Wonderland of frustration: Magnetic moments and itinerant electrons on the pyrochlore lattice

Abstract

Frustration, the inability to simultaneously minimise all interaction energy terms, has proven fruitful in the quest for new and exotic phases of matter. For example in strongly correlated electron systems, frustration has given rise to a wide range of intriguing quantum phenomena, including spin liquids, fractionalisation and long-range entanglement. In this thesis, we investigate the effects of frustration in magnetic systems on the pyrochlore lattice, looking at both the conventional frustrated magnetism as well as a less studied form of frustration known as kinetic frustration. We start at half-filling, where the magnetic moments in a Mott insulator are localised and interact through Heisenberg exchange. Classically, the nearest-neighbour pyrochlore Heisenberg antiferromagnet is a well-established spin liquid down to zero temperature. We investigate the stability of this spin liquid phase upon including further-neighbour interactions, finding that any second and/or third nearest-neighbour interaction leads to ordering. In particular, we characterise a new kind of ordered state, where pairs of sublattices order in antiparallel spirals. We then proceed to dope an S = 1/2 pyrochlore magnet deep in the Mott regime. The presence of itinerant carriers in large-U Hubbard models near half-filling is known to influence the spin correlations in the system, with Nagaoka’s theorem being a notable example. Nagaoka’s theorem is, however, restricted to bipartite lattices, and non-bipartite lattices, such as the triangular lattice, have shown antiferromagnetic spin correlations upon doping. On the pyrochlore lattice, we find that this kinetic frustration has the potential to induce a spin liquid, and more particularly, the long sought-after resonating valence bond spin liquid originally proposed by Anderson. We prove that the resonating valence bond spin liquid is the ground state of the infinite-U Hubbard model in the thermodynamic limit. Based on numerical computations, we find the result also to hold for finite systems and upon adding small antiferromagnetic and ferromagnetic Heisenberg exchange. While much attention was devoted to the emergence of new states from geometrically frustrated exchange, our work demonstrates that kinetic energy frustration in doped Mott insulators may be pivotal to stabilising robust quantum spin liquids in real materials. Having started with studying frustration of magnetic exchange interactions, followed by a study of kinetic frustration from doping a Mott insulator, we move on to study the potentially interesting phenomena that may emerge when itinerant and magnetically frustrated degrees of freedom are simultaneously present in the system. This is a question of great importance that has been raised in the Kondo context for a long time. While such systems are notoriously difficult to make progress on, we study here a simplified context of coupled itinerant and localised magnetic moments – in particular, we study the behaviour of a single tight-binding electron moving on a one-dimensional chain in a static spin background. We focus on how the electron may localise and demonstrate how the number of states localised by the system’s geometry shows quasiperiodic patterns. These patterns have corresponding effects on transport and hybridisation properties and provide an unusual example of how quasiperiodicity may emerge in a clean lattice system.