Revealing quantum phase string effect in a doped Mott insulator: A tensor network state study

We apply the fermionic tensor network (TN) state method to understand the strongly correlated nature in a doped Mott insulator. We conduct a comparative study of the $\ensuremath{\sigma}t\text{\ensuremath{-}}J$ model, in which the no-double-occupancy constraint remains unchanged but the quantum phase string effect associated with doped holes is precisely switched off. Thus, the ground state of the $\ensuremath{\sigma}t\text{\ensuremath{-}}J$ model can serve as a well-controlled reference state of the standard $t\text{\ensuremath{-}}J$ model. In the absence of the phase string, the spin long-range antiferromagnetic (AFM) order is found to be essentially decoupled from the doped holes, and the latter contribute to a Fermi-liquid-like compressibility and a coherent single-particle propagation with a markedly reduced pairing tendency. In contrast, our TN calculations of the $t\text{\ensuremath{-}}J$ model indicate that the AFM order decreases much faster with doping and the single-particle propagation of doped holes gets substantially suppressed, concurrently with a much stronger charge compressibility at small doping and significantly amplified Cooper pairing tendencies. These findings demonstrate that quantum many-body interference from phase strings plays a pivotal role in the $t\text{\ensuremath{-}}J$ model, mediating long-range entanglement between spin and charge degrees of freedom, which might hold the key mechanism for high-${T}_c$ cuprates.