Ground states of long-range interacting fermions in one spatial dimension

We explore systematically the ground state properties of one dimensional fermions with long-range interactions decaying in a power law $\sim1/r^\alpha$ through the density matrix renormalization group algorithm. By comparing values of Luttinger liquid parameters precisely measured in two different ways, we show convincing evidence that Luttinger liquid theory is valid if $\alpha$ is larger than some threshold, otherwise the theory breakdown. Combining analysis on structure factor, charge gap and charge stiffness, we determine how the metal-insulator transition point develops as the interaction range is continuously tuned. A region in the range of $0\leq\alpha\leq1$ and small interactions has finite charge gaps, but, interestingly, it shows metallic nature at the same time. From these, we obtain approximate phase diagrams for the entire parameter space and for band fillings equal 1/2 and 1/3. Finally, we compare certain bosonization and field theory formula with our quasi-exact numerical results, from which disagreements are found.