Kronig-Penney model of 1d localization

Numerical experiments » Kronig-Penney model of 1d localization

  • Click on figure  to see the distribution of -log(T)
    Click on figure to see the distribution of -log(T)
  • Click on figure to see results of extended Kronig-Penney
    Click on figure to see results of extended Kronig-Penney

Numerical experiments with the disordered Kronig-Penney model

The Kronig-Penney model is a 1d (quantum) system with a series of identical obstacles, all with the same reflection and transmission coefficients. In the disordered version, the relative distances are random variables with a dispersion much larger than the wavelength, so that one can use a random uniform distribution of phase between consecutive scatterers.

The script computes the total transmission using the transfer matrix method. Here, matrices are 2x2. For the scattering by an element, is has the simplest form [[1/t,ir/t],[-ir/t,1/t]] where r is the (real) reflection coefficient and t the (real) transmission coefficient. It corresponds to the simplest scattering matrix S=[[ir,t],[t,ir]] compatible with unitarity S^{dagger}S=1 and time-reversal symmetry (S symmetric), provided that r**2+t**2=1. For free propagation between scatterers, the transfer matrix is diagonal [[exp(i*phi),0],[0,exp(-i*phi)]] with phi a random variable uniformly chosen in [0,2*pi].
The script simply chain scattering and free propagation events and computes the global transmission, or the transmission along the chain.

Usage (4 parameters): L r nr savepsi

   L: system size
   r: reflection coefficient
   nr: number of disorder realizations
   switch savepsi: 1 for saving the log(transmission) along each sample
                                 (with a small number of realizations)
                                 the results are saved in the file log_local_transmission.dat
                            0 for saving only the log(total transmission)
                                 for each sample (with a large number of realizations)
                                 the results are saved in the file log_transmission.dat                    

In addition, few average values are printed
One goal of this script is to demonstrate that Anderson exponential localization is the generic behavior, that is that the transmission decreases exponentially with the system size L, in contrast with classical dynamics where it decays like 1/L.
The localization length is given by  1/|log(1-r**2)| approx 1/r**2 for small r.                    

1. Thin medium

We take r=0.1 (reflection for the wave AMPLITUDE). The probability of reflection is r**2=0.01 on each obstacle. If we take 10 obstacles, we expect altogether a large total transmission. Run the script with:
   python  10 0.1 10 1              
See that it is indeed the case.
To get more data on the distribution of transmission:
   python  10 0.1 10000 0
Build an histogram of -log(transmission). What is the theoretical prediction?  Are fluctuations strong?

2. Thick medium

We still take r=0.1, but now with 1000 obstacles:
    python  1000 0.1 10 1
Look at the log(transmission) vs. position, then transmission vs. position. What do you conclude about fluctuations?

In order to be more quantitative, run plenty of realizations
     python  1000 0.1 1000 0 
or even (takes slighly more than 1 minute on a decent computer)
     python  1000 0.1 10000 0
Build an histogram of log(T) (T is the transmission). You should find something similar to the figure below. The theoretical prediction for the localization length is xi=1/|log(1-r**2)| approx 1/r**2.
How does it compare with the numerical results? What do you think of fluctuations?

3. Fluctuations in a thick medium

Still for r=0.1 and 1000 obstacles:

  • What is the most probable log(T)?
  • What is the average transmission? Estimate it with 100 disorder realizations, and rerun again with 100 other realizations. What do you observe? How many realizations are needed to obtain a decent estimate of the average transmission.
  • Compare the most probable and the average transmission.
  • In the limit of large thickness t=L/xi, the distribution of log(T) is approximately normal, given by  P(log(T)) = frac{1}{2sqrt{pi t}} exp(-frac{(log T+t)^2}{4t})

How does it compare with your results? At any thickness, the DMPK method makes it possible to compute exactly the full distribution, see DMPK equations . Compare with your results.

4. Universality of the fluctuations

The fluctuations and the distributions are supposed not to depend on the microscopic details of the model, but only on the thickness t=L/xi, provided xi is larger than any microscopic length (here 1).
Check that the same distribution of transmission is obtained for r=0.5, for t=10.
One may also consider several extensions of the Kronig-Penney model:

  • Obstacles are not all identical, but with a random distribution of reflection coefficient r (for example a normal distribution).
  • Phases are not uniformly distributed in [-pi,+pi] but in a smaller interval, such as [-pi/4,pi/4].
  • Phases are distributed with a normal distribution, with sigma=pi/4.

Modify the script to run these extended Kronig-Penney models and compute the transmission distributions at t=L/xi=10 (xi may depend on the model).


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