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

A method to construct flat band model is exhibited. If a Gram matrix built upon a set of vectors is regarded as a Hamiltonian of certain physical system, then we can control the ground state degeneracy by modifying the vectors. If the vectors are chosen to be certain subsets of coherent states, the resulting Hamiltonians describe single-particle lattice models with a gauge field. The massive degeneracy of the lowest band in these models is a universal property independent from the shape of lattice, which is controled by the completeness of the subsets of coherent states. The excitation spectrums show Hofstadter-butterfly-like patterns which vary when the lattice changes. The models also feature ground state wave functions in universal form, which is like Landau lowest levels, but the dynamics differs. Experimental realization of the models is promising.