Part 2 of this book dealt with the magnetically ordered and quantum paramagnetic phases of models of N-component quantum rotors. In this chapter we consider models of Heisenberg spins: these directly represent the spin fluctuations of physical electrons in insulators or other systems with an energy gap towards charged excitations (e.g., certain quantum Hall states). We describe the conditions under which certain models of Heisenberg spins reduce to N = 3 quantum rotor models, thus providing the physical motivation for studying the latter models. We shall also discuss the physical properties of Heisenberg spin models under conditions in which they do not map onto the rotor models of Part 2.
We begin by showing how to set up a coherent state path integral for Heisenberg spin systems. Then we consider the properties of ferromagnets in which the ground state is the fully polarized state with all spins parallel and the total spin takes its maximum possible value. The discussion of antiferromagnets, in which the ground state has negligible total spin, then follows. Finally, we consider more complex situations with partial uniform polarization of the spins, which is accompanied by a certain `canted' order in dimensions d > 1.