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At sufficiently low temperatures, interacting electron systems tend todevelop orders. Exceptions are quantum critical point (QCP) and quantum spinliquid (QSL), where fluctuations prevent the highly entangled quantum matter toan ordered state down to the lowest temperature. While the ramification ofthese states may have appeared in high-temperature superconductors, ultra-coldatoms, frustrated magnets and quantum Moir\'e materials, their unbiasedpresence remain elusive in microscopic two-dimensional lattice models. Here, weshow by means of large-scale quantum Monte Carlo simulations of correlatedelectrons on the $\pi$-flux square lattice subjected to extended Hubbardinteraction, that a Gross-Neveu QCP separating massless Dirac fermions and ancolumnar valence bond solid at finite interaction, and a possible Dirac QSL atthe infinite yet tractable interaction limit emerge in a coherent sequence.These unexpected novel quantum states resides in this simple-looking model,unifying ingredients including emergent symmetry, deconfined fractionalizationand the dynamic coupling between emergent matter and gauge fields, will haveprofound implications both in quantum many-body theory and understanding of theaforementioned experimental systems.

Dirac fermions with plaquette interactions. II. SU(4) phase diagram with Gross-Neveu criticality and quantum spin liquid