Computational Physics

(advanced)

 
Program of the course Target The course aims to give to the student a hint and some working experience on the more commonly used advanced techniques for simulating quantum many-body systems. Contents Numerical solution of the one dimensional Schroedinger Equation. Density Functional Theory and Kohn-Sham equations. Numerical implementation of the Local Density Approximation (LDA) for finite systems. Hartree-Fock methods. Overview of existing numerical packages for quantum chemistry and materials science applications. Monte Carlo methods. The variational method for the resolution of the many-body Schroedinger equation. Many-body wavefunctions. Feenberg expansion. Cusp conditions. Bosonic many-body systems (4He, molecular hydrogen). Fermionic systems (3He, electrons, nucleons). Shadow Wave Functions for inhomogeneous systems. Imaginary time projection methods. Approximate Green’s Function. Diffusion Monte Carlo. Importance sampling. Application to many Boson systems. Fermion systems and the sign problem. Constrained approximations (fixed node, fixed phase). Transient estimation. Fermion Monte Carlo. Path based methods. Reptation Monte Carlo and Path Integral Ground State Monte Carlo. Finite temperature problem. Sampling of the thermal density matrix by paths. Path Integral Monte Carlo method. Serial and Parallel computing. Basic concepts of parallel computing. Star and toroidal networks. Architecture based parallelization strategies. The Aurora HPC facility. Background The students must have a sound background in statistical physics and quantum mechanics. Some knowledge of a programming language (C or Fortran) will be certainly useful. Class organization The course will consist of standard classes in which the theory is exposed, and of extesive “hands-on” sessions in which the students will have to write their own codes on assigned problems. Exams At the end of the course the student will choose among the following options to pass the exam: Standard exam (thorough interview about the course program); Reading, comprehension and exposition of a research paper present in the literature on a topic in quantum simulations not already treated in the course; Implementation of a code and presentation of a student’s paper on a topic not already treated in the course. Some reference textbooks S. E. Koonin, D. C. Meredith, Computational Physics - Fortran Version, Oxford U.P.
M.H. Kalos, P.A. Whitlok, Monte Carlo Methods, Volume 1, J. Wiley & Sons 1986
E. Lipparini, Modern Many Particle Physics, World Scientific 2003