Gapped spin liquid with Z2\mathbb{Z}_2-topological order for kagome Heisenberg model

We apply symmetric tensor network state (TNS) to study the nearest neighbor spin-1/2 antiferromagnetic Heisenberg model on Kagome lattice. Our method keeps track of the global and gauge symmetries in TNS update procedure and in tensor renormalization group (TRG) calculation. 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 Heisenberg model on Kagome lattice is a gapped spin liquid with the $\mathbb{Z}_2$-topological order (or toric code type), which has a long correlation length $\xi\sim 10$ unit cell length. We justify that the TRG method can handle very large systems with over thousands of spins. Such a long $\xi$ explains the gapless behaviors observed in simulations on smaller systems with less than 300 spins or shorter than 10 unit cell length. We also discuss experimental implications of the topological excitations encoded in our symmetric tensors.Comment: 10 pages, 7 figure

Anomalous Hall effect in type-I Weyl metals beyond the noncrossing approximation

We study the anomalous Hall effect (AHE) in tilted Weyl metals with Gaussian disorder due to the crossed X and {\Psi} diagrams in this work. The importance of such diagrams to the AHE has been demonstrated recently in 2D massive Dirac model and Rashba ferromagnets. It was shown that the inclusion of such diagrams dramatically changes the total AHE in such systems. In this work, we show that the contributions from the X and {\Psi} diagrams to the AHE in tilted Weyl metals are of the same order of the non-crossing diagram we studied in a previous work, but with opposite sign. The total contribution of the X and {\Psi} diagrams cancels the majority of the contribution from the non-crossing diagram in tilted Weyl metals, similar to the 2D massive Dirac model. We also discussed the difference of the contributions from the crossed diagrams between 2D massive Dirac model and the tilted Weyl metals. At last, we discussed the experimental relevance of observing the AHE due to the X and {\Psi} diagrams in type-I Weyl metal Co3Sn2S2.Comment: This paper is a sequential work of our recently published work PRB 107, 125106(2023