Topological entanglement entropy in Gutzwiller projected spin liquids

TL;DR

This paper investigates the topological entanglement entropy of Gutzwiller projected spin liquids, revealing conditions for Z2 topological order and emphasizing the importance of wave function sign structure over amplitude.

Contribution

It demonstrates that the sign of the RVB wave function primarily encodes topological entanglement, and clarifies when Z2 topological order appears in Gutzwiller projected states.

Findings

Z2 topological order is absent in the studied spin liquid state from Ref..2.

A clear TEE of .693147... (ln2) is observed in a triangular lattice RVB state.

Sign structure, not amplitude, dominates the topological entanglement entropy.

Abstract

The topological entanglement entropy(TEE) of Gutzwiller projected RVB state is studied with Monte Carlo simulation. New tricks are proposed to improve the convergence of TEE, which enable us to show that the spin liquid state studied in Ref.\cite{Vishwanath} actually does not support $Z_{2}$ topological order, a conclusion that is consistent with the information drawn from the inspection of the topological degeneracy on the same state. We find both a long ranged RVB amplitude and an approximate Marshall sign structure are at the origin of the suppression of vison gap in this spin liquid state. On the other hand, robust signature of $Z_{2}$ topological order, i.e., a TEE of $\ln2$, is clearly demonstrated for a Gutzwiiler projected RVB state on triangular lattice which is evolved from the RVB state proposed originally by Anderson\cite{Sorella}. We also find that it is the sign, rather than the amplitude of the RVB wave function, that dominates the TEE and is responsible for a positive value of TEE, which implies that the nonlocal entanglement in the RVB state is mainly encoded in the sign of the RVB wave function. Our results indicate that some information that is important for the topological property of a RVB state is missed in the effective field theory description and that a $Z_{2}$ gauge structure in the saddle point action is not enough for the RVB state to exhibit topological order, even if its spin correlation is extremely short ranged.