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 ${J}_{\ensuremath{\perp}}\ensuremath{\gg}{J}_{\ensuremath{\Vert}}$ 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 ${T}_{c}(h)$ 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 ${T}_{c}\ensuremath{\sim}(h\ensuremath{-}{h}_{c1}{)}^{2/3}$ for $h\ensuremath{\sim}{h}_{c1}$. We discuss the experimental consequences for compounds such as ${\mathrm{Cu}}_{2}({\mathrm{C}}_{5}{\mathrm{H}}_{12}{\mathrm{N}}_{2}{)}_{2}{\mathrm{Cl}}_{4}$.

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