Using the quantum Monte Carlo loop algorithm we have determined the phase diagram of the $1/5$-depleted Heisenberg square lattice model representing Ca${\mathrm{V}}_4$${\mathrm{O}}_9$ as a function of the ratio of the two different couplings: bonds within a plaquette and dimer bonds between plaquettes. At isotropic coupling long range order survives the depletion, but the system is close to the quantum critical point. Already a small frustration can thus drive the system into the quantum disordered phase and explain the spin gap behavior of Ca${\mathrm{V}}_4$${\mathrm{O}}_9$.

• Received 15 November 1995

DOI:https://doi.org/10.1103/PhysRevLett.76.3822