Chapter 13
Heisenberg spins: ferromagnets and antiferromagnets

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.

• On the last line of the paragraph containing (13.8), replace the phrase "are twice the Pauli matrices" by "are 1/2 times the Pauli matrices".
• In Eqn. (13.16) replace (t+Dt) in the first two lines by (t-Dt).
• In Eqn. (13.17) the S_B appears outside the t integral.
• In Eqn. (13.18) the right hand side should be integrated over t.
• In the first line of Eqn. (13.51), the coefficient of L² is 2dJa^d. 
• In the second line of Eqn. (13.51), a factor of 2 should be inserted before the H.L term.
• The paragraph following (13.77) should begin: However, even at the level of the action S_z (defined with a short-distance regularization which preserves the invariance under (13.76)), and ignoring.......