model.

The article is organized as follows. In section II, we introduce the semi-classical approach to simple quantum antiferromagnets, and we explain the importance of the spin Berry phase and how one can include it in a semi-classical description to obtain the correct theory. In particular, we show how it leads to the celebrated Haldane conjecture. The existence of end excitations as a natural consequence of the low-energy effective theory of these systems is discussed. One-dimensional quantum spin systems are of great interest at present because they provide some of the simplest realizations of symmetry-protected topological (SPT) phases in strongly correlated systems.

The limitations of the semi-classical approach when applied to systems with frustrated interactions are discussed in section III, where we introduce the alternative idea of constructing variational wavefunctions directly. We introduce Anderson’s famous idea of the RVB wavefunction for spin $1/2$ systems and discuss how this idea can be implemented in practice. The difficulty of incorporating the $SU(2)$ spin algebra in the usual many-body perturbation theory is noted, and the trick of representing spins by particles (fermions or bosons) with [*constraints*]{} to avoid this difficulty is introduced. The non-trivial $SU(2)$ gauge structure in the fermion representation of RVB 