Dimer stability region in a frustrated quantum Heisenberg antiferromagnet

We study the stability region for the columnar dimer state proposed as a candidate ground state for the square-lattice quantum antiferromagnet with first- and second-neighbor antiferromagnetic couplings (${\mathit{J}}_1$-${\mathit{J}}_2$ model). We use a boson representation of the spin operators suited to the perturbative expansion around a dimer ground state. At lowest order, the columnar dimer is found to be stable only at the classical critical value ${\mathit{J}}_2$/${\mathit{J}}_2$=1/2. However, we show that the leading anharmonic corrections stabilize the dimerized phase in a region of a finite width around ${\mathit{J}}_2$/${\mathit{J}}_1$=1/2. A comparison of the ground-state energies shows that among the possible dimerized states the columnar dimer is the most favorable candidate to separate the two ordered states in the S=1/2 antiferromagnetic with first- and second-neighbor exchange.