Exploring quantum phase transitions by the cross derivative of the ground state energy

TL;DR

This paper extends the classical Gibbs free energy cross derivative method to quantum systems, demonstrating its effectiveness in identifying Gaussian-type quantum phase transitions in the spin-1 XXZ chain through finite-size analysis.

Contribution

It introduces a novel application of the cross derivative of the Gibbs free energy to quantum phase transitions, providing a new tool for analyzing higher-order transitions.

Findings

Clear valley structures in the cross derivative indicate phase transitions.

Finite-size extrapolation shows logarithmic divergence of valley depth.

Critical points and exponents are accurately estimated, matching literature.

Abstract

In this work, the cross derivative of the Gibbs free energy, initially proposed for phase transitions in classical spin models [Phys. Rev. B 101, 165123 (2020)], is extended for quantum systems. We take the spin-1 XXZ chain with anisotropies as an example to demonstrate its effectiveness and convenience for the Gaussian-type quantum phase transitions therein. These higher-order transitions are very challenging to determine by conventional methods. From the cross derivative with respect to the two anisotropic strengths, a single valley structure is observed clearly in each system size. The finite-size extrapolation of the valley depth shows a perfect logarithmic divergence, signaling the onset of a phase transition. Meanwhile, the critical point and the critical exponent for the correlation length are obtained by a power-law fitting of the valley location in each size. The results are well consistent with the best estimations in the literature. Its application to other quantum systems with continuous phase transitions is also discussed briefly.