Spin-gap phase of a quantum spin system on a honeycomb lattice

We study a quantum spin system on a honeycomb lattice, when it includes frustration and distortion in antiferromagnetic (AF) exchange interactions. We transform the spin system onto a nonlinear $\sigma$ model (NLSM), preserving the original spin degrees of freedom. Assisted by a renormalization-group argument, the NLSM provides a ground-state phase diagram consisting of an ordered AF phase and a disordered spin-gap phase. The spin-gap phase extends from a strong frustration regime to a strong distortion regime, showing that the disordered ground states are essentially the same in both regimes. In the spin-1/2 case, the spin-gap phase for spin system on a honeycomb lattice is larger than that for the $J_1\text{−}{J}_2$ model on a square lattice.