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

TL;DR

This paper uses symmetric tensor network states to demonstrate that the ground state of the Kagome Heisenberg model is a gapped $ ext{Z}_2$ topological spin liquid, revealing long correlation lengths and topological order.

Contribution

The study introduces a symmetry-preserving tensor network approach combined with modular matrices to identify a gapped $ ext{Z}_2$ spin liquid in the Kagome Heisenberg model, with implications for understanding topological phases.

Findings

Ground state is a gapped $ ext{Z}_2$ topological spin liquid.

Correlation length is approximately 10 unit cells.

Method can handle large systems with thousands of spins.

Abstract

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.