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

Max A. Metlitski, Lukasz Fidkowski, Xie Chen, Ashvin Vishwanath

Abstract

Three dimensional topological superconductors with time reversal symmetry (class DIII) are indexed by an integer $ν$, the number of surface Majorana cones, according to the free fermion classification. The superfluid B phase of He$^3$ realizes the $ν=1$ 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, $ν=2m$, 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 $T^2=\pm i$.

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 , the number of surface Majorana cones, according to the free fermion classification. The superfluid B phase of He realizes the phase. Recently, it has been argued that this classification is reduced in the presence of interactions to Z. 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, , 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. Finally, we discuss in detail the fermionic analog of Kramers time reversal symmetry, which allows semions of some surface topological orders to transform as .

Paper Structure

This paper contains 22 sections, 87 equations, 1 figure, 8 tables.

Table of Contents

  1. Introduction
  2. Topological Superconductors in 3D
  3. Vortices and the Eight Fold Way
  4. The $m=1$ surface with strength $k$ vortices.
  5. Vortices on the surface of a $\nu=2m$ TSc.
  6. Surface Topological Order from Vortex Condensation
  7. The STO of the $\nu=2$ topological superconductor (single Dirac cone) from vortex condensation
  8. Vortex Statistics
  9. Vortex condensation: topological order
  10. STO for other $\nu=\pm2 \,({\rm mod }\,8)$ topological superconductors; 16-fold way
  11. Fermionic Kramers doublets and $T^2=\pm i$ time reversal action
  12. Examples of $T^2 = \pm i$ defects
  13. Vortices on the superfluid surface of a $\nu = 2$ TSc
  14. Monopoles in the bulk of a $\nu = 2$ TSc with $U(1)$ symmetry
  15. Conclusions
  16. ...and 7 more sections

Figures (1)

  • Figure 1: Slab geometry with superfluidity on the top surface (which breaks both $U(1)$ and $T$ symmetry, but preserves the combination $S$ in Eq. (\ref{['eq:S']})), and insulator on the bottom surface which breaks $T$, for a topological superconductor with $\nu=2m$. Vortices on the top surface trap gauge flux that leaks down to the bottom surface. The statistics of well separated flux-vortex composites piercing the slab is described by the $\nu_{\rm {Kitaev}}=\nu/2$ topological order (Ising for $\nu_{\rm {Kitaev}} = 1$, $U(1)_4$ for $\nu_{\rm {Kitaev}} = 2$, $U(1)_2\times U(1)_2$ for $\nu_{\rm {Kitaev}} = 4$). To extract the intrinsic, time reversal invariant vortex statisics associated with the top surface, the contribution of the bottom surface must be subtracted, resulting in vortex statistics described by a $\nu_{\rm{Kitaev}} \times U(1)_{-16/\nu}$ theory. Subsequently, the vortices of strength $16/\nu$ are condensed to give the surface topological order.