Quantum Mediated Effective Interactions For Spatially Complex Systems

Abstract

Magnetic interactions between classical or quantum spin degrees of freedom in a condensed matter system are mediated by particles, which come in two flavors, fermions or bosons. Such a magnetic system can be put on a discrete lattice and one can ask about the nature of the ground state resulting from minimizing the magnetic interactions. More often than not, a ground state, defined by specifying the spin orientation or the spin state at every site on the discrete lattice, is complex. Complex meaning that the spin arrangement in real space is complicated or the ground state has properties (excitations) that are not common place. The origin of this complexity can be attributed to the nature of the discrete lattice on which the spins live, the nature of the mediating quantum particle - fermion versus a boson, and to the nature of the spins themselves - classical versus quantum. In this thesis, we present examples of how non-trivial, spatially (both real and spin space) complex ground states can arise due to quantum (fermion/boson) mediated interactions between spin degrees of freedom. Chapters 2, 3, 4 and 7 solve model Hamiltonians for quantum spins, where the interactions are boson mediated, and the resulting ground states are spatially complex - meaning that they break translational invariance (some states in Chapter 7 also break time reversal invariance). Chapters 4 and 5 introduce and solve model Hamiltonians for classical spin degrees of freedom, where the interactions are fermion mediated, and the resulting ground states have complex and beautiful spin arrangements in both real and spin space. To solve the model Hamiltonians in this thesis and to arrive at spatially complex ground states, our central technique is to use Effective Hamiltonians - which under certain approximations, are a good mimicry of model Hamiltonians. The true usefulness of Effective Hamiltonians lies in the fact that they are much easier to solve, compared to the model Hamiltonians. To solve these Effective Hamiltonians, we develop a host of new numerical and theoretical tools. Each of these tools are more generic than the specific problems they have been applied to in this thesis, and a discerning reader will immediately see their broader applicability to a variety of other problems in condensed matter physics.