Quantum Phase Transitions detected by a local probe using Time Correlations and Violations of Leggett-Garg Inequalities

TL;DR

This paper presents a method to identify quantum phase transitions in many-body systems using local time correlations and Leggett-Garg inequalities, enabling detection of both finite and infinite order transitions through local measurements.

Contribution

The authors introduce a novel approach using local time correlations and Leggett-Garg inequalities to detect quantum phase transitions, including those of infinite order like Kosterlitz-Thouless.

Findings

Finite-order transitions show singularities in time correlations at criticality.

Leggett-Garg inequalities are violated near quantum critical points.

Infinite-order transition exhibits maximal violation of Leggett-Garg inequalities.

Abstract

In the present paper we introduce a way of identifying quantum phase transitions of many-body systems by means of local time correlations and Leggett-Garg inequalities. This procedure allows to experimentally determine the quantum critical points not only of finite-order transitions but also those of infinite order, as the Kosterlitz-Thouless transition that is not always easy to detect with current methods. By means of simple analytical arguments for a general spin-$1 / 2$ Hamiltonian, and matrix product simulations of one-dimensional $X X Z$ and anisotropic $X Y$ models, we argue that finite-order quantum phase transitions can be determined by singularities of the time correlations or their derivatives at criticality. The same features are exhibited by corresponding Leggett-Garg functions, which noticeably indicate violation of the Leggett-Garg inequalities for early times and all the Hamiltonian parameters considered. In addition, we find that the infinite-order transition of the $X X Z$ model at the isotropic point can be revealed by the maximal violation of the Leggett-Garg inequalities. We thus show that quantum phase transitions can be identified by purely local measurements, and that many-body systems constitute important candidates to observe experimentally the violation of Leggett-Garg inequalities.