Theoretical studies of quantum magnets and strongly correlated metals

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The goal of this research program is to discover and study theoretical frameworks which capture the general underlying and unifying physical principles of quantum materials such as frustrated magnets and strongly correlated metals. Geometrically frustrated magnets are characterized by the geometry of their magnetic ion sub-lattice: the magnetic moments ("spins") are located at the vertices of triangles or tetrahedra. Generally, a pair of spins has a particular relative orientation which minimizes their interaction energy. A system is frustrated when there is no arrangement of the spins that simultaneously minimizes the interaction energy between all pairs of spins. A famous example is the rare-earth pyrochlore crystals, where the magnetic rare earth ions are located on the vertices of a connected network of tetrahedra. Many different phenomena have been observed in pyrochlore crystals, including ordinary ferromagnetism (a state where the spins are all aligned with each other), anti-ferromagnetism (in which the spins alternate their orientation), spin ice (systems with spins obeying the ice-rule: two spins pointing into and two spins pointing out of each tetrahedron), quantum spin-ice (spin-ice states with large quantum fluctuations) and spin-liquid (a state that has macroscropic quantum entanglement). The proposed research will develop and use a suite of computational and analytical methods. The foundation for all approaches is the model, and in crystals with a high underlying symmetry, completely general models will be derived using the methods of group theory (symmetry); these same methods can greatly reduce the complexity of the models. Computational methods include exact diagonalization (which yields exact solutions for systems with a small number of spins) and classical spin simulations. Strongly correlated metals are conductors in which there are strong interactions between electrons, yielding a rich phase diagram that includes metal-to-insulator transitions, superconductivity or magnetic order. The electronic correlations can be modeled by the Hubbard interaction: a strong, constant repulsion that inhibits more than one conduction electron from occupying each atomic site. Although conceptually simple, this model presents difficult computational challenges, even in two-dimenstions. Recent advances by our group at MUN, which combine analytic and computational methods in a procedure called "Algorithmic Matsubara Integration" have considerably decreased computing times to the point where exact results up to sixth order in perturbation theory are within reach. This proposal aims to fulfill this task. Beyond two-dimensions, the Hubbard model has been applied to many systems, especially those with transition elements or rare earths. Among these are the metallic rare-earth pyrochlores such as Pr2Ir2O7; the longer term goal of this work is to apply our numerical techniques to this system.