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Motivated by experiments exploring the physics of neutral atoms in artificialmagnetic fields, we study the ground state of bosons interacting with longrange dipolar interactions on a two-leg ladder. Using two complimentaryvariational approaches, valid for weak interactions, we find rich physicsdriven by the long range forces. Generically, long range interactions tend todestroy the Meissner phase in favor of modulated density wave phases. Nearestneighbor interactions produce a novel interleg charge density wave phase, wherethe total density remains uniform, but the density on each leg of the ladder ismodulating in space, out-of-phase with one another. At weak magnetic fields,next nearest neighbor interactions lead to a fully modulated biased ladderphase, where all the particles are on one leg of the ladder, and the density ismodulating in space. This state simultaneously breaks $Z_{2}$ reflectionsymmetry and $U(1)$ symmetry associated with translation in real space. Forvalues of the flux near $\phi = \pi$, we find that a switching effect occursfor arbitrarily weak interactions, where the density modulates in space, butthe chiral current changes sign on every plaquette. Arbitrarily weak attractiveinteractions along the rungs destroy the Meissner phase completely, in favor ofa modulated density wave phase. Varying magnetic field produces a cascade offirst order transitions between modulated density wave states with differentwave-vectors, which manifests itself as discrete jumps in the chiral current.Polarizing the dipoles along the ladder direction yields a region of phasespace where a stable biased ladder phase occurs even at arbitrarily weakmagnetic fields. We discuss the experimental consequences of our work, inparticular, how the interleg charge density wave can manifest itself in recentexperiments on bosons in synthetic dimensions.

Bosons with long range interactions on two-leg ladders in artificial magnetic fields