Clustered Wigner-crystal phases of cold polar molecules in arrays of one-dimensional tubes

We analyze theoretically polar molecules confined in planar arrays of one-dimensional tubes. In the classical limit, if the number of tubes is finite, new types of “clustered Wigner crystals” with increasingly many molecules per unit cell can be stabilized by tuning the in-plane angle between the dipolar moments and the tube direction. Quantum mechanically these phases melt into distinct “clustered Luttinger liquids.” We calculate the phase diagram of the system and study the quantum melting of the clustered phases. We find that the requirements for exploring these phases are reachable in current experiments and discuss possible experimental signatures.

Figure 1

(a) Proposed setup to observe cluster formation of polar molecules. (b) The classical phase diagram for two tubes as a function of the tilting angle $\phi$ and the particle density $\rho$. Lobe shaped phases consisting of clusters with $q$ particles per tube emerge. The phase separated regions are indicated by the shaded layer surrounded with speckles. Inset: Optimized cluster configuration in one unit cell for $\phi =1.01$ (upper graph) and $\phi =0.99$ (lower graph) at $\rho =0.7$, corresponding to $q=2$ and $q=3$, respectively.

Figure 2

Phase diagram in the tilting angle $\phi$ and the particle density $\rho$ plane for (a) $n_T=3$, (b) $n_T=4$, and (c) $n_T=8$ tubes. The phase separation regions are indicated by shaded layers.

Figure 3

Particle density $\rho \left(x\right)$ in the lower (red squares) and the upper (blue triangles) tube for dipolar strength $\gamma =8$, left column, and $\gamma =50$, right column, for a system of length $L=24\delta$, density $\rho =0.5{\delta }^{-1}$, and lattice spacing $a=\delta /4$. From top to bottom the tilt angle takes the values $\phi =\{1.02,0.99,0.98,0.97,0.96\}$. For $\gamma =50$ pronounced clusters with $q=\{1,2,3,4,6\}$ are found. The data is obtained by DMRG for systems with open boundary conditions.

Figure 4

Quantum phase diagram for a two tube system of length $L=24\delta$, density $\rho =0.5{\delta }^{-1}$, and lattice spacing $a=\delta /4$, as a function of the tilting angle $\phi$ and the ratio between the kinetic and the interaction energy $1/\gamma$. Right of the asterisks, connected by lines, cluster formation can be observed. To the left of this line, the ground state is a $q=1$ Luttinger liquid.