Universal geometric entanglement close to quantum phase transitions

Abstract

Under successive Renormalization Group transformations applied to a quantum state $|\Psi ⟩$ of finite correlation length $\xi$, there is typically a loss of entanglement after each iteration. How good it is then to replace $|\Psi ⟩$ 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 $L$.

4 pages, 3 figures

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