Exact Results on Itinerant Ferromagnetism in Strongly Correlated Single- and Multiorbital Systems
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Date
2021-10-27
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Johns Hopkins University
Abstract
The study of magnetism is one of the oldest areas of physics, yet achieving a microscopic understanding remains difficult. Itinerant magnetism, which considers mobile electrons rather than localized spins, is particularly challenging to study both analytically and numerically, as magnetic order in itinerant systems typically results from strong interactions. As a result, the electrons are highly correlated and cannot be described well by simple noninteracting states using mean field or perturbative approaches. Further, many-electron problems are difficult to simulate efficiently due to, for example, the large Hilbert space required by exact diagonalization and numerical instabilities resulting from the fermion sign problem in quantum Monte Carlo methods. Thus, exact results on itinerant ferromagnetism are valuable both to understand the limits and features of systems that can lead to magnetism as well as to serve as accurate starting points for further study. From mean field arguments, ferromagnetism is expected to arise in systems where Coulomb interactions are stronger than the kinetic energy scale. Although these arguments are inaccurate in general, they correctly identify two directions where provable examples of ferromagnetism can be found: the strong coupling limit and the flat band limit.
By studying Hubbard models in both single- and multiorbital systems, we find exact results extending provable ferromagnetism in both the strong coupling and flat band limits. We show in the strong coupling limit that Nagaoka ferromagnetism can be found in a large class of systems including finite regular lattices in more than one dimension by using a connection to the generalized 15 puzzle on arbitrary graphs. In a $t_{2g}$ multiorbital system, we analyze the stability of the Nagaoka-type ferromagnetic state in the presence of multiple holes and emphasize the ability of Hund's coupling to stabilize the state. We further emphasize the value of Hund's coupling in forming ferromagnetism in our study of a multiorbital flat band model. We show that the presence of Hund's coupling allows for an exact description in terms of a percolation problem, where the ferromagnetic phase transition at varying flat band filling can be efficiently studied through Monte Carlo simulation.
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Itinerant ferromagnetism, Hund's coupling, Flat bands, Nagaoka's theorem