Direct evidence for a gapless $Z_2$ spin liquid by frustrating Néel 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 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 nonmagnetic 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.

Figure 1

Energies per site for (a) the $S=0$ ground state, (b) the $S=1$ state with $k=(\pi ,0)$, and (c) the $S=2$ with $k=(0,0)$ versus the variance for $J_2/J_1=0.5$. The results with $p=0$, 1, and 2 are reported for $L=6$, 8, and 10, and with only $p=0$ and $p=1$ for $L=14$ and $L=18$. The variance extrapolated results are also shown.

Figure 2

Spin gap for the $S=1$ excitation at $k=(\pi ,0)$ (left panel) and the $S=2$ excitation with $k=(0,0)$ (right panel) for the $6\times 6$ cluster. Results for $p=0$, 1, and 2 Lanczos steps are reported, together with the extrapolated and the exact ones.

Figure 3

The $S=2$ spin gap as a function of the system size for the variational wave function and the Lanczos extrapolation for two values of $J_2$. The thermodynamic extrapolations are consistent with a vanishing gap within the error bars, i.e., ${\Delta }_2=-0.04\left(5\right)$ and $-0.07\left(7\right)$ for $J_2/J_1=0.5$ and $0.55$, respectively. The DMRG results on $2L\times L$ cylinders (with open boundary conditions along $x$ and periodic along $y$) for the $S=1$ excitation are also shown.[13] Exact results (stars) of the $S=2$ gap and the lowest $S=1$ gap on the $6\times 6$ cluster (with periodic boundary conditions) are reported.

Figure 4

The $S=1$ with $k=(\pi ,0)$ spin gap as a function of the system size for the variational wave function and the Lanczos extrapolation for two values of $J_2$ (upper panel). The behavior of the extrapolated gap as a function of $J_2/J_1$ is reported in the lower panel, the line is a guide to the eye. The Lanczos extrapolated gap as a function of $L$ for different values of $J_2$ are also reported in the inset.