UC Riverside

In this dissertation we present the results of a theoretical investigation of spin models on two-dimensional and quasi one-dimensional lattices, all unified under the concept of quantum frustrated antiferromagnetism, and all discussing various aspects of the antiferromagnetic Heisenberg model on the kagomé lattice. In the Introduction (Chapter 1), we discuss at some length such concepts as frustration and superexchange, among others, which are of common relevance in the rest of the chapters. In Chapter 2, we study the effect of Dzyaloshinskii--Moriya (DM) interactions on the zero-temperature magnetic susceptibility of systems whose low energy can be described by short-range valence bond states. Our work shows that this treatment is consistent with the experimentally observed non-vanishing susceptibility -- in the specified temperature limit -- of the spin-1/2 kagomé antiferromagnetic compound ZnCu₃(OH)₆Cl₂, also known as herbertsmithite. Although the objective of this work is explaining the aforementioned characteristic of the experimental system, our methods are more general and we apply them to the checkerboard and Shastry-Sutherland lattices as well. In Chapter 3, we discuss our findings in the study of ghost-mediated domain wall interactions in the diamondback ladder. These domain walls are the the spin excitations -- the kinks and the antikinks -- separating the ground states along one chain of the ladder. While as individual entities an antikink is energy costly and a kink energy free, our study finds that both interact via the ghosts that they produce in the opposite side of the ladder from where they are located. Through the study of these ghosts, we find that domain walls proliferate in the system above a critical value of the system's coupling constants. It is this proliferation that makes their treatment as free, non-interacting particles impossible, so we study here their interactions both quantitatively and qualitatively, in a region where the latter are yet not very strong, namely below the critical point. Based on the calculated two-body interaction potential, domain walls interact attractively (repulsively) when separated at even (odd) distances, with a strength that decays as 1/s^p, where s is their separation and p<1. We also consider higher-order interactions. In the last chapter, Chapter 4, we present our study of the spin-1 kagomé Heisenberg antiferromagnet. Our approach is to first consider an SU(2)-symmetric parent Hamiltonian with known ground states on the S=1 kagomé lattice, in which nearest-neighbor Heisenberg interactions are already present. We then enhance these interactions by an additional Heisenberg term added perturbatively in order to move the system closer to a pure Heisenberg antiferromagnet. The results of this enhancement is obtaining a description of the system in terms of an effective Hamiltonian, namely a transverse field Ising AF on the triangular lattice. Based on the particular values of this effective Hamiltonian, our system is found to be in the order-by-disorder phase.