Low Energy Theories of Quantum Magnets: Emergent Descriptions and Order by Singularity [HBNI Th204]

Abstract

This thesis explores the notion of emergence in quantum magnets. Emergence describes properties of a system that do not resemble the behavior of its microscopic constituents. We argue that a quantum magnet at low energies behaves as if it were a particle mov- ing on an abstract space — an emergent description of the magnet. This space consists of all classical minimum-energy configurations of the magnet. A striking feature of this description is that a highly interacting problem reduces to a non-interacting one. Further- more, we argue that the emergent properties are qualitatively distinct in unfrustrated and frustrated magnets. For an unfrustrated magnet, the emergent particle is free. That is, the particle explores every part of the space uniformly. However, in the case of a frustrated magnet the particle may not move uniformly over the space. The probability of finding the particle may be unevenly distributed taking different values at different points. In ex- treme cases, the probability distribution may be sharply peaked such that the particle is essentially localized at some point. In terms of the physics of the frustrated magnet, this maps to ordering in a certain classical configuration. This is a consequence of quantum effects in the magnet. In this thesis, we investigate mechanisms by which such ordering takes place. We discuss them in the language of the effective particle picture. As ordering corresponds to localization of the particle, we present two distinct mechanisms that bring about this localization: a) order by potential: quantum fluctuations generate a potential on the space where the particle moves. If the potential has a sufficiently deep minimum, the particle localizes in its vicinity. b) Order by singularity: if the space is self-intersecting in nature, the particle localizes at the self-intersection point due to bound state formation — a consequence of quantum interference and geometry of the underlying space. We explore these ideas in a few magnetic clusters: the symmetric XY quadrumer, the Kitaev square, the Kitaev tetrahedron, and the one dimensional spin-S Kitaev model.