Abstract:

This thesis considers two topics in magnetism, the first involving classical spins and the second quantum spins. A theme running through this work is how geometric constraints and frustration can substantially influence the qualitative physics.

The first topic[1] is the magnetization process of spin ice. Spin ice in a magnetic field in the [111] crystallographic direction displays two magnetization plateaux, one at saturation and an intermediate one with finite entropy. We study the crossovers between the different regimes from the viewpoint of (entropically) interacting defects. We develop an analytical theory for the nearest-neighbor spin ice model, which covers most of the magnetization curve. We find that the entropy is non-monotonic, exhibiting a giant spike between the two plateaux. The intermediate plateau and crossover region are described by a two-dimensional monomer-dimer model with tunable fugacities. At low fields, we develop mean-field and renormalization group treatments for the extended "string" defects which restore three-dimensionality.

The second topic[2] is the construction of a family of rotationally invariant, local. S=1/2 Klein Hamiltonians on various lattices that exhibit ground state manifolds spanned by nearest-neighbor valence bond states. We show that with selected perturbations such models can be driven into phases modeled by well understood quantum dimer models on the corresponding lattices. Specifically; we show that the perturbation procedure is arbitrarily well controlled by a new parameter which is the extent of decoration of a reference lattice. This strategy leads to Hamiltonians that exhibit (i)?Z2 RVB phases in two dimensions, (ii)?U (1) RVB phases with a gapless "photon" in three dimensions, and (iii)?a Cantor deconfined region in two dimensions. We also construct two models on the pyrochlore lattice, one model exhibiting a Z2 RVB phase and the other a U (1) RVB phase. This construction provides a proof of principle that topological phases can be realized in a local. SU(2)-invariant spin model.