Spin Topological Field Theory and Fermionic Matrix Product States

Abstract

We study state-sum constructions of G-equivariant spin-TQFTs and their relationship to Matrix Product States. We show that in the Neveu-Schwarz, Ramond, and twisted sectors, the states of the theory are generalized Matrix Product States. We apply our results to revisit the classification of fermionic Short-Range-Entangled phases with a unitary symmetry G and determine the group law on the set of such phases. Interesting subtleties appear when the total symmetry group is a nontrivial extension of G by fermion parity.

References in corpus (7)

• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology
• Fermionic Matrix Product States and One-Dimensional Topological Phases
• D-branes and K-theory in 2D topological field theory
• Short-range entanglement and invertible field theories
• Matrix product states and equivariant topological field theories for bosonic symmetry-protected topological phases in (1+1) dimensions
• Topological Field Theory and Matrix Product States

Cited by in corpus (30)

• Orbifold groupoids
• Fermionic SPT phases in higher dimensions and bosonization
• Fermionic Lieb-Schultz-Mattis Theorems and Weak Symmetry-Protected Phases
• Codimension-2 defects and higher symmetries in (3+1)D topological phases
• Fermionic projected entangled-pair states and topological phases
• Fermionization of fusion category symmetries in 1+1 dimensions
• Lieb-Schultz-Mattis type theorems for Majorana models with discrete symmetries
• Rotation Symmetry-Protected Topological Phases of Fermions
• Full Commuting Projector Hamiltonians of Interacting Symmetry-Protected Topological Phases of Fermions
• The classification of symmetry protected topological phases of one-dimensional fermion systems
• Fermionic quantum cellular automata and generalized matrix product unitaries
• Fermionic Matrix Product States and One-Dimensional Short-Range Entangled Phases with Anti-Unitary Symmetries
• Free and Interacting Short-Range Entangled Phases of Fermions: Beyond the Ten-Fold Way
• Unwinding Fermionic SPT Phases: Supersymmetry Extension
• Topological Field Theory and Matrix Product States
• Locally Purified Density Operators for Symmetry-Protected Topological Phases in Mixed States
• Boundary supersymmetry of 1+1 d fermionic SPT phases
• Generalized Thouless Pumps in 1+1-dimensional Interacting Fermionic Systems
• Diagrammatic State Sums for 2D Pin-Minus TQFTs
• On ${\mathbb{Z}}_2$-indices for ground states of fermionic chains
• Supersymmetric boundaries of one-dimensional phases of fermions beyond symmetry-protected topological states
• Non-local Order Parameters and Quantum Entanglement for Fermionic Topological Field Theories
• Characterization and Classification of Fermionic Symmetry Enriched Topological Phases
• Duality and Stacking of Bosonic and Fermionic SPT Phases
• Stacking Group Structure of Fermionic Symmetry-Protected Topological Phases
• Lieb-Robinson Bounds on Entanglement Gaps from Symmetry-Protected Topology
• Les Houches Lecture Notes on Tensor Networks
• Tensor network representations of fermionic crystalline topological phases on two-dimensional lattices
• Non-local order parameters for fermion chains via the partial transpose
• Symmetric entanglers for non-invertible SPT phases