  We investigate the efficacy of shared purity, a measure of quantum
correlation that is independent of the separability-entanglement paradigm, as a
quantum phase transition indicator in comparison with concurrence, a bipartite
entanglement measure. The order parameters are investigated for thermal states,
pseudo-thermal states and more, of the systems considered. In the case of the
one-dimensional $J_1-J_2$ Heisenberg quantum spin model and the one-dimensional
transverse-field quantum Ising model, shared purity turns out to be as
effective as concurrence in indicating quantum phase transitions. In the
two-dimensional $J_1-J_2$ Heisenberg quantum spin model, shared purity
indicates the two quantum phase transitions present in the model, while
concurrence detects only one of them. Moreover, we find diverging finite-size
scaling exponents for the order parameters near the transitions in odd- and
even-sized systems governed by the one-dimensional \(J_1-J_2\) model, as had
previously been reported for quantum spins on odd- and even-legged ladders. It
is plausible that the divergence is related to a M{\"o}bius strip-like boundary
condition required for odd-sized systems, while for even-sized systems, the
usual periodic boundary condition is sufficient.
