Just twenty years ago, quantum magnetism was a dead-end track of condensed matter physics. Now it is one of the most promising fields to look for new states of matter. It all started with the unforeseen discovery of high-temperature superconductivity in a family of cuprates in 1986, and the suggestion by Anderson [1] that, although superconductivity is achieved by doping a standard antiferromagnet, what actually happens is that doped holes drive the magnetic background into another state known as a resonating-valence bond (RVB) state, a superposition of configurations in which spins form singlets pairs but change partner from one configuration to the other. This gave a new and strong impetus to the search for new materials with unconventional magnetic properties and to a profound understanding of the connection between magnetism and superconductivity.

The RVB state is a particular example of a more general class of states called spin liquids, in which the constituent localized moments are highly correlated but still fluctuate strongly down to a temperature of absolute zero [2]. As a result, remarkable collective phenomena such as emergent gauge fields and fractional particle excitations may develop. After a long period dominated by the prejudice that spin liquids might be impossible to stabilize in the real world, there is now an increasing evidence that they can be obtained in both simple models and real materials (oxides with kagome geometry in particular, a very active field at present).

Today, spin liquids represent one of the main stream of Solid State physics (hundreds of papers are published every year on this subject), since quantum fluctuations may lead to unconventional (i.e., non-classical) phases of matter. Just to restrict to few important issues, we would like to mention the possibility to have topological order (required for future "topological quantum computing"), fractional excitations, or new classes of critical phenomena, similarly to what appears in the celebrated fractional quantum Hall effect.

Given their inherently non-classical nature, standard mean-field approaches or perturbation expansions cannot be used and new techniques must be conceived. One key trick to capture the low-energy properties of spin liquids is to express the original spin (or electron) operators in term of auxiliary degrees of freedom, like for example fermionic or bosonic spinon operators (which are spin-1/2 but charge-neutral objects). Within this new representation, some non-trivial mean-field approaches become possible, containing spinon hopping and pairing [3]. Most importantly, a gauge structure naturally emerges, so that the original model is equivalent to a model with particles (fermions or bosons) interacting through gauge fields [4]. In turn, the gauge degrees of freedom may profoundly modify the initial "bare" spinons into strongly interacting objects (confinement or deconfinement, statistical transmutation, etc.).

In the last years, it became clear that gauge theories provide the natural description of spin liquid states [5]. Analytical approaches that can study the validity of the mean-field predictions are very complex and often require to consider limiting cases, whose results may be beyond the physical regime (e.g., taking a large number of "flavors" N with an SU(N) symmetry, instead of N=2). Therefore, non-perturbative approaches are needed. Numerical simulations represent an undisputed occasion to verify theoretical predictions based upon mean-field approximations and obtain evidences for refining or constructing new theories.

In this respect, important results are based upon variational Monte Carlo methods [6,7,8], which give the unique opportunity to assess the stability of gauge theories in a non-perturbative way, by reinserting part of the gauge fluctuations that are neglected in mean-field approximations. This kind of approach, based upon the definition of correlated wave functions, is pivoting in any future understanding of spin liquid phases. Moreover, an increasing evidence for spin liquid ground states is now available from other numerical techniques, like density-matrix renormalization group or its recent descendants, based upon tensor network approaches, e.g., pair-entangled projected states (PEPS) [9,10,11,12]. Progress to understand the connections between PEPS and RVB wave functions will also help the understanding of many physical properties, like the description of various topological sectors.

Today we have arrived to a crossroad: from one side, theoretical approaches have been settled down and some important predictions have been obtained; on the other side, it is now crucial to verify these predictions on microscopic models and realistic systems. Moreover, new challenges appear at the horizon, in order to give precise statements that may be experimentally tested.

We believe that the present school is very timely and may provide a reference point for fixing what we already know on this subject and what are the next important questions to be answered in order to have a complete characterization of spin liquid phases.

The total duration of the school will be two weeks.

We plan to have 6 main courses, covering important aspects of quantum spin liquids that emerged in the last few years. Lectures will take place in the morning (3 hours per day, from Monday to Friday), while tutorials and (few) seminars will be in the afternoon, leaving time for free discussions among participants and lecturers. We also plan to have two poster sessions (one per week), where participants may present their work.

M.P.A. Fisher will give an introductory course on lattice gauge theories to describe strongly-correlated systems.

M. Hermele will give an introductory course on slave-particle approaches to frustrated quantum spin models and spin liquid classification (topological order and PSG).

J.I. Cirac will give a course on pair-entangled projected states (PEPS), which represent a powerful and promising way to simulate correlated electron and spin models on the lattice.

D. Poilblanc or R. Moessner will give a course on gauge theories to describe quantum dimer models, together with their numerical simulations.

S. Sorella will give a course on the variational Monte Carlo technique to simulate projected wave functions based upon slave-particle approaches.

A. Lauchli will give a course on exact diagonalization and density-matrix renormalization group approaches to detect quantum spin liquids.