Spurious Strange Correlators in Symmetry-Protected Topological Phases

TL;DR

This paper reveals how improper reference states can cause false positives in strange correlator diagnostics of SPT phases, emphasizing the importance of choosing appropriate references to accurately identify topological order.

Contribution

It systematically classifies mechanisms leading to spurious signals in strange correlators for trivial SPT phases in 1D systems, guiding correct SPT detection.

Findings

Spurious long-range correlators can arise from high-dimensional irreps in the transfer matrix.

Phase mismatches in symmetry representations can cause false positives.

Symmetry breaking and long-range order can lead to misleading correlator signals.

Abstract

Strange correlator is a powerful tool widely used in detecting symmetry-protected topological (SPT) phases. However, the result of strange correlator crucially relies on the adoption of the reference state. In this work, we report that an ill-chosen reference state can induce spurious long-range strange correlators in trivial SPT phases, leading to false positives in SPT diagnosis. Focusing on 1D gapped bosonic/spin systems described by matrix product states (MPS), we trace the origin of these spurious signals in trivial SPT phases to the magnitude-degeneracy of the transfer matrix. We systematically classify three distinct mechanisms responsible for such degeneracy, each substantiated by concrete examples: (1) the presence of high-dimensional irreducible representations (abbreviated as \emph{irrep}) in the eigenspace corresponding to the entanglment spectrum (entanglement space); (2) a phase mismatch in symmetry representations between the target and reference states; and (3) long-range order arising from symmetry breaking. Our findings clarify the importance of the choice of proper reference states, providing a guideline to avoid pitfalls and correctly identify SPT order using strange correlators.