[Paper Review] 2+1D symmetry-topological-order from local symmetric operators in 1+1D

Kansei Inamura, Xiao-Gang Wen|arXiv (Cornell University)|Oct 9, 2023

Advanced Chemical Physics Studies4 citations

TL;DR

This paper proposes a novel method to compute 2+1D symmetry-topological order (symmetry-TO) from local symmetric operators in 1+1D systems using commutant patch operators—extended operators built from local symmetric operators that commute away from their boundary. The key result is that topological invariants, including non-modular ones like Borromean and Whitehead link invariants, can be computed as contracted products of these patch operators, establishing a direct algebraic link between 1+1D symmetric systems and 2+1D symmetry-TO.

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.

Motivation & Objective

• To establish a systematic framework for deriving 2+1D symmetry-topological order (symmetry-TO) from local symmetric operators in 1+1D quantum systems.
• To identify commutant patch operators as the central algebraic structure encoding topological invariants in 2+1D symmetry-TO.
• To demonstrate that topological invariants beyond modular data—such as link invariants for Borromean rings and Whitehead links—can be computed from 1+1D patch operators.
• To show that the algebra of commutant patch operators realizes the full structure of 2+1D symmetry-TO, providing a concrete computational tool.
• To propose that commutant patch operators serve as order parameters for gapped phases with finite symmetries, linking symmetry structure to topological order.

Proposed method

• Define commutant patch operators as extended operators constructed from products and sums of local symmetric operators in 1+1D systems.
• Ensure that commutant patch operators commute with all local symmetric operators away from their boundary, ensuring topological protection.
• Use the commutant patch operators to compute topological invariants in 2+1D by contracting their products in a way that mirrors anyon diagram computations.
• Construct explicit formulae for topological invariants—including non-modular ones—using the patch operator algebra.
• Leverage the holographic isomorphism between low-energy theories and bulk topological orders with gapped boundaries to relate 1+1D symmetric systems to 2+1D symmetry-TO.
• Establish a correspondence between ribbon operators on the rough boundary of Kitaev’s quantum double model and the 1+1D patch operators, validating the construction.

Experimental results

Research questions

• RQ1 How can 2+1D symmetry-topological order be systematically derived from local symmetric operators in 1+1D systems?
• RQ2 What algebraic structure in 1+1D encodes the full topological invariant content of 2+1D symmetry-TO, including non-modular invariants?
• RQ3 Can commutant patch operators serve as order parameters for gapped phases with finite symmetries?
• RQ4 How do non-modular topological invariants—such as those associated with the Borromean rings and Whitehead link—emerge from 1+1D symmetric systems?
• RQ5 What is the precise mathematical and physical correspondence between 1+1D patch operators and 2+1D anyonic excitations in symmetry-TO?

Key findings

• Commutant patch operators in 1+1D systems are shown to encode the full algebraic structure of 2+1D symmetry-topological order.
• Topological invariants such as the Borromean ring and Whitehead link invariants can be computed as contracted products of commutant patch operators in 1+1D.
• The construction provides explicit formulae for these invariants in terms of patch operators, extending beyond the modular data of a theory.
• The algebra of commutant patch operators is demonstrated to be isomorphic to the fusion and braiding rules of anyons in 2+1D symmetry-TO.
• A direct correspondence is established between ribbon operators on the rough boundary of Kitaev’s quantum double model and 1+1D patch operators, validating the framework.
• The theory confirms that commutant patch operators can act as order parameters for gapped phases with finite group symmetries, particularly in the context of holo-equivalent generalized symmetries.