Polar molecules in frustrated triangular ladders

Polar molecules in geometrically frustrated lattices may result in a very rich landscape of quantum phases, due to the nontrivial interplay between frustration, and two- and possibly three-body intersite interactions. In this paper we illustrate this intriguing physics for the case of hard-core polar molecules in frustrated triangular ladders. Whereas commensurate lattice fillings result in gapped phases with bond order and/or density-wave order, at incommensurate fillings we find chiral, two-component, and pair superfluids. We show as well that, remarkably, polar molecules in frustrated lattices allow for the observation of bond-ordered supersolids.

Figure 1

Triangular ladder lattice with two- and three-body interactions $V$ and $W$, respectively. The hopping along the rungs and the legs are $t>0$ and $t^{\prime }<0$, respectively.

Figure 2

Phase diagram of model (1) as a function of ${\mu }^{\prime }$ and $V=W$ for $t^{\prime }=-t=-1$. Here ${\mu }^{\prime }=\frac{9+4\mu }{20W+18}$ is the scaled chemical potential such that ${\mu }^{\prime }=0$ (1) corresponds to the vacuum (full) state. Solid (red) curves mark the boundaries of gapped phases, such as BO and CDW phases at $\rho =1/2$ and BO plus CDW phases at $\rho =2/3$. The CSF phase is present for all ${\mu }^{\prime }$ (except for half filling) and weak interactions. The black circles separate the CSF phase from other superfluid phases. The 2SF-SS transition is marked by green circles. The shaded region corresponds to the CSS phase. The PSF phase is a small region appearing in the upper part of the BO plus CDW phase.

Figure 3

Cut through the phase diagram of Fig. 2 for $V=W=2.6$. We depict as a function of $\rho$ the (a) chemical potential $\mu$ ($L=120$), (b) central charge $c$ ($L=180$), (c) entanglement entropy, (d) entanglement gap, (e) chirality $\kappa$, (f) maximum of $S_{\text{CDW}}\left(k\right)$, and (g) maximum of $S_{\text{BO}}\left(k\right)$ [both (f) and (g) are obtained by means of a linear extrapolation to $L\to \infty$ using the two largest system sizes $L=120$ and $180]$. (h) $S_{\text{CDW}}\left(k\right)$ as a function of (quasi)momentum $k$ and filling $\rho$ ($L=120$).

Figure 4

Momentum distribution $n\left(k\right)$ for different densities ($V=W=2.6, L=180$). The single curves have been shifted for clarity. In the 2SF ($\rho =0.33, 0.4, 0.73$, and $0.8$) and the CSF ($\rho =0.26$) phases, $n\left(k\right)$ exhibits two peaks at incommensurate momenta $\pm k$. When approaching the SS phase the momentum distribution broadens but still shows two maxima (for $\rho =0.4, 0.43, 0.6$, and $0.66$), characterizing the ICSS region. However, at intermediate fillings ($\rho =0.53$) only one maximum is observed (CSS region).

Figure 5

Analytical prediction of the phase diagram for filling $\rho \to 0$ as a function of the frustration $J_2/J_1$ and the interaction $\Delta =V=W$. The inset shows the phase boundaries for $J_1^z=3\Delta$ and $J_2^z=\Delta$, which correspond to the particular case of $\rho \to 1$ of model (1).

Figure 6

Phase diagram of model (8) as a function of $J_2/J_1$ and the effective density ${\rho }^{\prime }$.

Figure 7

Cut through the phase diagram of Fig. 2 at large interactions, $V=W=8.0$, and $|t^{\prime }|=t=1$. We depict as a function of $\rho$ the (a) chemical potential $\mu$ ($L=120$), (b) central charge $c$ ($L=180$), (c) entanglement entropy, (d) entanglement gap, (e) maximum of $S_{\text{CDW}}\left(k\right)$, and (f) maximum of $S_{\text{BO}}\left(k\right)$. (g) Structure factor $S_{\text{CDW}}\left(k\right)$ as a function of (quasi)momentum $k$ and $\rho$ ($L=120$).