate times where the squared displacement is computed

The script produces <r^2(t)> vs. t in the file squared_displacement.dat. It also opens 3 windows showing the cumulative (or column, i.e. summed over the third direction) densities in the xy, xz and yz planes. It is useful to control that the wavepacket did not touch the edges (if it touches significantly, the expectation values are rotten).

time_propagation_anderson_model_3d.py 12 10.0 5 5. 50

Look at squared_displacement.dat. Observe that the dynamics is approximately diffusive at short time. Estimate the diffusion coefficient.

If one gets closer to the critical disorder strength W_c=16.53, the dynamics is diffusive at short time then slows down because of interference. It may remain diffusive at long time (in the delocalized regime, albeit with a strongly reduced diffusion coefficient), or <r^2(t)> may tend to a constant (in the localized regime). In any case, it is very difficult to determine the asymptotic regime which is reached very slowly. Run for example

time_propagation_anderson_model_3d.py 16 12.0 1 10. 50
time_propagation_anderson_model_3d.py 16 16.0 1 10. 50
time_propagation_anderson_model_3d.py 16 20.0 1 10. 50

at increasing disorder strengths W=12, 16, 20. Be careful not to overwrite the squared_displacement.dat file as it will be needed (each run takes about 1 minute).

The sciript addiitonally fit the squared displacement with a power law with additional background. This is known to be more or less correct in the critical regime with exponent 2/3. The fit is printed in the third column in squared_displacement.dat. Check that the fit are not too bad. From the (printed as output) value of the exponent, deduce the approximate value of W_c.

N.B.: There is an additional script view_density.py to visualize as a color plot a 2d spatial density.

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Warning: For some stupid reason, the CMS of this Web site refuses files with the .py extension. Thus, you have to manually rename the file, changing the final _py to .py