Spurious Strange Correlators in Symmetry-Protected Topological Phases

TL;DR

Spurious strange correlators can lead to false positives when diagnosing SPT order with strange correlators if the reference state is not chosen carefully. The authors develop an MPS-based transfer-matrix framework showing that long-range strange correlations are tied to magnitude-degeneracy of the transfer matrix $M_ extomega$, and they categorize three degeneracy mechanisms: high-dimensional irreps, phase mismatch, and symmetry breaking. They provide rigorous theorems, explicit lattice-model examples (including spin-2 AKLT with $SO(3)$), and practical guidelines for selecting reference states to avoid spurious signals. This work enhances the reliability of strange correlators as a diagnostic tool for 1D SPT order and suggests directions for extending the framework to fermions, higher dimensions, and subsystem symmetries.

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. Using matrix product state (MPS) representation, 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 in the 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.

Figures (2)

• Figure 1: The formulation of $\expval{O^aO^b}_S$ articulated utilizing a Matrix Product State (MPS) representation. The tensor denoted as $M$, which is depicted with three interconnected lines, corresponds to the $M$ tensor within the MPS framework, characterized by two horizontal virtual legs and a single vertical physical leg. The black dots are indicative of direct product states, each defined by an individual physical index. The interconnecting lines signify the process of index contraction. The definitions of $M_a$, $M_b$, and $M_\omega$ are each enclosed within a rectangular boundary.
• Figure 2: AKLT models construction: Two spin-$S/2$ particles project onto spin-$S$ at each site; adjacent sites form spin singlets.

Theorems & Definitions (17)

• Theorem 1
• Definition 1: Magnitude-degeneracy
• Theorem 2: Origin of long-range strange correlator
• Theorem 3
• Example 1: $SO(3)$ $D=3$ irrep
• Example 2: $\mathbb Z_2$ phase mismatch
• Example 3: GHZ state
• Theorem 1
• proof
• Theorem 2: Magnitude-degeneracy implies long-range strange correlator