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Injecting a single hole into a one-dimensional Heisenberg spin chain isprobably the simplest case of doping a Mott insulator. The motion of such asingle hole will generally induce a many-body phase shift, which can beidentified by an exact sign structure of the model known as the phase string.We show that the sign structure is nontrivial even in this simplest problem,which is responsible for the essential properties of Mott physics. We find thatthe characteristic momentum structure, the Luttinger liquid behavior, and thequantum phase interference of the hole under a periodic boundary condition, canall be attributed to it. We use the density matrix renormalization group (DMRG)numerical simulation to make a comparative study of the $t$-$J$ chain and amodel in which the sign structure is switched off. We further show that the keyDMRG results can be reproduced by a variational wave function withincorporating the correct sign structure. Physical implications of the signstructure for doped Mott insulators in general are also discussed.

Exact sign structure of the $t$-$J$ chain and the single hole ground state