Diamagnetism of doped two-leg ladders and probing the nature of their commensurate phases

Abstract

We study the magnetic orbital effect of a doped two-leg ladder in the presence of a magnetic field component perpendicular to the ladder plane. Combining both low-energy approach (bosonization) and numerical simulations (density-matrix renormalization group) on the strong coupling limit ($t\text{\ensuremath{-}}J$ model), a rich phase diagram is established as a function of hole doping and magnetic flux. Above a critical flux, the spin gap is destroyed and a Luttinger liquid phase is stabilized. Above a second critical flux, a reentrance of the spin gap at high magnetic flux is found. Interestingly, the phase transitions are associated with a change of sign of the orbital susceptibility. Focusing on the small magnetic field regime, the spin-gapped superconducting phase is robust, but immediately acquires algebraic transverse (i.e., along rungs) current correlations which are commensurate with the $4{k}_{F}$ density correlations. In addition, we have computed the zero-field orbital susceptibility for a large range of doping and interaction ratio $J∕t$: we found strong anomalies at low $J∕t$ only in the vicinity of the commensurate fillings corresponding to $\ensuremath{\delta}=1∕4$ and $1∕2$. Furthermore, the behavior of the orbital susceptibility reveals that the nature of these insulating phases is different: while for $\ensuremath{\delta}=1∕4$ a $4{k}_{F}$ charge density wave is confirmed, the $\ensuremath{\delta}=1∕2$ phase is shown to be a bond order wave.