We consider a two-leg boson ladder in an artiﬁcial U(1) gauge ﬁeld and study its phase diagram. In absence of interchain interaction we find a transition from commensurate Meissner to incommensurate Vortex state at increasing interchain hopping till a critical value above which the Meissner state is stable at any ﬂux[1,2]. For ﬂux close to π, and below the critical hopping, we observe the formation of a second incommensuration in the Mott Vortex state[1,2] that could be detectable in current experiments. We then show that, in the presence of interleg attractive interaction, the ﬂux induced Vortex state can be melted by dislocations. For increasing ﬂux, instead of the Meissner to Vortex transition in the commensurate-incommensurate universality class, ﬁrst an Ising transition from the Meissner state to a charge density wave takes place, then, at higher ﬂux, the melted Vortex phase is established via a disorder point where incommensuration develops in the rung current correlation function and in momentum distribution. Finally, the quasi-long range ordered Vortex phase is recovered for suﬃciently small interaction. Our predictions for the observables, such as the spin current and the static structure factor, could be tested in current experiments with cold atoms in bosonic ladders.
 M. Di Dio, S. De Palo, E. Orignac, R. Citro, and M.-L. Chiofalo, Phys. Rev. B 92, 060506 (2015)
 E. Orignac, R. Citro, M. Di Dio, S. De Palo, and M. L. Chiofalo, New J. Phys. 18, 055017 (2016)
 E. Orignac, R. Citro, M. Di Dio and S. De Palo, arxiv 1703.07742 (2017)