We study a family of Hamiltonians of fermions hopping on a set of lattices in the presence of a background gauge field. The lattices are constructed by decorating the root lattices of various Lie algebras with their minuscule representations. The Hamiltonians are, in momentum space, themselves elements of the Lie algebras in these same representations. We describe various interesting aspects of the spectra, which exhibit a family resemblance to the Dirac spectrum, and in many cases are able to relate them to known facts about the relevant Lie algebras. Interestingly, various realizable lattices such as the kagomé and pyrochlore can be given this Lie algebraic interpretation, and the particular flux Hamiltonians arise as mean-field Hamiltonians for spin-1/2 Heisenberg models on these lattices.
Bibliographical noteFunding Information:
R. Shankar thanks Professors Greg Moore and Siddhartha Sahi from Rutgers and Professor Greg Zuckerman from Yale for helpful discussions, the National Science Foundation for grant DMR-0354517, and the Princeton Center for Theoretical Physics for its hospitality during the course of this work. F. Burnell and S.L. Sondhi thank Shoibal Chakravarty for a prior collaboration which inspired this project. F. Burnell acknowledges the support of NSERC. S.L. Sondhi acknowledge support from NSF Grant No. DMR 0213706.
- Flux phases
- Large N Heisenberg model
- Lattice gauge theory
- Lie algebras
- Quantum magnetism