  We study the ergodic and non-ergodic behaviors of a clean
Jaynes-Cummings-Hubbard chain for different parameters based on the average
level spacings and the generalized fractal dimensions of eigenstates by using
exact diagonalization. It can be found that a transition from ergodicity to
non-ergodicity phases happens when the atom-photon detuning is large, and the
non-ergodic phases maybe exist in the thermodynamic limit. We also find that
the non-ergodic phase violates the eigenstate thermalization hypothesis.
Finally, we study the many-body multifractality of the ground state and find
that the derivative of the generalized fractal dimensions can determine the
critical point of the Superfluid-Mott-insulation phase transition in a small
range of parameters under different boundary conditions and there is no
ergodicity for the ground state.
