We apply the generalized Wigner function formalism to detect and characterize
a range of quantum phase transitions in several cyclic, finite-length,
spin-21 one-dimensional spin-chain models, viz., the Ising and
anisotropic XY models in a transverse field, and the XXZ anisotropic
Heisenberg model. We make use of the finite system size to provide an
exhaustive exploration of each system's single-site, bipartite and
multi-partite correlation functions. In turn, we are able to demonstrate the
utility of phase-space techniques in witnessing and characterizing first-,
second- and infinite-order quantum phase transitions, while also enabling an
in-depth analysis of the correlations present within critical systems. We also
highlight the method's ability to capture other features of spin systems such
as ground-state factorization and critical system scaling. Finally, we
demonstrate the generalized Wigner function's utility for state verification by
determining the state of each system and their constituent sub-systems at
points of interest across the quantum phase transitions, enabling interesting
features of critical systems to be intuitively analyzed.Comment: 20 pages, 8 figure