Orbital effect of a magnetic field on two-leg Hubbard ladder

We study the effect of a magnetic field on the dimensional crossover of weakly coupled two-leg Hubbard ladders under pressure. Our model is based on the perturbative renormalization approach (PRG) with two cut-off parameters, the bandwidth E-0 and a characteristic magnetic energy omega(c). We determine the temperature-pressure phase diagram for different values of the magnetic field and discuss the relative stability of the d-wave superconducting phase (SCd) and the two dimensional Fermi liquid phase (2D) which appear at zero field. We show that the field induces a reduction in the effective dimensionality of the system and confines the electron motion within the ladder. In fact, we find that with increasing magnetic field, the isolated ladder phase gets wider at the expense of the SCd phase which disappears at a critical magnetic field H-c. The superconducting transition temperature T-c is found to decrease as the field increases up to H-c for which T-c falls to zero. Concerning the 2D phase, we show that it is destroyed for omega(c) greater than the crossover temperature at which the system crosses to 2D phase at zero magnetic field.