Quantum criticality of a $\mathbb{Z}_{3}$ symmetric spin chain with long-range interactions

TL;DR

This study uses large-scale numerical methods to map the phase diagram of a long-range interacting quantum spin chain, revealing a critical threshold and two universality classes, advancing understanding of quantum phase transitions.

Contribution

First non-perturbative numerical determination of the critical long-range interaction threshold in a $ ext{Z}_3$ symmetric spin chain, clarifying universality class distinctions.

Findings

Identified the critical long-range interaction power $ ext{alpha}_c \, \approx 1.43$.

Mapped the phase diagram showing shift of critical points with interaction range.

Established the existence of two universality classes separated by $ ext{alpha}_c$.

Abstract

Based on large-scale density matrix renormalization group techniques, we investigate the critical behaviors of quantum three-state Potts chains with long-range interactions. Using fidelity susceptibility as an indicator, we obtain a complete phase diagram of the system. The results show that as the long-range interaction power $\alpha$ increases, the critical points $f_{c}^{*}$ shift towards lower values. In addition, the critical threshold $\alpha_{c}(\approx 1.43$) of the long-range interaction power is obtained for the first time by a non-perturbative numerical method. This indicates that the critical behavior of the system can be naturally divided into two distinct universality classes, namely the long-range ($\alpha \textless \alpha_c$) and short-range ($\alpha \textgreater \alpha_c$) universality classes, qualitatively consistent with the classical $\phi^{3}$ effective field theory. This work provides a useful reference for further research on phase transitions in quantum spin chains with long-range interaction.