Quantum Monte Carlo methods are among the most successful approaches for computing the properties of interacting quantum many-body systems. For a variety of problems in Physics (but also in Chemistry), quantum Monte Carlo techniques have been employed to provide answers in cases where standard methods, such as Hartree-Fock or Density-Functional Theory, have been shown to be inadequate, and post-Hartree-Fock methods are too computationally expensive. The principal problem, which manifests itself in different forms, is that of describing electron correlation, in either molecules, materials or effective low-energy models (e.g., Hubbard or Heisenberg), beyond mean-field or pure density-functional theories. Difficult systems include strong correlation problems, often associated with partially filled and highly localized d and f shells, or else even simpler molecular dissociation problems. Similarly, in materials, the narrow band systems of the transition metals and rare-earth metals are very difficult to describe. There are a number of important outstanding problems that have for decades resisted solution, most prominently the many fermion problem, but also quantum spin systems with frustrating or competing interactions. In the last 10 years, an increasing effort devoted to define new computational algorithms and quantum Monte Carlo techniques have been demonstrated to be one of the most promising way to solve the problem of correlation in fermionic and bosonic systems.
The tutorial is needed to promote Monte Carlo methods in the context of all problems where strong electron-electron correlation is important to generate new quantum phases (such as spin liquids, supersolids, exotic superconductors, etc). In particular, we would like to boost and encourage the Monte Carlo methods in the new generation of scientists working in correlated materials.
The scope of this school is two-fold; from one side, we want to give a detailed description (with technical implementations during the hands-on tutorials) of basic quantum Monte Carlo methods, which have been successfully applied in several problems arising from Condensed Matter or cold atomic matter trapped in optical lattices. In this regard, we will describe few selected examples of Monte Carlo algorithms, such as variational, fixed-node and diffusion, path integral and auxiliary-field techniques.
From the other side, we would like to highlight important physical applications of these methods to unveil new and interesting phenomena in correlated systems of fermions and bosons. Applications of the variational Monte Carlo and auxiliary-field techniques in lattice models will be focused on the possibility to describe spin liquids at zero temperature. These pure quantum phases escape any mean-field description and require the inclusion of strong electron-electron repulsion that can be achieved by Monte Carlo methods.