Geometric Entanglement in Valance-Bond-Solid state

TL;DR

This paper analyzes the geometric entanglement in valence-bond-solid states, revealing its saturation behavior, divergence in the thermodynamic limit, and scaling with spin, highlighting differences from bipartite entanglement.

Contribution

It provides a simplified method for calculating geometric entanglement in VBS states and explores its scaling and divergence properties.

Findings

GE saturates with increasing particles

GE diverges in the thermodynamic limit

Scaling of GE depends on spin parity

Abstract

Multipartite entanglement, measured by the geometric entanglement(GE), is discussed for integer spin Valance-Bond-Solid (VBS) state respectively with periodic boundary condition(PBC) and open boundary condition(OBC) in this paper. The optimization in the definition of geometric entanglement can be reduced greatly by exploring the symmetry of VBS state, and then the fully separable state can be determined explicitly. Numerical evaluation for GE by the random simulation is also implemented in order to demonstrate the validity of the reductions. Our calculations show that GE is saturated by a finite value with the increment of particle number, that means that the total entanglement for VBS state would be divergent under the thermodynamic limit. Moreover it is found that the scaling behavior of GE with spin number $s$ is fitted as $\alpha\log (s+\tfrac{\beta}{s}+\gamma)+\delta$, in which the values of the parameters $\alpha, \beta, \gamma, \sigma$ are only dependent on the parity of spin $s$. A comparison with entanglement entropy of VBS state is also made, in order to demonstrate the essential differences between multipartite and bipartite entanglement in this model.