Frustration through anisotropy in quantum magnets

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This research proposal aims to improve our theoretical understanding of magnetic materials. Such materials, and condensed matter physics more broadly, offer an unparalleled window into the realm of quantum and classical many-body phenomena. Through the study of such systems one can hope to understand the rich tapestry of emergent and collective behaviour that can lurk inside even everyday materials. In particular, we aim to improve our understanding of magnets that are both highly anisotropic and highly frustrated. Anisotropy, refers to the physics of the material being dependent on how the microscopic magnetic moments of the atoms are oriented relative to the crystal lattice -- a consequence of a relativistic effect, spin-orbit coupling. Frustration, refers to the incompatibility of multiple tendencies; when it is not possible to satisfy all of the interactions between the magnetic moments at the same time. Combining these two, frustration through anisotropy is when the origin of the frustration is due to the anisotropy in the magnetic interactions, and not, say, through the geometric arrangement of the atoms in the crystal. Through anisotropy, frustration can be generated in a rich variety of ways, each of which has the potential to lead to new and interesting phases of matter. This proposal attacks this topic from two different, but related, directions. First, we address the microscopic origins of magnetic interactions in one of the largest and most varied classes of anisotropic magnets, those where the magnetic ion is a rare-earth element. We will theoretically investigate what kinds of magnetic interactions are possible in a variety of different crystal lattices and structures that are relevant for real materials, with the goal of identifying new and unexplored kinds of frustration. Second, we will develop new methods for understanding the excitations of anisotropic magnets, i.e. the kinds of states they take on when they absorb energy, as happens when they are probed by light or neutrons in many experimental studies. This effort will be pursued though both a quantitative, computational perspective using a kind of series expansion technique, as well as via a more analytical, mathematical route involving the theory of systems that exhibit intrinsic loss or gain of particles. This research will be important for the theoretical and experimental condensed matter physics community, as it will provide new ways to understand known frustrated anisotropic materials, as well as motivate new materials to explore. These new materials, and an understanding of their physics, should provide insights into fundamental questions of classical and quantum many-body physics, and perhaps inspire new technological applications.