Coherent states, Gram matrix, and Hofstadter butterfly with flat band

Through a concrete example, we illustrate that the correspondence between Gram matrices and Hamiltonians may open up grand opportunities to discover novel systems. We find that the Gram matrices of certain subsets of coherent states, whose associating eigenvalues form a lattice on the complex plane, can be interpreted as Hamiltonians which govern the dynamics of a particle hopping around the lattice under a gauge field. The completeness of the subsets of coherent states guarantees massive degeneracy of the ground states of the Hamiltonians, which is independent from the shape of the lattice, while different types of lattice result in different Hofstadter-butterfly-like patterns in the excitation spectrum. The models also feature ground state wave functions in universal form whose dynamics is special. Numerical evidence suggests that it is highly possible to study the models experimentally.