Diagnosing the Non-Abelian Kitaev Quantum Spin Liquid via Modular Commutator

We argue that a non-abelian quantum spin liquid phase can be probed by numerical calculations of modular commutator. Modular commutator is a new measure of quantum entanglement that is closely related with the chiral central charge [1,2], which determines the boundary energy flow and quantized thermal Hall effect of a non-abelian quantum spin liquid. We compute the modular commutator for the Kitaev’s chiral spin liquid by employing exact diagonalization. Our calculations show that the modular commutator is half-quantized in the phase of Kitaev's chiral spin liquid as expected from the chiral central charge. We point out subtleties with exact diagonalization on a 24-site cluster in connection with topological degeneracy on a torus geometry and discuss how to resolve the subtleties by using symmetries. Our results suggest that the modular commutator is a useful probe for non-abelian quantum spin liquids.

[1] "Chiral central charge from a single bulk wave function", Isaac H. Kim, Bowen Shi, Kohtaro Kato, and Victor V. Albert, Phys. Rev. Lett. 128, 176402 (2022).

[2] "From entanglement generated dynamics to the gravitational anomaly and chiral central charge", Ruihua Fan, arXiv:2206.02823 (2022).