Z$_{2}$ spin liquid phase on the kagome lattice: a new saddle point
We have performed large scale variational search for the best RVB ansatz for the spin-$\frac{1}{2}$ kagome antiferromagnet with both nearest-neighboring(NN) and next-nearest-neighboring(NNN) exchanges assuming only translational symmetry. We find the best RVB state is always fully symmetric and has an mean field ansatz that is gauge equivalent to a previous proposed Z$_{2}$ spin liquid ansatz. The Z$_{2}$ state is found to be slightly more stable than the extensively studied U(1) gapless Dirac spin liquid state in both the $J_{2}\neq0$ and the $J_{2}=0$ case and to possess a small spinon gap for $J_{2}<0.2$. The breaking of the U(1) gauge symmetry in the Z$_{2}$ state is found to increase with $J_{2}$ and to be quite substantial around $J_{2}=0.15$. However, we find the Z$_{2}$ state is always very close to the gapless U(1) Dirac spin liquid state, although they have very different RVB parameters. We argue the kagome antiferromagnet should be better understood as a near critical system, rather than a system deep inside a gapped spin liquid phase with well established Z$_{2}$ topological order.