World-line and determinantal quantum Monte Carlo methods for spins, phonons and electrons

In this chapter we will concentrate primarily on world-line methods with loop updates, for spins and also for spin-phonon systems, as well as on the auxiliary field quantum Monte Carlo (QMC) method. Both approaches are based on a path integral formulation of the partition function which maps a d-dimensional quantum system onto a d + 1 dimensional classical system. The additional dimension is nothing but the imaginary time. World-line based approaches for quantum spin systems offer a simple realization of the mapping from quantum to classical, and allow for new approaches to phonons, as recently developed. Auxiliary field QMC methods provide access to fermionic systems both at finite temperature and in the ground state. An important example is the Hirsch-Fye approach that allows for an efficient simulation of impurity models, such as the Kondo and Anderson models, and is widely used in the domain of dynamical mean field theories (DMFT).