# Discrete and generalized phase space techniques in critical quantum spin chains

## TL;DR

This paper introduces phase space techniques using Wigner functions to detect and distinguish various quantum phase transitions in critical spin-1/2 chains, linking them to thermodynamic properties and quantum correlations.

## Contribution

It develops a general formula connecting phase space methods with thermodynamic quantities and applies it to spin models, enabling detection of different quantum phase transitions.

## Key findings

- Phase space techniques can detect first, second, and infinite-order phase transitions.
- Factorization phenomena in the XY model are detectable via the square root of the reduced density matrix.
- Phase space methods are experimentally promising for studying many-body quantum systems.

## Abstract

We apply the Wigner function formalism from quantum optics via two approaches, Wootters' discrete Wigner function and the generalized Wigner function, to detect quantum phase transitions in critical spin-$\tfrac{1}{2}$ systems. We develop a general formula relating the phase space techniques and the thermodynamical quantities of spin models, which we apply to single, bipartite and multi-partite systems governed by the $XY$ and the $XXZ$ models. Our approach allows us to introduce a novel way to represent, detect, and distinguish first-, second- and infinite-order quantum phase transitions. Furthermore, we show that the factorization phenomena of the $XY$ model is only directly detectable by quantities based on the square root of the bipartite reduced density matrix. We establish that phase space techniques provide a simple, experimentally promising tool in the study of many-body systems and we discuss their relation with measures of quantum correlations and quantum coherence.