Diagnosing $SO(5)$ Symmetry and First-Order Transition in the $J-Q_3$ Model via Entanglement Entropy

TL;DR

This paper uses entanglement entropy scaling to identify an emergent SO(5) symmetry and determine the nature of the phase transition in the $J-Q_3$ model, revealing it as a weak first-order transition.

Contribution

It introduces a novel entanglement entropy analysis method to detect symmetry breaking and characterize phase transitions, especially weak first-order ones.

Findings

Identifies four Goldstone modes indicating SO(5) symmetry breaking.

Shows the transition is a weak first-order, not a deconfined quantum critical point.

Provides a new approach to distinguish broken symmetry states from critical states.

Abstract

We study the scaling behavior of the R\'enyi entanglement entropy with smooth boundaries at the phase transition point of the two-dimensional $J-Q_3$ model. Using the recently developed scaling formula [Deng {\it et al.}, Phys. Rev. B {\textbf{108}, 125144 (2023)}], we find a subleading logarithmic term with a coefficient showing that the number of Goldstone modes is four, indicating the existence of the spontaneous symmetry breaking from an emergent $SO(5)$ to $O(4)$ in the thermodynamic limit, but restored in a finite size. This result shows that the believed deconfined quantum critical point of the $J-Q_{3}$ model is a weak first-order transition point. Our work provides a new way to distinguish a state with spontaneously broken continuous symmetry from a critical state. The method is particularly useful in identifying weak first-order phase transitions, which are hard to determine using conventional methods.