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We investigate the efficacy of shared purity, a measure of quantumcorrelation that is independent of the separability-entanglement paradigm, as aquantum phase transition indicator in comparison with concurrence, a bipartiteentanglement measure. The order parameters are investigated for thermal states,pseudo-thermal states and more, of the systems considered. In the case of theone-dimensional $J_1-J_2$ Heisenberg quantum spin model and the one-dimensionaltransverse-field quantum Ising model, shared purity turns out to be aseffective as concurrence in indicating quantum phase transitions. In thetwo-dimensional $J_1-J_2$ Heisenberg quantum spin model, shared purityindicates the two quantum phase transitions present in the model, whileconcurrence detects only one of them. Moreover, we find diverging finite-sizescaling exponents for the order parameters near the transitions in odd- andeven-sized systems governed by the one-dimensional $J_1-J_2$ model, as hadpreviously been reported for quantum spins on odd- and even-legged ladders. Itis plausible that the divergence is related to a M{\"o}bius strip-like boundarycondition required for odd-sized systems, while for even-sized systems, theusual periodic boundary condition is sufficient.

Shared purity and concurrence of a mixture of ground and low-lying excited states as indicators of quantum phase transitions