[Paper Review] Characterization and Classification of Fermionic Symmetry Enriched Topological Phases

David Aasen, Parsa Bonderson+1 more|arXiv (Cornell University)|Sep 22, 2021

Topological Materials and Phenomena33 citations

TL;DR

The paper develops a fermionic symmetry enriched topological (FSET) phase framework using Gf-crossed braided tensor categories, classifying symmetry fractionalization and defects for (2+1)D fermionic topological orders with on-site unitary fermionic symmetry Gf.

ABSTRACT

We examine the interplay of symmetry and topological order in $2+1$ dimensional fermionic topological phases of matter. We define fermionic topological symmetries acting on the emergent topological effective theory described using braided tensor category theory. Connecting this to the ${\\cal G}^{\ m f}$ fermionic symmetry of the microscopic physical system, we characterize and classify symmetry fractionalization in fermionic topological phases. We find that the physical fermion provides constraints that result in a tiered structure of obstructions and classification of fractionalization with respect to the physical fermions, the quasiparticles, and the vortices. The fractionalization of the (bosonic) symmetry $G= {\\cal G}^{\ m f}/\\mathbb{Z}_2^{\ m f}$ on the physical fermions is essentially the central extension of $G$ by the $\\mathbb{Z}_2^{\ m f}$ fermion parity conservation that yields the fermionic symmetry ${\\cal G}^{\ m f}$. We develop an algebraic theory of ${\\cal G}^{\ m f}$ symmetry defects for fermionic topological phases using $G$-crossed braided tensor category theory. This formalism allows us to fully characterize and classify $2+1$ dimensional fermionic symmetry enriched topological phases with on-site unitary fermionic symmetry group ${\\cal G}^{\ m f}$. We first apply this formalism to extract the minimal data specifying a general fermionic symmetry protected topological phase, and demonstrate that such phases with fixed ${\\cal G}^{\ m f}$ form a group under fermionic stacking. Then we analyze general fermionic symmetry enriched topological phases and find their classification is given torsorially by the classification of the symmetry fractionalization of quasiparticles combined with the classification of fermionic symmetry protected topological phases. We illustrate our results by detailing a number of examples, including all the invertible fermionic topological phases.

Motivation & Objective

• Motivate and formalize the interplay between symmetry and topological order in 2+1D fermionic systems with a physical fermion.
• Extend braided tensor category theory to fermionic topological orders by introducing fermionic symmetry and Gf-crossed structures.
• Classify symmetry fractionalization and defect structures for fermionic topological phases, including obstructions and torsor classifications.
• Construct a framework for stacking, condensing, and combining fermionic SPT and SET phases consistent with fermion parity.
• Provide explicit examples, including all invertible fermionic topological phases, to illustrate the classification.

Proposed method

• Adopt braided tensor category (BTC) and super-modular/fermionic MTC (FMTC) formalisms to describe fermionic topological orders.
• Introduce fermionic topological symmetry and the fermionic symmetry group Gf as a Z2 central extension of the physical symmetry G by Zf2.
• Develop Gf-crossed BTCs to model fusion, braiding, symmetry action, and fractionalization of quasiparticles, vortices, and defects.
• Determine obstructions to symmetry fractionalization and classify fractionalization patterns via torsors (cohomology groups) H2 and H3.
• Construct a base theory from Zf2 × Z2 FSPT phases to efficiently generate general FSPT phases and analyze their stacking structure.
• Provide a three-stage classification of (2+1)D FSET phases via quasiparticle/vortex fractionalization, defect theory, and FSPT torsors.

Experimental results

Research questions

• RQ1 What are the obstructions to fermionic symmetry fractionalization in (2+1)D fermionic topological phases?
• RQ2 How can Gf-crossed braided tensor category theory classify symmetry defects and fractionalization in fermionic SET phases?
• RQ3 How does fermion parity constraints shape the relation between bosonic and fermionic symmetry actions and fractionalization?
• RQ4 What is the group structure of fermionic SPT phases under stacking, and how does this feed into FSET classifications?
• RQ5 Can explicit examples (including invertible FMTCs) illustrate the proposed FSET classification framework?

Key findings

• Fermionic symmetry fractionalization is governed by obstructions and torsors that reflect the Z2 fermion parity, yielding a refined hierarchy compared to bosonic cases.
• The physical fermion constrains the possible symmetry actions and fractionalization patterns, linking the fermionic symmetry Gf to a central extension of G by Zf2.
• Gf-crossed defect theory captures the fusion, braiding, and action of symmetry on quasiparticles and vortices, enabling complete characterization of FSET phases.
• FSPT phases with fixed Gf form a group under fermionic stacking, consistent with pairing and condensing physical fermions, and this group combines with quasiparticle fractionalization to classify FSET phases.
• The framework recovers known classifications in examples and provides a structured, torsor-based description of both FSPT and FSET phases, including all invertible FMTCs.