Vanishing spectral weight in a one-hole-doped antiferromagnet: A rigorous result

By explicitly tracking the Marshall sign, a phase string induced by hopping is revealed in the one-hole-doped $t-J$ model. It is rigorously shown that such a phase string cannot be eliminated through low-lying spin excitations, and it either causes the spectral weight $Z=0$, or leads to the localization of the doped hole. Implications for finite doping are also discussed.