Interaction effects on 3D topological superconductors: surface topological order from vortex condensation, the 16 fold way and fermionic Kramers doublets

Abstract

Three dimensional topological superconductors with time reversal symmetry (class DIII) are indexed by an integer $\nu$, the number of surface Majorana cones, according to the free fermion classification. The superfluid B phase of He${}^3$ realizes the $\nu =1$ phase. Recently, it has been argued that this classification is reduced in the presence of interactions to Z${}_{16}$. This was argued from the symmetry respecting surface topological orders of these states, which provide a non-perturbative definition of the bulk topological phase. Here, we verify this conclusion by focusing on the even index case, $\nu =2m$, where a vortex condensation approach can be used to explicitly derive the surface topological orders. We show a direct relation to the well known result on one dimensional topological superconductors (class BDI), where interactions reduce the free fermion classification from Z down to Z${}_8$. Finally, we discuss in detail the fermionic analog of Kramers time reversal symmetry, which allows semions of some surface topological orders to transform as $T^2=\pm i$.

36 pages, 1 figure

Cited by in corpus (35)

• Classification of topological quantum matter with symmetries
• Fermion Path Integrals And Topological Phases
• Classification of Interacting Topological Floquet Phases in One Dimension
• The "Parity" Anomaly On An Unorientable Manifold
• Boson Condensation in Topologically Ordered Quantum Liquids
• Global Anomalies on the Hilbert Space
• Symmetric Mass Generation
• Anomalies and Symmetry Fractionalization
• Composite Dirac liquids: parent states for symmetric surface topological order
• On time-reversal anomaly of 2+1d topological phases
• Anomaly Obstructions to Symmetry Preserving Gapped Phases
• Dai-Freed theorem and topological phases of matter
• Comments on Symmetric Mass Generation in 2d and 4d
• Bulk-boundary correspondence for three-dimensional symmetry-protected topological phases
• More on time-reversal anomaly of 2+1d topological phases
• All-fermion electrodynamics and fermion number anomaly inflow
• Realizing topological surface states in a lower-dimensional flat band
• Fractional $S$-duality, Classification of Fractional Topological Insulators and Surface Topological Order
• A study of time reversal symmetry of abelian anyons
• String flux mechanism for fractionalization in topologically ordered phases
• Anomaly and Cobordism Constraints Beyond the Standard Model: Topological Force
• Standard Model as the topological material
• Boundary supersymmetry of 1+1 d fermionic SPT phases
• Anomaly indicators and bulk-boundary correspondences for 3D interacting topological crystalline phases with mirror and continuous symmetries
• Higgs-Confinement Transitions in QCD from Symmetry Protected Topological Phases
• Gapped Boundary Phases of Topological Insulators via Weak Coupling
• Anomaly and Cobordism Constraints Beyond Grand Unification: Energy Hierarchy
• Geometric Approach to 3D Interfaces at Strong Coupling
• Gauge interactions and topological phases of matter
• Characterization and Classification of Fermionic Symmetry Enriched Topological Phases
• Lattice construction of exotic invertible topological phases
• Commuting projector models for (3+1)d topological superconductors via string net of (1+1)d topological superconductors
• Gauging Defects in Quantum Spin Systems
• Models for $d$ Wave Topological Superconductors and Quantum Anomalous Hall Effect with Arbitrary Large Chern Numbers
• Generalized symmetry enriched criticality in (3+1)d