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One dimensional free-fermions and hard-core bosons are often considered to beequivalent. Indeed, when restricted to nearest-neighbor hopping on a chain theparticles cannot exchange themselves, and therefore hardly experience their ownstatistics. Apart from the off-diagonal correlations which depends on theso-called Jordan-Wigner string, real-space observables are similar forfree-fermions and hard-core bosons on a chain. Interestingly, by coupling onlytwo chains, thus forming a two-leg ladder, particle exchange becomes allowed,and leads to a totally different physics between free-fermions and hard-corebosons. Using a combination of analytical (strong coupling, field theory,renormalization group) and numerical (quantum Monte Carlo, density-matrixrenormalization group) approaches, we study the apparently simple butnon-trivial model of hard-core bosons hopping in a two-leg ladder geometry. Athalf-filling, while a band insulator appears for fermions at large interchainhopping tperp >2t only, a Mott gap opens up for bosons as soon as tperp\neq0through a Kosterlitz-Thouless transition. Away from half-filling, the situationis even more interesting since a gapless Luttinger liquid mode emerges in thesymmetric sector with a non-trivial filling-dependent Luttinger parameter1/2\leq Ks \leq 1. Consequences for experiments in cold atoms, spin ladders ina magnetic field, as well as disorder effects are discussed. In particular, aquantum phase transition is expected at finite disorder strength between a 1Dsuperfluid and an insulating Bose glass phase.

Phase diagram of hard-core bosons on clean and disordered 2-leg ladders: Mott insulator - Luttinger liquid - Bose glass