Effective potential and effective Hamiltonian in quantum statistical mechanics for condensed matter physics

An overview on the theoretic formalism and up-to-date applications in quantum condensed matter physics of the effective potential and effective hamiltonian methods is given. The main steps of their unified derivation by the pure-quantum self-consistent harmonic approximation (PQSCHA) are reported and explained. For a given quantum system the PQSCHA yields an effective classical hamiltonian with dependence on the Planck constant and on temperature and classical-like expressions for the averages of observables. The power of this approach has been demonstrated through applications in different fields, as soliton theory, rare-gas crystals, and magnetism.