Computational Physics II
• Assignment 3 - LAPACK and Landau Levels

Due: March 11

Code

• I used the LAPACK routine zheev to find the eigenvalues and eigenvectors of the given matrix. This was read in by my code saved in the following format:
  
  
  1+0i 2+1i 1+-1i 1+0i 5+-1i
  2+-1i 4+0i 0+-1i 3+0i 1+2i
  1+1i 0+1i 0+0i 1+-1i 2+1i
  1+0i 3+0i 1+1i 4+0i 0+1i
  5+1i 1+-2i 2+-1i 0+-1i 1+0i
  


• Notice that even if there is a zero coefficient in either the real or imaginary, it must be explicitly set as such or the reader will fail.
• Using the FORTRAN LAPACK routine zheev, my program outputted the following information:
  
  
  1.000000 + 0.000000i    2.000000 + 1.000000i    1.000000 - 1.000000i    1.000000 + 0.000000i    5.000000 - 1.000000i
  2.000000 - 1.000000i    4.000000 + 0.000000i    0.000000 - 1.000000i    3.000000 + 0.000000i    1.000000 + 2.000000i     
  1.000000 + 1.000000i    0.000000 + 1.000000i    0.000000 + 0.000000i    1.000000 - 1.000000i    2.000000 + 1.000000i     
  1.000000 + 0.000000i    3.000000 + 0.000000i    1.000000 + 1.000000i    4.000000 + 0.000000i    0.000000 + 1.000000i     
  5.000000 + 1.000000i    1.000000 - 2.000000i    2.000000 - 1.000000i    0.000000 - 1.000000i    1.000000 + 0.000000i  
     
  Eigenvalues:
  -5.019797
  -1.469186
  1.710129
  5.326501
  9.452353
  
  Eigenvectors:
  0.598840 - 0.178647i    -0.387301 + 0.084144i    0.095841 - 0.079969i    0.374510 - 0.305771i    -0.439672 - 0.098740i   
  -0.015695 + 0.290233i    0.104960 + 0.220637i    -0.169048 + 0.577941i    -0.394975 - 0.175311i    -0.486816 - 0.263627i   
  0.198941 + 0.046840i    0.752282 + 0.321292i    -0.194756 - 0.346781i    0.244012 + 0.129537i    -0.176521 - 0.152983i   
  -0.078035 - 0.027454i    -0.065562 - 0.331207i    0.113228 - 0.609117i    -0.524643 + 0.014365i    -0.352812 - 0.308841i   
  -0.690203 + 0.000000i    -0.008196 - 0.000000i    0.282481 + 0.000000i    0.477210 + 0.000000i    -0.464788 + 0.000000i   

2. Landau Levels

• Now I will use this method to study the eigenstates of the lowest Landau level in a 2D periodic box in the x-y plane with an extern transverse magnetic field in the z direction.
• The parameters that I used were the ones suggested by the assignment :: 48x48 grid, lattic spacing of 0.125.
• As a general trend, adding disorder to the crystal lattice caused the Landau levels to broaden. Although their exactness decreased somewhat, the jumps were still very apparent.
• Increasing the magnetic field strength caused the Landau levels to become more degenerate. For instance, with a magnetic potential of 1 in the x direction, the first Landau level ranged from eignvalue 1 to 31 with a disorder of 0.3. For the same disorder but a doubled field strength, the first Landau level extended to eigenvalue 63.
• One particularly interesting note is the localization properties within the Landau levels.
• This set of images shows the wave functions for the lower edge of a Landau level, the center, and the upper edge for a disorder of 0:

Low Edge Eigenstate of the First Landau Level

Center Eigenstate of the First Landau Level

High Edge Eigenstate of the First Landau Level

• Notice that all of the solutions have the periodic property we expect, with relatively no localization.
• If we introduce disorder however, we find that this delocalization breaks down. When disorder is introduced, the Landau levels themselves broaden slightly, seemingly attempting to break the degeneracy. The degeneracy is not broken, but it is easily noticed that the Landau levels do not appear as "flat" as the did without disorder.
• Here I have chosen again the upper and lower edges and the center of the Landau level, but this time with a disorder of 0.3:
  
  

Low Edge Eigenstate of the First Landau Level




Center Eigenstate of the First Landau Level




High Edge Eigenstate of the First Landau Level



• This time only the center portion of the Landau level maintains the delocalization property. This means that only a small subset of the eigenstates in each Landau level are responsible for carrying the Hall current (an actual guess is difficult to obtain considering the noise in the wave functions. but in many cases it was only a few).
• The Quantum Hall Effect predicts the behavior that was seen in the exercises. The QHE predicts that Landau levels combined with impurities should exhibit a limited range of delocalized states. The plots above show that this is the case, where the edges of the Landau levels are localized and the center part is delocalized.