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Under successive Renormalization Group transformations applied to a quantumstate $\ket{\Psi}$ of finite correlation length $\xi$, there is typically aloss 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 wegive a quantitative answer to this question by providing first analytical andgeneral proofs that, for translationally invariant quantum systems in onespatial 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 $\epsilon$ isa cut-off at short distances. Moreover, the situation at criticality is alsodiscussed and an upper bound on the critical global geometric entanglement isprovided in terms of a logarithmic function of $L$.

Universal geometric entanglement close to quantum phase transitions