Gap Opening Transition and Fractal Ground State Phase Diagram in One Dimensional Fermions with Long Range Interaction : Mott Transition as a Quantum Phase Transition of Infinite Order

The metal-insulator transition in one dimensional fermionic systems with long-range interaction is investigated. We have focused on an excitation spectrum by the exact diagonalization technique in sectors with different momentum quantum numbers. At rational fillings, we have demonstrated gap opening transitions from the Tomonaga-Luttinger liquid to the Mott insulator associated with a discrete symmetry breaking by changing the interaction strength. Finite interaction range is crucial to have the Mott transition at a rational filling away from the half filling. It is consistent with the strong coupling picture where the Mott gap exists at any rational fillings with sufficiently strong interaction. The critical regions as a quantum phase transition are also investigated numerically. Non-analytic behavior of the Mott gap is the characteristic in the weak coupling. It is of the order of the interaction in the strong coupling. It implies that the metal-insulator transition of the model is of the infinite order as a quantum phase transition at zero temperature. Fractal nature of the ground state phase diagram is also revealed.