Requirements for the admission to the module exam:

For the signature on the form "Anmeldung zur Modulprüfung im Masterstudiengang", each student has to select one topic from the list below and present the results in a talk of about 20 minutes. The talks can be prepared and presented in groups of two students.

The presentation should include:

• motivation; theoretical background; the specific numerical approach
• the code (the interesting parts of the code should be explained in detail)
• results and interpretation; what are the limitations of the numerical method?

Please let me know (via e-mail or after the lectures) the topic of your choice. If necessary, we can split each of the projects into several projects.

Projects:

• rule N (elementary cellular automata)
  - random initial configurations with p the probability for z_i = 1; calculate the average < z_i(t) > for different values of p and N;
  - generalizations: extended neighbourhoods, z_i in {0,1,2}, etc.; find rules with "interesting" behaviour;
  [Bieler]
• Game of Life
  - visualize the evolution of various initial configurations (oscillators, gliders, etc.);
  - starting from a random configuration, calculate the number of living cells as function of iteration;
  - variations of the rules: are there any other interesting sets of rules?
  - generalization to other lattices: 2d-hexagonal, 2d-triangular, etc.;
  [Kahl]
• ASEP
  - calculate the fundamental diagram (flow vs. density) for different variants of the model (open/periodic boundary conditions, sequential/parallel update, etc.)
  [Itak, Overberg]
• Nagel-Schreckenberg model
  - calculate the fundamental diagram (flow vs. density)
  [Mai, Sandt]
• two-dimensional Ising model
  - Monte-Carlo/Metropolis algorithm: calculate the critical exponents for the phase transition (paramagnet-ferromagnet)
  [Winkowski]
• classical XY-model
  - Monte-Carlo/Metropolis algorithm
  - thermodynamics; T_c
  [Walter, Wassmer], [Angaji, Keisers]
• Fermi-Pasta-Ulam problem
  - harmonic chain + anharmonic potential
  - numerical solution of the set of differential equation
  - energy transport/equilibration
  [Bruch, Gresista], [Hernández, Teja]
• N-body gravitational systems
  - periodic orbits of the three-body problem
  - numerical simulation of ~10-20 masses [Neis, Ruchnewitz]
• Schrödinger equation for many-electron atoms
  [Kalhöfer, von Schoeler]
• spin correlations of quantum spin models I
  - afm Heisenberg chain, T=0
  - study the influence of geometric frustration on the spin-correlation
  [Krahe]
• spin correlations of quantum spin models II
  - afm Heisenberg chain, temperature dependence of spin-correlations
  - comparison with afm Ising model (via classical Monte-Carlo/Metropolis algorithm)
• anisotropic (quantum) Heisenberg models
  - XXZ- and XY model
  - Jordan-Wigner transformation; relation to fermionic tight-binding models
  [Michael]
• Kitaev clusters (this can be split up in various separate projects)
  - geometry: honeycomb lattice, ladders
  - spectrum of eigenenergies
  - spin-correlations
  - expectation value of plaquette operators
  - eigenenergies from the Majorana-Fermion approach: comparison with full diagonalization
  [Bezvershenko, Joy]
• entanglement entropy
  - afm Heisenberg chain, T=0
  - dependence on length of subsystems
  [Biertz, van der Feltz]
• logarithmic negativity
  - afm Heisenberg chain, T=0
  - comparison with entanglement entropy
  - are there entangled states for which the logarithmic negativity gives zero?
  [Hopfer, Titz]
• Lanczos algorithm
  - implement the Lanczos algorithm for the one-dimensional Heisenberg chain
  - optional: symmetries; sparse matrices
  - comparison with the results from the full diagonalization
  [Tanul]
• iterative diagonalization
• single-impurity Anderson model
  - full diagonalization of a small cluster
  - use the symmetries of the model to reduce the size of the matrices
  [Henze, Kaufhold]

(more projects to follow)

schedule:

• Monday, June 17
  - S. Bieler: rule N
  - P. Kahl: Game of Life
• Wednesday, June 19
  - Y. Itak, F. Overberg: ASEP
  - M. Mai, R. Sandt: Nagel-Schreckenberg model
• Monday, June 24
  - Y. Winkowski: two-dimensional Ising model
  - T. Walter, J. Wassmer: classical XY-model I
  - A. Angaji, J. Keisers: classical XY-model II
• Wednesday, June 26
  - N. Bruch, L. Gresista: Fermi-Pasta-Ulam problem I
• Monday, July 1
  - A. Hernández, K. Teja: Fermi-Pasta-Ulam problem II
  - S. Kalhöfer, K. von Schoeler: Schrödinger equation for many-electron atoms
  - T. Krahe: spin correlations of quantum spin models I
• Wednesday, July 3
  - A. Bezvershenko, A. Joy: Kitaev clusters
  - R. Biertz, W. van der Feltz: entanglement entropy
  - Tanul: Lanczos algorithm
• Monday, July 8
  - K. Hopfer, M. Titz: logarithmic negativity
  - F. Henze, L. Kaufhold: single-impurity Anderson model
• Wednesday, July 10
  - M. Neis, D. Ruchnewitz: N-body gravitational systems
  
• date not yet fixed
  - F. Michael: anisotropic (quantum) Heisenberg models