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We apply the generalized Wigner function formalism to detect and characterizea range of quantum phase transitions in several cyclic, finite-length,spin-$\frac{1}{2}$ one-dimensional spin-chain models, viz., the Ising andanisotropic $XY$ models in a transverse field, and the $XXZ$ anisotropicHeisenberg model. We make use of the finite system size to provide anexhaustive exploration of each system's single-site, bipartite andmulti-partite correlation functions. In turn, we are able to demonstrate theutility of phase-space techniques in witnessing and characterizing first-,second- and infinite-order quantum phase transitions, while also enabling anin-depth analysis of the correlations present within critical systems. We alsohighlight the method's ability to capture other features of spin systems suchas ground-state factorization and critical system scaling. Finally, wedemonstrate the generalized Wigner function's utility for state verification bydetermining the state of each system and their constituent sub-systems atpoints of interest across the quantum phase transitions, enabling interestingfeatures of critical systems to be intuitively analyzed.

Generalized Phase-Space Techniques to Explore Quantum Phase Transitions in Critical Quantum Spin Systems