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An Entanglement-Complexity Generalization of the Geometric Entanglement

We propose a class of generalizations of the geometric entanglement for pure states by exploiting the matrix product state formalism. This generalization is completely divested from the notion of separability and can be freely tuned as a function of the bond dimension to target states which vary in entanglement complexity. We first demonstrate its value in a toy spin-1 model where, unlike the conventional geometric entanglement, it successfully identifies the AKLT ground state. We then investigate the phase diagram of a Haldane chain with uniaxial and rhombic anisotropies, revealing that the generalized geometric entanglement can successfully detect all its phases and their entanglement complexity. Finally we investigate the disordered spin-$1/2$ Heisenberg model, where we find that differences in generalized geometric entanglements can be used as lucrative signatures of the ergodic-localized entanglement transition.