Geometric entanglement and quantum phase transitions in two-dimensional quantum lattice models

TL;DR

This paper introduces a systematic method to compute geometric entanglement in 2D quantum lattice models and uses it to analyze phase transitions, revealing complex behaviors of entanglement across different models.

Contribution

It develops a new computational approach for geometric entanglement in 2D models and classifies various types of entanglement behavior at phase transitions.

Findings

GE can be continuous with maximum at the transition point in some models.

Different types of GE behavior are linked to dual symmetry and polarized phases.

The method successfully analyzes multiple 2D quantum models and their phase transitions.

Abstract

Geometric entanglement(GE), as a measure of multipartite entanglement, has been investigated as a universal tool to detect phase transitions in quantum many-body lattice models. We outline a systematic method to compute GE for two-dimensional (2D) quantum many-body lattice models based on the translational invariant structure of infinite projected entangled pair state (iPEPS) representations. By employing this method, the $q$-state quantum Potts model on the square lattice with $q \in \{2, 3 ,4, 5\}$ is investigated as a prototypical example. Further, we have explored three 2D Heisenberg models, such as the spin-$\frac{1}{2}$ XXX and antiferromagnetic anisotropic XYX models in an external magnetic field, and the spin-1 antiferromagnetic XXZ model. We find that continuous GE does not guarantee a continuous phase transition across a phase transition point. We observe and thus classify three different types of continuous GE across a phase transition point: (i) GE is continuous with maximum value at the transition point and the phase transition is continuous, (ii) GE is continuous with maximum value at the transition point but the phase transition is discontinuous, and (iii) GE is continuous with non-maximum value at the transition point and the phase transition is continuous. For the models under consideration we find that the second and the third types are related to a point of dual symmetry and a fully polarized phase, respectively.