One-Dimensional Quantum Liquids with Power-Law Interactions: The Luttinger Staircase

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 (TLL) theory. We obtain an accurate analytical expression linking the TLL 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 TLL unstable towards the formation of a cascade of lattice solids with fractional filling, a “Luttinger staircase.” Several of these quantum phases and phase transitions are realized with ground state polar molecules and weakly bound magnetic Feshbach molecules.

Figure 1

(a) Experimental setup (sketch): an array of 1D polar molecular gases is formed along $x$ (green tubes); molecules are polarized perpendicular to $x$. (b1) Ground state configuration in the solid phase with filling $1/p=1/3$. (b2) Soliton and antisoliton excitations with repulsive interactions with $1/p=1$. (b3) A breather, $1/p=1$ (text).

Figure 2

(a) Commensurate phase diagram for bosons with dipolar interactions $\beta =3$ and lattice depth $U_L=0.1$. Physical configurations correspond to commensurate fillings $n\lambda /2\equiv 1/p$, with $p\in \mathbb{N}$ (horizontal lines are guides to the eye for $p\le 10$). Quantum phase transitions from a TLL to a lattice solid [or Mott insulator (MI)] occur for each $1/p$ at the position of the dots on the continuous line, while gray (red and blue) dots on dashed lines indicate crossovers. MI and M2 indicate MI with solitonic and breather excitations, respectively (see Fig. 1). (b) Phase diagram at commensurate filling $1/p=1/3$ in the $U_L$ vs $n{R}_3$ plane. Continuous line: quantum phase transition between a TLL and a lattice solid. The phase diagram for fermions is identical to the one for bosons, except the TLL is always a CDW.

Figure 3

(a) TLL parameter $K$ vs the dimensionless interaction strength $n^{\beta -2}{R}_{\beta }$ for dipolar interactions $\beta =3$. Line: analytic result Eq. (4). Squares and dots: quantum Monte Carlo (QMC) results of Refs. 7, 9, respectively. Inset: $K$ vs $n^{\beta -2}{R}_{\beta }$ with $\beta =2$. Continuous line: Eq. (4). Dashed line: exact Calogero-Sutherland model. (b) TLL velocity $v$ vs $n{R}_3$ for $\beta =3$, with $v_0\equiv \hslash /(\sqrt{2}m{R}_3)$. Line and dots: analytic and QMC results of Ref. 8, respectively.