Fermionic defects of topological phases and logical gates
arXiv:2211.12394 · doi:10.21468/SciPostPhys.15.1.028
Abstract
We discuss the codimension-1 defects of (2+1)D bosonic topological phases, where the defects can support fermionic degrees of freedom. We refer to such defects as fermionic defects, and introduce a certain subclass of invertible fermionic defects called "gauged Gu-Wen SPT defects" that can shift self-statistics of anyons. We derive a canonical form of a general fermionic invertible defect, in terms of the fusion of a gauged Gu-Wen SPT defect and a bosonic invertible defect decoupled from fermions on the defect. We then derive the fusion rule of generic invertible fermionic defects. The gauged Gu-Wen SPT defects give rise to interesting logical gates of stabilizer codes in the presence of additional ancilla fermions. For example, we find a realization of the CZ logical gate on the (2+1)D toric code stacked with a (2+1)D ancilla trivial atomic insulator, which is implemented by a finite depth circuit. We also investigate a gapped fermionic interface between (2+1)D bosonic topological phases realized on the boundary of the (3+1)D Walker-Wang model. In that case, the gapped interface can shift the chiral central charge of the (2+1)D phase. Among these fermionic interfaces, we study an interesting example where the (3+1)D phase has a spatial reflection symmetry, and the fermionic interface is supported on a reflection plane that interpolates a (2+1)D surface topological order and its orientation-reversal. We construct a (3+1)D exactly solvable Hamiltonian realizing this setup, and find that the model generates the classification of the (3+1)D invertible phase with spatial reflection symmetry and fermion parity on the reflection plane. We make contact with an effective field theory, known in literature as the exotic invertible phase with spacetime higher-group symmetry.
38 pages, 16 figures
References in corpus (22)
- Non-Abelian Anyons and Topological Quantum Computation
- Exotic non-Abelian anyons from conventional fractional quantum Hall states
- Genons, twist defects, and projective non-Abelian braiding statistics
- Higher Gauging and Non-invertible Condensation Defects
- Fractionalizing Majorana fermions: non-abelian statistics on the edges of abelian quantum Hall states
- Topological boundary conditions in abelian Chern-Simons theory
- Superconducting Proximity Effect on the Edge of Fractional Topological Insulators
- Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology
- Projective non-Abelian Statistics of Dislocation Defects in a Z_N Rotor Model
- Fusion 2-categories and a state-sum invariant for 4-manifolds
- Higher central charges and topological boundaries in 2+1-dimensional TQFTs
- Pauli stabilizer models of twisted quantum doubles
- Codimension-2 defects and higher symmetries in (3+1)D topological phases
- Topological Order, Quantum Codes and Quantum Computation on Fractal Geometries
- Fermionic symmetry fractionalization in (2+1)D
- Locality-Preserving Logical Operators in Topological Stabiliser Codes
- Three-dimensional quantum cellular automata from chiral semion surface topological order and beyond
- Lorentz Symmetry Fractionalization and Dualities in (2+1)d
- Defects in the 3-dimensional toric code model form a braided fusion 2-category
- Symmetry-preserving boundary of (2+1)D fractional quantum Hall states
- A 3-categorical perspective on G-crossed braided categories
- Lattice construction of exotic invertible topological phases