1D Quantum Liquids with Power-Law Interactions: a Luttinger Staircase with Polar Molecules

We study one dimensional fermionic and bosonic gases with repulsive power-law interactions $1/|x|^{\beta}$, with $\beta>1$, in the framework of Tomonaga-Luttinger liquid (LL) theory. We obtain an accurate analytical expression linking the LL parameter to the microscopic Hamiltonian, for arbitrary $\beta$ and strength of the interactions. In the presence of a small periodic potential, power-law interactions make the LL unstable towards the formation of a cascade of lattice solids with fractional filling, thus forming a "Luttinger staircase". Several of these quantum phases and phase transitions are realized with groundstate polar molecules and weakly-bound magnetic Feshbach molecules.