Localization in a $t$-$J$ type ladder with translational symmetry

An explicit $\textit{spatial}$ localization of a hole is shown in a two-leg $t$-$J$ ladder in the presence of a staggered chemical potential, which still retains a translational symmetry, by density matrix renormalization group method. Delocalization can be recovered in the following cases, where either the hidden phase string effect is turned off or a finite next-nearest-neighbor hopping $t'$ is added to sufficiently weaken the phase string effect. In addition, two holes are always delocalized by forming a mobile bound pair, in contrast to the localized single holes, which points to a novel pairing mechanism as one of the essential properties of a doped Mott insulator.