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

TL;DR

This paper demonstrates how generalized phase-space techniques, specifically the Wigner function formalism, can effectively detect, characterize, and analyze quantum phase transitions and correlations in finite spin-chain models.

Contribution

It introduces the application of generalized Wigner functions to explore various quantum phase transitions and correlations in spin systems, providing a comprehensive phase-space analysis tool.

Findings

Successfully detects first, second, and infinite-order phase transitions.

Analyzes correlations and features like ground-state factorization.

Enables intuitive visualization of critical system properties.

Abstract

We apply the generalized Wigner function formalism to detect and characterize a range of quantum phase transitions in several cyclic, finite-length, spin-$\frac{1}{2}$ 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.