Universal geometric entanglement close to quantum phase transitions

TL;DR

This paper analytically investigates how the global geometric entanglement in one-dimensional quantum systems behaves near quantum phase transitions, revealing divergence patterns related to correlation length and criticality.

Contribution

It provides the first analytical proofs of the divergence of geometric entanglement near quantum critical points in one-dimensional systems.

Findings

Global geometric entanglement per region diverges as (c/12) log(ξ/ε) near criticality

At criticality, an upper bound on entanglement is established as a logarithmic function of region size L

Results connect entanglement measures with conformal field theory parameters like central charge c

Abstract

Under successive Renormalization Group transformations applied to a quantum state $\ket{\Psi}$ of finite correlation length $\xi$, there is typically a loss of entanglement after each iteration. How good it is then to replace $\ket{\Psi}$ by a product state at every step of the process? In this paper we give a quantitative answer to this question by providing first analytical and general proofs that, for translationally invariant quantum systems in one spatial dimension, the global geometric entanglement per region of size $L \gg \xi$ diverges with the correlation length as $(c/12) \log{(\xi/\epsilon)}$ close to a quantum critical point with central charge $c$, where $\epsilon$ is a cut-off at short distances. Moreover, the situation at criticality is also discussed and an upper bound on the critical global geometric entanglement is provided in terms of a logarithmic function of $L$.