Charge-spin mutual entanglement: A case study by exact diagonalization of the one hole doped $t$-$J$ loop

A doped Mott insulator exhibits peculiar properties associated with its singular sign structure. As a case study, we investigate the ground state and excitations of finite-size Heisenberg loops doped with one hole by exact diagonalization. We find that there appear a series of quantum critical points (QCPs), which separate regimes by distinct total momenta along the axis of the ratio $J/t$ ($J$ and $t$ denote the superexchange coupling and hopping integral, respectively). Each QCP involves a crystal momentum jump with level crossing or merging of lowest energy levels. In contrast to the conserved total momentum, however, a broad momentum distribution of \emph{individual} electrons is also found, indicating charge incoherence/translational symmetry breaking in violation of the one-to-one correspondence. Such a charge incoherence is further related to quantum fluctuations or the transverse part of ${\bf S}^2=3/4$ with $S^z=\pm 1/2$ in the one-hole ground state. Turning off the phase-string sign structure, by contrast, we show that the total momentum of the ground state reduces to null in the whole regime of $J/t$ with no more QCP or incoherence. We introduce the so-called charge-spin mutual entanglement to characterize these novel properties, with the entanglement spectrum providing additional information on the charge incoherence, which capture the nature of strong correlation due to the many-body quantum interference.