Geometrically frustrated antiferromagnets: statistical mechanics and dynamics

1.1 Introduction This chapter is intended to give an introduction to the theory of thermal fluctuations and their consequences for static and dynamic correlations in geometrically frustrated antiferromagnets, focusing on the semiclassical limit, and to discuss how our theoretical understanding leads to an explanation of some of the main observed properties of these systems. A central theme will be the fact that simple, classical models for highly frustrated magnets have a ground state degeneracy which is macroscopic, though accidental rather than a consequence of symmetries. We will be concerned in particular with: (i) the origin of this degeneracy and the possibility that it is lifted by thermal or quantum fluctuations; (ii) correlations within ground states; and (iii) low-temperature dynamics. We concentrate on Heisenberg models with large spin S, referring to the chapter by G. Misguich for a discussion of quantum spin liquids, and to the chapter by M. Gingras for an overview of geometrically frustrated Ising models in the context of spin ice materials. Several earlier reviews provide useful further reading, including [1], [2] and [3] for experimental background, and [4] and [5] for an alternative perspective on theory. To provide a comparison, it is useful to begin by recalling the behaviour of an unfrustrated antiferromagnet. To be definite, consider the Heisenberg model with nearest neighbour exchange J on a simple cubic lattice. As the lattice is bipartite – it can be separated into two interpenetrating sublattices, in such a way that sites of one sublattice have as their nearest neighbours only sites from the other sublattice – the classical ground states are two-sublattice Néel states, in which spins on one sublattice all have the same orientation, and those on the other sublattice have the opposite orientation. These states are unique up to global spin rotations, which are a symmetry of the model. Their only low energy excitations are long wavelength spinwaves. These are Goldstone modes – a consequence of the symmetry breaking in ground states – and have a frequency ω(k) that is linear in wavevector k at small k. This classical picture carries over to the quantum system, and for