Coupled ladders in a magnetic field

We investigate the phase transitions in two-legs ladder systems in the incommensurate phase, for which the gap is destroyed by a magnetic field (h(c1)<h) and the ladder is not yet totally saturated (h<h(c2)). We compute quantitatively the correlation functions as a function of the magnetic field for an isolated strong-coupling ladder J(perpendicular to)much greater than J(parallel to) and use it to study the phase transition occurring in a three-dimensional array of antiferromagnetically coupled ladders. The three-dimensional ordering is in the universality class of Bose condensation of hard-core bosons. We compute the critical temperature T-c(h) as well as various physical quantities such as the NMR relaxations rate. T-c has an unusual camel-like shape with a local minimum at h=(h(c1)+h(c2))/2 and behaves as T(c)similar to(h-h(c1))(2/3) for h similar to h(c1). We discuss the, experimental consequences for compounds such as Cu-2(C5H12N2)(2)Cl-4 [S0163-1829(99)06417-6].