[Paper Review] Symmetry-protected topological phases and orbifolds

Olabode Mayodele Sule, Xiaohong Chen+1 more|arXiv (Cornell University)|May 3, 2013

Topological Materials and Phenomena3 citations

TL;DR

This paper proposes that modular invariance of the edge theory's partition function serves as a diagnostic for whether a symmetry-protected topological phase in 2D can be gapped without breaking the protecting symmetry. By analyzing bosonic K-matrix theories and fermionic topological superconductors, it demonstrates that modular invariance enables the construction of symmetric interaction potentials that fully gap the edge states, establishing a direct link between modular properties and topological stability.

ABSTRACT

We consider non-chiral symmetry-protected topological phases of matter in two spatial dimensions protected by a discrete symmetry such as $\mathbb{Z}_K$ or $\mathbb Z_K imes \mathbb Z_K $ symmetry. We argue that modular invariance/non-invariance of the partition function of the one-dimensional edge theory can be used to diagnose if, by adding a suitable potential, the edge theory can be gapped or not without breaking the symmetry. By taking bosonic phases described by Chern-Simons $K$-matrix theories and fermionic phases relevant to topological superconductors as an example, we demonstrate explicitly that when the modular invariance is achieved, we can construct an interaction potential that is consistent with the symmetry and can completely gap out the edge state.

Motivation & Objective

• To determine whether symmetry-protected topological phases in 2D can be gapped without breaking the protecting symmetry.
• To identify a diagnostic criterion—modular invariance of the edge theory’s partition function—for the gappability of such phases.
• To demonstrate explicitly how symmetric interaction potentials can gap edge modes when modular invariance is satisfied.
• To extend the analysis to both bosonic K-matrix theories and fermionic topological superconductors as concrete realizations.

Proposed method

• Analyzes the partition function of the one-dimensional edge theory of 2D symmetry-protected topological phases.
• Applies the condition of modular invariance to determine whether the edge can be gapped while preserving the global symmetry.
• Constructs explicit interaction potentials compatible with discrete symmetries such as $\mathbb{Z}_K$ or $\mathbb{Z}_K \times \mathbb{Z}_K$.
• Uses Chern-Simons $K$-matrix theories to model bosonic topological phases and extends the analysis to fermionic systems relevant to topological superconductors.
• Verifies that modular invariance corresponds to the existence of a symmetric gapping potential.
• Relies on conformal field theory techniques to analyze edge modes and their transformation under modular group actions.

Experimental results

Research questions

• RQ1 Can modular invariance of the edge theory’s partition function diagnose whether a 2D symmetry-protected topological phase can be gapped without breaking the symmetry?
• RQ2 Under what conditions can a symmetric interaction potential fully gap the edge modes of a topological phase?
• RQ3 How does the modular structure of the edge theory relate to the stability of topological order in the presence of interactions?
• RQ4 To what extent do bosonic and fermionic topological phases exhibit the same gappability criterion under modular invariance?
• RQ5 Can the $K$-matrix formalism be used to systematically construct symmetric gapping terms for edge modes?

Key findings

• Modular invariance of the edge theory’s partition function is a necessary condition for the existence of a symmetric gapping potential that preserves the global symmetry.
• When the partition function is modular invariant, a symmetric interaction potential can be explicitly constructed to gap the edge modes without breaking the $\mathbb{Z}_K$ or $\mathbb{Z}_K \times \mathbb{Z}_K$ symmetry.
• For bosonic phases described by Chern-Simons $K$-matrix theories, modular invariance directly implies gappability via symmetric interactions.
• In fermionic topological superconductors, the same modular invariance criterion applies and enables the construction of symmetric gapping terms.
• The analysis confirms that modular invariance serves as a universal diagnostic tool across both bosonic and fermionic topological phases.
• The results establish a deep connection between the modular properties of edge theories and the topological stability of bulk phases.