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Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model

Lukasz Fidkowski, Xie Chen, Ashvin Vishwanath

TL;DR

The work demonstrates that fully symmetric, gapped surfaces of 3D topological superconductors in class DIII can host topological order that is anomalous in 2D. It constructs an exactly solvable Walker–Wang model realizing the non-Abelian SO(3)$_{6}$ surface order for odd $ u$ and an Abelian semion–fermion order for even $ u$, connecting these to the free-fermion $ u$-class via the sixteen-fold way and a 3D bulk with $T^{2}=-1$ electrons. By decorating and ungauging the Walker–Wang framework, the authors obtain onsite time-reversal symmetry and demonstrate how bulk $ ext{Z}_{2}$ topological order can be removed to yield a short-range entangled bulk with a symmetric surface termination. The results provide non-perturbative surface realizations for all free-fermion TScs, including the claim that $n=16$ yields a trivially gapped, $T$-symmetric surface, and they illuminate the deep connection between surface topological order and three-dimensional topological superconductivity.

Abstract

Three dimensional topological superconductors (TScs) protected by time reversal (T) symmetry are characterized by gapless Majorana cones on their surface. Free fermion phases with this symmetry (class DIII) are indexed by an integer n, of which n=1 is realized by the B-phase of superfluid Helium-3. Previously it was believed that the surface must be gapless unless time reversal symmetry is broken. Here we argue that a fully symmetric and gapped surface is possible in the presence of strong interactions, if a special type of topological order appears on the surface. The topological order realizes T symmetry in an anomalous way, one that is impossible to achieve in purely two dimensions. For odd n TScs, the surface topological order must be non-Abelian. We propose the simplest non-Abelian topological order that contains electron like excitations, SO(3)_6, with four quasiparticles, as a candidate surface state. Remarkably, this theory has a hidden T invariance which however is broken in any 2D realization. By explicitly constructing an exactly soluble Walker-Wang model we show that it can be realized at the surface of a short ranged entangled 3D fermionic phase protected by T symmetry, with bulk electrons trasforming as Kramers pairs, i.e. T^2=-1 under time reversal. We also propose an Abelian theory, the semion-fermion topological order, to realize an even n TSc surface, for which an explicit model is derived using a coupled layer construction. We argue that this is related to the n=2 TSc, and use this to build candidate surface topological orders for n=4 and n=8 TScs. The latter is equivalent to the three fermion state which is the surface topological order of a Z2 bosonic topological phase protected by T invariance. One particular consequence of this is that an n=16 TSc admits a trivially gapped T-symmetric surface.

Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model

TL;DR

The work demonstrates that fully symmetric, gapped surfaces of 3D topological superconductors in class DIII can host topological order that is anomalous in 2D. It constructs an exactly solvable Walker–Wang model realizing the non-Abelian SO(3) surface order for odd and an Abelian semion–fermion order for even , connecting these to the free-fermion -class via the sixteen-fold way and a 3D bulk with electrons. By decorating and ungauging the Walker–Wang framework, the authors obtain onsite time-reversal symmetry and demonstrate how bulk topological order can be removed to yield a short-range entangled bulk with a symmetric surface termination. The results provide non-perturbative surface realizations for all free-fermion TScs, including the claim that yields a trivially gapped, -symmetric surface, and they illuminate the deep connection between surface topological order and three-dimensional topological superconductivity.

Abstract

Three dimensional topological superconductors (TScs) protected by time reversal (T) symmetry are characterized by gapless Majorana cones on their surface. Free fermion phases with this symmetry (class DIII) are indexed by an integer n, of which n=1 is realized by the B-phase of superfluid Helium-3. Previously it was believed that the surface must be gapless unless time reversal symmetry is broken. Here we argue that a fully symmetric and gapped surface is possible in the presence of strong interactions, if a special type of topological order appears on the surface. The topological order realizes T symmetry in an anomalous way, one that is impossible to achieve in purely two dimensions. For odd n TScs, the surface topological order must be non-Abelian. We propose the simplest non-Abelian topological order that contains electron like excitations, SO(3)_6, with four quasiparticles, as a candidate surface state. Remarkably, this theory has a hidden T invariance which however is broken in any 2D realization. By explicitly constructing an exactly soluble Walker-Wang model we show that it can be realized at the surface of a short ranged entangled 3D fermionic phase protected by T symmetry, with bulk electrons trasforming as Kramers pairs, i.e. T^2=-1 under time reversal. We also propose an Abelian theory, the semion-fermion topological order, to realize an even n TSc surface, for which an explicit model is derived using a coupled layer construction. We argue that this is related to the n=2 TSc, and use this to build candidate surface topological orders for n=4 and n=8 TScs. The latter is equivalent to the three fermion state which is the surface topological order of a Z2 bosonic topological phase protected by T invariance. One particular consequence of this is that an n=16 TSc admits a trivially gapped T-symmetric surface.

Paper Structure

This paper contains 16 sections, 21 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. $SO(3)_{6}$ Topological Order and Walker-Wang Construction of the 3D Phase
  3. $SO(3)_{6}$ Topological Order of Surface State
  4. Time Reversal Symmetry and $SO(3)_{6}$ Topological Order
  5. Emergence of $\mathcal{T}^{2}=-1$ for fermions in the Walker-Wang model:
  6. Variants of ${\cal T}$action on $SO(3)_{6}$:
  7. Review of Standard Walker-Wang construction
  8. Improved Walker Wang Model - Onsite ${\cal T}$ symmetry and ungauging the bulk topological order.
  9. Exactly soluble model with onsite ${\cal T}$-symmetry
  10. Ungauging the bulk $\mathbb{Z}_{2}$ topological order.
  11. Abelian Surface Topological Orders and the sixteen fold way
  12. The Semion-Fermion State and Time reversal symmetry
  13. Connecting the Abelian Semion-Fermion State to free fermion TScs
  14. Remaining Abelian topological orders and the sixteen-fold way
  15. Conclusion
  16. ...and 1 more sections

Figures (7)

  • Figure 1: Braiding statistics and fusion data for $SO(3)_{6}$.
  • Figure 2: Graphical definition of $F$ symbol, together with some identities satisfied by $F$.
  • Figure 3: (color online) Origin of ${\mathcal{T}}^{2}=-1$ in Walker Wang models of SO(3)$_{6}$ surface topological order. The additional phase factor shown in red depends on the orientation of the vertices and is required to ensure time reversal symmetry. This leads to ${\mathcal{T}}^{2}=-1$ for fermions in this model.
  • Figure 4: Trivalent resolution of the cubic lattice WW.
  • Figure 5: Plaquette term in Walker-Wang model (taken from WW).
  • ...and 2 more figures