Population-imbalance instability in a Bose-Hubbard ladder in the presence of a magnetic flux

We consider a two-leg Bose-Hubbard ladder in the presence of a magnetic flux. We make use of Gross-Pitaevskii, Bogoliubov, bosonization, and renormalization group approaches to reveal a structure of ground-state phase diagrams in a weak-coupling regime relevant to cold atom experiments. It is found that, except for a certain flux $\varphi =\pi$, the system shows different properties as changing hoppings, which also leads to a quantum phase transition similar to the ferromagnetic $XXZ$ model. This implies that population-imbalance instability occurs for certain parameter regimes. On the other hand, for $\varphi =\pi$, it is shown that an umklapp process caused by commensurability of a magnetic flux stabilizes a superfluid with chirality and the system does not experience such a phase transition.

Figure 1

A schematic figure of a two-leg Bose-Hubbard model with a flux $\varphi$. In the gauge chosen here, the flux effect appears only in rung hoppings.

Figure 2

Changes of topology in the lower band at a certain $\varphi \ne \pi$. The direction of the arrows means the reduction of $K/J$. In a strong-enough $K/J$, the band has a single minimum, while in the opposite limit, the band of double-well structure forms. In between there exists a critical point in which the band bottom becomes quartic in $k$. The double-well structure is always maintained at $\varphi =\pi$ regardless of values of $K/J$.

Figure 3

The absolute value of density difference at $\varphi =\frac{\pi }{2}$ in the biased-ladder phase. At the boundary between biased-ladder ($m\ne 0$) and Meissner phases ($m=0$), the density difference disappears. On the other hand, at the boundary between incommensurate vortex and biased-ladder phases, an infinite number of degeneracy in density difference emerges due to the emergent symmetry as shown in Sec. 4a1.

Figure 4

Schematic behavior of the Bogoliubov spectrum predicted by Eq. (68). The excitation is linear in small $k$, which should be interpreted as the TLL spectrum. On the other hand, a local minimum exists in the vicinity of $k=-Q$ originating from the $Z_2$ symmetry breaking.

Figure 5

Schematic phase diagram (a) for $Q\ne \pi /2$ and (b) for $Q=\pi /2$. An emergent $\text{SU}\left(2\right)$ symmetry shows up at the boundary between incommensurate vortex and biased-ladder phases, which is marked with the red circle in (a).