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Quantum phase transitions from the cluster-charge interaction, which iscomposed of competing short- and long-range interactions, are investigated on a$\pi$-flux lattice by using the mean-field theory and determinant quantum MonteCarlo (DQMC) simulations. Both methods identify a plaquette-dimer phase, whichdevelops from a finite interaction strength. While its signature in DQMC isrelatively weak, a obvious antiferromagnetic transition is revealed in the spinstructure factor instead. The corresponding critical interaction and exponentsare readily obtained by finite-size scalings, with the plaquette-dimerstructure factor that can also be well scaled. These results suggest a possibledeconfined quantum critical point between the plaquette-dimer andantiferromagnetic phases driven by the cluster-charge interaction on a$\pi$-flux lattice.

Quantum phase transitions from competing short- and long-range interactions on a $\pi$-flux lattice