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

TL;DR

This paper compares shared purity and concurrence as indicators of quantum phase transitions across various spin models, demonstrating shared purity's effectiveness and revealing size-dependent divergence in finite-size scaling.

Contribution

It introduces shared purity as a quantum correlation measure capable of detecting phase transitions, showing its effectiveness across different models and system sizes.

Findings

Shared purity effectively indicates quantum phase transitions.

Shared purity detects all transitions in 2D J1-J2 model, unlike concurrence.

Finite-size scaling exponents diverge differently for odd and even systems.

Abstract

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.