Computational Physics II
• Assignment 4 - ARPACK and Complex Magnetic Fields

Due: March 24

Code

• I used the ARPACK routine zndrv1 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 4+-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
  4+1i 1+-2i 2+-1i 0+-1i 5+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 ARPACK routine zndrv1.f file as a template, I modified it to allow me to set up the matrix that would be used in the calculation and to output the relevant returned information from the actual fortran methods. Here is the output of my system:
  
  
  1.000000 + 0.000000i    2.000000 + 1.000000i    1.000000 - 1.000000i    1.000000 + 0.000000i    4.000000 - 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     
  4.000000 + 1.000000i    1.000000 - 2.000000i    2.000000 - 1.000000i    0.000000 - 1.000000i    5.000000 + 0.000000i  
  
  

  Eigenvalue: 10.184309
  Eigenvector:
  -0.001363 + 0.399179i
  -0.213689 + 0.427193i
  -0.119137 + 0.205082i
  -0.257253 + 0.301403i
  0.097270 + 0.624284i
  

  Eigenvalue: 6.026641
  Eigenvector:
  -0.373861 + 0.118100i
  0.201138 + 0.475623i
  -0.106655 + -0.158483i
  0.302881 + 0.445085i
  -0.377802 + -0.332450i
  

  Eigenvalue: 2.027527
  Eigenvector:
  0.018282 + -0.078824i
  -0.539077 + -0.084593i
  0.339477 + -0.270763i
  0.642255 + 0.130838i
  0.005814 + 0.278121i
  

  Eigenvalue: -2.777356
  Eigenvector:
  -0.034712 + 0.656263i
  -0.363554 + -0.209421i
  -0.202900 + 0.232164i
  0.161687 + -0.042477i
  0.230621 + -0.464626i
  

  Eigenvalue: -1.461121
  Eigenvector:
  0.469392 + -0.167566i
  -0.102233 + -0.144443i
  -0.727474 + -0.306880i
  0.049394 + 0.305024i
  -0.030523 + 0.011845i

     


• There is a lot going on under the hood that isn't show here explicitly. I actually checked the eigen vectors to make sure they were indeed eigen vectors of the system with the appropriate eigen values.
• The file test_arpack.c in the code archive holds the C code to set the matrix to be used and to output the results. The file test_znsimp.f holds the FORTRAN code to perform the eigen decomposition.

2. Complex Magnetic Field

• Now I will use this method to study the eigenstates of the various magnetic field and electric potential settings. I will perform six tests. The first three will be for a magnetic potential centered on the lattice and cyclical. For this configuration, I will vary the electric potential to be zero, have a crystal structure, and then a randomized offset to the crystal. The next three will have the same electric potential but the magnetic potential will be changed to a single link.
• The parameters that I used were the ones suggested by the assignment :: 64x64 grid, lattic spacing of 0.125.
• Here are the sets of images I generated. Each row represents a series of eigenvectors that were solved for using the methods from Section 1 to find the solution to Schrödinger's Equation in the appropriate fields with varied magnetic field strength, and each column represents a ground or excited state (with the ground state being the first)

Potential Set to Zero :: Different Links

Potential with Crystal Structure :: Different Links

Potential with Randomized Crystal :: Different Links

---

Potential Set to Zero :: Single Links

Potential with Crystal Structure :: Single Links

Potential with Randomized Crystal :: Single Links

• In general the magnetic field tends to distort the wave function solution no matter what the potential. However, the effects are more present in the zero potential and the crystal structured potential than in the single link random struture. The differed link magnetic potential however, does begin to break the effects of the eletrical potential. As the magnetic potential increases, the wave function begins to converge to a more stable looking state (i.e. one with less 'bumps'). This effect is not as strong in the single link case again however.
• The file mag2d_arpack.c in the code archive holds the C code to set the matrix to be used and to output the results to a file that gnuplot can use. The file zndrv1_mag2d.f holds the FORTRAN code to perform the eigen decomposition.