2+1D symmetry-topological-order from local symmetric operators in 1+1D

TL;DR

The paper develops a framework to derive 2+1D symmetry-topological-order data directly from 1+1D systems by using commutant, symmetric transparent connectable patch operators. It provides a concrete computational scheme, including twisted evaluation/coevaluation, to extract topological invariants such as quantum dimensions, topological spins, and the modular S-matrix, and demonstrates that invariants beyond modular data (e.g., Borromean rings and Whitehead link) can be obtained. The authors validate the approach by reconstructing Kitaev's quantum double topological orders for general finite groups G, deriving explicit patch-operator constructions for G and recovering all bulk anyon data. They argue that patch operators function as order/disorder parameters for gapped phases with finite symmetries and discuss broader implications, including potential extensions to anomalies, higher dimensions, and fusion-category symmetries.

Abstract

A generalized symmetry (defined by the algebra of local symmetric operators) can go beyond group or higher group description. A theory of generalized symmetry (up to holo-equivalence) was developed in terms of symmetry-TO -- a bosonic topological order (TO) with gappable boundary in one higher dimension. We propose a general method to compute the 2+1D symmetry-TO from the local symmetric operators in 1+1D systems. Our theory is based on the commutant patch operators, which are extended operators constructed as products and sums of local symmetric operators. A commutant patch operator commutes with all local symmetric operators away from its boundary. We argue that topological invariants associated with anyon diagrams in 2+1D can be computed as contracted products of commutant patch operators in 1+1D. In particular, we give concrete formulae for several topological invariants in terms of commutant patch operators. Topological invariants computed from patch operators include those beyond modular data, such as the link invariants associated with the Borromean rings and the Whitehead link. These results suggest that the algebra of commutant patch operators is described by 2+1D symmetry-TO. Based on our analysis, we also argue briefly that the commutant patch operators would serve as order parameters for gapped phases with finite symmetries.

Figures (8)

• Figure 1: To define a homomorphism between quantum field theories that preserves properties of excitations, Ref. KZ150201690 introduced the isomorphism $\varepsilon$ (a low energy equivalence that preserves, for example, low energy partition functions CW221214432) between two (gapped or gapless) quantum field theories, $\underline{ {\cal C} }$ and ${\cal C} \boxtimes_{\EuScript{M}} \widetilde{ {\cal R} }$, where the bulk topological order $\EuScript{M}$ and the gapped boundary $\widetilde{ {\cal R} }$ have infinite energy gaps. The equivalence $\varepsilon$ exposes the emergent symmetry described by the fusion higher category $\widetilde{ {\cal R} }$ and/or the symmetry-TO $\EuScript{M}$ in quantum field theory $\underline{ {\cal C} }$.
• Figure 2: A 2+1D topological order on a slab gives rise to a 1+1D system with finite symmetry. The symmetry action in 1+1D is implemented by inserting a topological line on the topological boundary of the slab. In the figure, time is supposed to go up.
• Figure 3: The relation between anyons in a 2+1D topological order and patch operators in 1+1D.
• Figure 4: The patch operators commute with the symmetry operators.
• Figure 5: A bulk anyon line becomes a sum of topological lines on the topological boundary.