Universal geometric entanglement close to quantum phase transitions

Abstract

Under successive Renormalization Group transformations applied to a quantum state $\ketΨ$ of finite correlation length $ξ$, there is typically a loss of entanglement after each iteration. How good it is then to replace $\ketΨ$ 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 $L \gg ξ$ diverges with the correlation length as $(c/12) \log{(ξ/ε)}$ close to a quantum critical point with central charge $c$, where $ε$ 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 $L$.

4 pages, 3 figures