Gapped spin liquid with Z2 topological order for the kagome Heisenberg model

We apply the symmetric tensor network state (TNS) to study the nearest- neighbor spin-1/2 antiferromagnetic Heisenberg model on the kagome lattice. Our method keeps track of the global and gauge symmetries in the TNS update procedure and in tensor renormalization group (TRG) calculations. We also introduce a very sensitive probe for the gap of the ground state-the modular matrices, which can also determine the topological order if the ground state is gapped. We find that the ground state of the Heisenberg model on the kagome lattice is a gapped spin liquid with the Z(2) topological order (or toric code type), which has a long correlation length xi similar to 10 unit cells. We justify that the TRG method can handle very large systems with thousands of spins. Such a long. explains the gapless behaviors observed in simulations on smaller systems with less than 300 spins or shorter than the length of 10 unit cells. We also discuss experimental implications of the topological excitations encoded in our symmetric tensors.