Vortex-Hole Duality: A Unified Picture of Weak- and Strong-Coupling Regimes of Bosonic Ladders with Flux

Two-leg bosonic ladders with flux harbor a remarkable vortex-hole duality between the weak-coupling vortex lattice superfluids and strong-coupling charge-density-wave crystals. The strong-coupling crystalline states, which are realized in the vicinity of $\pi$ flux, are independent of particle statistics, and are related to the incompressible fractional quantum Hall states in the thin-cylinder limit. These fully gapped ground states, away from $\pi$ flux, develop nonzero chiral (spin) currents. Contact-interacting quantum gases permit exploration of this vortex-hole duality in experiments.

Figure 1

Microscopic structures of vortex-hole dual configurations of weak (a), (c), (e) and strong-coupling (b), (d), (f) ground states of bosonic ladders with flux. Dual configurations are (a) ${\rho }_V=1/2$ vs (b) ${\rho }_H=1/2$, (c) ${\rho }_V=1/3$ vs (d) ${\rho }_H=1/3$, and (e) ${\rho }_V=1/4$ vs (f) ${\rho }_H=1/4$. Note, that in (a) particle densities are uniform along the ladder, and in (b) particle currents do not show modulations. In contrast, in (c) and (e) particle densities show modulations, similar to particle currents in (d) and (f).

Figure 2

The dependence of vortex density on flux ${\rho }_V(\varphi )$ in the weak-coupling classical JJ limit and the hole density dependence on chemical potential ${\rho }_H(\mu )$ in the strong-coupling $U=\infty$, $\varphi =\pi$ limit [32] due to vortex-hole duality exhibit a remarkable similarity. With doubling $J_{\perp }/J$ from (a) to (b) fewer plateaus are formed in ${\rho }_V(\varphi )$ and ${\rho }_H(\mu )$ curves.

Figure 3

Ground state phase diagram in the parameter space of $\mu$ and $J_{\perp }$ for $U=\infty$ and $\varphi =\pi$. For clarity only the most stable fractional CDW phases are shown. Inset shows the equation of state $\rho =\rho (\mu )$ of the $U=\infty$ ladder at $\pi$ flux for (from left to right) $J_{\perp }=J$, $J/2$ and 0.

Figure 4

Ground state phases in parameter space of flux and density for $U=\infty$ and $J_{\perp }=J$. Dots indicate CDW states at rational values of $\rho >1/2$ at $\varphi =\pi$ [53]. Inset shows corresponding phase diagram for $J\ll J_{\perp }$ when number of legs $M\to \infty$.