Strange correlator and string order parameter for non-invertible symmetry protected topological phases in 1+1d

TL;DR

This paper develops new tools like strange correlators and string order parameters to characterize non-invertible symmetry protected topological phases in 1+1d quantum systems, revealing their long-range order and algebraic structures.

Contribution

It introduces a systematic construction of strange correlators and string order parameters for NISPTs, linking them to boundary algebra and categorical data, with tensor network illustrations.

Findings

Strange correlator shows long-range order between different NISPTs.

String order parameter derived from symmetry operators and charge decorations.

Entanglement spectrum degeneracy linked to irreducible representations of interface algebra.

Abstract

In this paper, we construct strange correlators and string order parameters for non-invertible symmetry protected topological phases (NISPTs) in 1+1d quantum lattice spin models. The strange correlator exhibits long-range order when evaluated between two distinct NISPTs and decays exponentially otherwise. We show that strange charged operators inserted into the strange correlator are linked to the interface algebra (boundary tube algebra) and are non-trivial when all its irreducible representations have dimensions greater than one. We discuss the generalization to higher dimensions. The string order parameter is obtained by contracting the truncated symmetry operator with charge decoration operators, which are determined by the NISPT action tensors. We illustrate the above construction using the three NISPTs of $\text{Rep}(D_8)$ and demonstrate the extraction of categorical data via tensor networks, particularly through the ZX calculus. Finally, we show that the entanglement spectrum degeneracy is determined by the irreducible representations of the interface algebra when assuming non-invertible symmetry on-site condition.