Direct evidence for a gapless $Z_2$ spin liquid by frustrating N\'eel antiferromagnetism

By direct calculations of the spin gap in the frustrated Heisenberg model on the square lattice, with nearest- ($J_1$) and next-nearest-neighbor ($J_2$) super-exchange couplings, we provide a solid evidence that the spin-liquid phase in the frustrated regime $0.45 \lesssim J_2/J_1 \lesssim 0.6$ is gapless. Our numerical method is based on a variational wave function that is {\it systematically} improved by the application of few Lanczos steps and allows us to obtain reliable extrapolations in the thermodynamic limit. The peculiar nature of the non-magnetic state is unveiled by the existence of S=1 gapless excitations at $k=(\pi,0)$ and $(0,\pi)$. The magnetic transition can be described and interpreted by a variational state that is built from Abrikosov fermions having a $Z_2$ gauge structure and four Dirac points in the spinon spectrum.