Aspects of frustrated magnetism and topological order

In the first part we study O(N) spins on geometrically frustrated pyrochlore and planar pyrochlore lattices. For classical spins the strong frustration causes a lack of ordering for N ≠ 2 at all temperatures. Nevertheless at low temperatures the system is strongly correlated. We find degrees of freedom that are relevant in this regime and find that the correlations are dipolar up to distance ∼ 1/ T and then decay exponentially. We use various approaches---most notably, we show that this holds to any order in the 1/N expansion. Then we look at spin evolution under precessional and Monte-Carlo dynamics and find that in both cases the dynamical correlations decay exponentially in time with a scale tau ∼ 1/T and that the long distance and long time correlations are described by the simple Langevin evolution of the degrees of freedom found in the static case. Finally we look at the effects of quantum fluctuations by studying the quantum rotor model on the pyrochlore lattice. Using quantum Monte-Carlo simulations and analytical arguments we find that for every N there is a region in quantum coupling---temperature plane with long range nematic order.; In the second part we study stability of topological order in Kitaev's model under arbitrary small local perturbations. Using the Fredenhagen-Marcu order parameter [13] we show in quantum perturbation theory that in two or more dimensions, the topologically ordered phase is distinct from more conventional phases. Further we show that this distinction survives to positive temperatures in three or more dimensions and that the topological phase is also distinct from the high temperature phase. We extend this result to odd Ising gauge theories and note that the relation between these and dimer models strongly suggests that this result also holds for the Ising RVB phases found in the latter [16] and in the related spin models.