Identifying Symmetry-Protected Topological Order by Entanglement Entropy

TL;DR

This paper introduces a method using entanglement entropy and non-Abelian symmetries within DMRG to identify and distinguish symmetry-protected topological phases in spin-1 chains and ladders, confirming phase transitions and critical points.

Contribution

It develops a novel approach to detect SPT order by calculating an entanglement gap from multiplet entanglement spectra, leveraging symmetry implementation in DMRG.

Findings

Successfully distinguishes SPT and trivial phases via entanglement gap.

Confirms SPT phase existence in spin-1 tube models.

Identifies continuous phase transition with central charge c=3.

Abstract

According to the classification using projective representations of the SO(3) group, there exist two topologically distinct gapped phases in spin-1 chains. The symmetry-protected topological (SPT) phase possesses half-integer projective representations of the SO(3) group, while the trivial phase possesses integer linear representations. In the present work, we implement non-Abelian symmetries in the density matrix renormalization group (DMRG) method, allowing us to keep track of (and also control) the virtual bond representations, and to readily distinguish the SPT phase from the trivial one by evaluating the multiplet entanglement spectrum. In particular, using the entropies $S^I$ ($S^H$) of integer (half-integer) representations, we can define an entanglement gap $G = S^I - S^H$, which equals 1 in the SPT phase, and $-1$ in the trivial phase. As application of our proposal, we study the spin-1 models on various 1D and quasi-1D lattices, including the bilinear-biquadratic model on the single chain, and the Heisenberg model on a two-leg ladder and a three-leg tube. Among others, we confirm the existence of an SPT phase in the spin-1 tube model, and reveal that the phase transition between the SPT and the trivial phase is a continuous one. The transition point is found to be critical, with conformal central charge $c=3$ determined by fits to the block entanglement entropy.