  We consider quantum states under the renormalization-group (RG)
transformations introduced by Verstraete et al. [Phys. Rev. Lett. 94, 140601
(2005)] and propose a quantification of entanglement under such RG (via the
geometric measure of entanglement). We examine the resulting entanglement under
RG for the ground states of "matrix-product-state" (MPS) Hamiltonians
constructed by Wolf et al. [Phys. Rev. Lett. 97, 110403 (2006)] that possess
quantum phase transitions. We find that near critical points, the ground-state
entanglement exhibits singular behavior. The singular behavior within finite
steps of RG obeys a scaling hypothesis and reveals the correlation length
exponent. However, under the infinite steps of RG transformation, the singular
behavior is rendered different and is universal only when there is an
underlying conformal-field-theory description of the critical point.
