Syllabus
The aim of the course is to illustrate selected general principles on simple examples, both in tight-binding lattice models and in the direct space.
Hydrogen molecule as the simplest example of strong electron-electron correlations: Hubbard model with just two orbitals, shortcomings of the Hartree–Fock approximation illustrated by comparison with the exact solution (spin contamination, overestimated tendency toward magnetic states), relationship between the Hubbard model (charge and spin) and the Heisenberg model (spin only) – origin of magnetism in strong correlations among electrons.
Magnetic impurity in a metal: Anderson impurity model, approximate description of the ground state in the limit of strong Coulomb interaction using the Gunnarsson–Schönhammer method, similarity to the hydrogen molecule, response to the homogeneous magnetic field and the static susceptibility, Kondo effect.
Correlated metal: Hubbard model, Gutzwiller wave function and Gutzwiller approximation, suppression of the electron mobility due to electron-electron repulsion, metal–insulator transition.
Helium atom and helium-like ions: analytical properties of explicitly correlated wave functions in Coulomb systems, Jastrow correlation factor, Hylleraas variational method (or how to hold the hydrogen anion together with just a pen and paper), amount of correlations in orthohelium and parahelium.
Variational Monte Carlo: evaluation of quantum-mechanical expectation values by means of stochastic integration, Metropolis algorithm for generation of random numbers with complicated multivariate distributions, elementary properties of Markov chains that represent the basis of the Metropolis algorithm.
Diffusion Monte Carlo: projection of the exact ground state from an approximate wave function, Feynman–Kac formula, stochastic evaluation of the path integral, fermion sign problem, examples of applications.
Annotation
At the start of the course we recall the Hartree–Fock approximation that, apart from the Pauli principle, neglects all other correlations among particles. Using applications to a few simple systems we illustrate the accuracy as well as the weaknesses of this approximation.
For the ground state of the helium atom we construct a considerably more accurate wave function that takes into account correlations among the two electrons and that still allows for evaluation of the approximate ground-state energy by analytical means. Numerical methods will applied for analogous correlated wave functions.