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 ${(h}_{c1\mathrm{}}<h)$ and the ladder is not yet totally saturated $(h<\mathrm{}{h}_{c2\mathrm{}})$. We compute quantitatively the correlation functions as a function of the magnetic field for an isolated strong-coupling ladder $J_{\perp }\gg J_{\Vert }$ 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 $T_c\mathrm{}\left(h\right)\mathrm{}$ 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 ${h=(h}_{c1\mathrm{}}{+h}_{c2\mathrm{}})/2\mathrm{}$ and behaves as $T_c\sim (h-h_{c1\mathrm{}}{)}^{2/3}$ for $h\sim h_{c1\mathrm{}}$. We discuss the experimental consequences for compounds such as ${\mathrm{Cu}}_2({\mathrm{C}}_5{\mathrm{H}}_{12}{\mathrm{N}}_2{)}_2{\mathrm{Cl}}_4$.