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We consider a two-leg Bose-Hubbard ladder in the presence of a magnetic flux.We make use of Gross-Pitaevskii, Bogoliubov, bosonization, and renormalizationgroup approaches to reveal a structure of ground-state phase diagrams in aweak-coupling regime relevant to cold atom experiments. It is found that exceptfor a certain flux $\phi=\pi$, the system shows different properties aschanging hoppings, which also leads to a quantum phase transition similar tothe ferromagnetic XXZ model. This implies that population-imbalance instabilityoccurs for certain parameter regimes. On the other hand, for $\phi=\pi$, it isshown that an umklapp process caused by commensurability of a magnetic fluxstabilizes a superfluid with chirality and the system does not experience sucha phase transition.

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