Dimensional Reduction and Topological Invariants of Symmetry-Protected Topological Phases

Abstract

We review the dimensional reduction procedure in the group cohomology classification of bosonic SPT phases with finite abelian unitary symmetry group. We then extend this to include general reductions of arbitrary dimensions and also extend the procedure to fermionic SPT phases described by the Gu-Wen super-cohomology model. We then show that we can define topological invariants as partition functions on certain closed orientable/spin manifolds equipped with a flat connection. The invariants are able to distinguish all phases described within the respective models. Finally, we establish a connection to invariants obtained from braiding statistics of the corresponding gauged theories.

arXiv comment37 pages, 17 figures. v2 published: misprints fixed & references added

References in corpus (28)

• Symmetry protected topological orders and the group cohomology of their symmetry group
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• Topological phases of fermions in one dimension
• Braiding statistics approach to symmetry-protected topological phases
• Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect
• 2D symmetry protected topological orders and their protected gapless edge excitations
• Complete classification of 1D gapped quantum phases in interacting spin systems
• Fermionic Symmetry Protected Topological Phases and Cobordisms
• Symmetry-protected topological orders for interacting fermions -- Fermionic topological nonlinear $\sigma$ models and a special group supercohomology theory
• Topological phases protected by point group symmetry
• Spin TQFTs and fermionic phases of matter
• State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter
• The effect of interactions on 2D fermionic symmetry-protected topological phases with Z2 symmetry
• Non-Abelian String and Particle Braiding in Topological Order: Modular SL(3,Z) Representation and 3+1D Twisted Gauge Theory
• Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions
• Topological invariants for gauge theories and symmetry-protected topological phases
• Many-body topological invariants for fermionic symmetry-protected topological phases
• Generalized modular transformations in 3+1D topologically ordered phases and triple linking invariant of loop braiding
• Symmetry-protected topological invariants of symmetry-protected topological phases of interacting bosons and fermions
• Construction of bosonic symmetry-protected-trivial states and their topological invariants via $G\times SO(\infty )$ non-linear $\sigma$-models
• Many-body topological invariants in fermionic symmetry protected topological phases: Cases of point group symmetries
• Interacting fermionic symmetry-protected topological phases in two dimensions
• Universal symmetry-protected topological invariants for symmetry-protected topological states
• Detecting two dimensional symmetry protected topological order in a ground state wave function
• Loop braiding statistics in exactly soluble 3D lattice models
• Braiding statistics and classification of two-dimensional charge-$2m$ superconductors
• Unoriented 3d TFTs

Cited by in corpus (34)

• Orbifold groupoids
• Global Anomalies on the Hilbert Space
• Bosonic topological phases of matter: bulk-boundary correspondence, SPT invariants and gauging
• Construction and classification of symmetry protected topological phases in interacting fermion systems
• Generalized Charges, Part I: Invertible Symmetries and Higher Representations
• Classification of (2+1)D invertible fermionic topological phases with symmetry
• Generalized homology and Atiyah-Hirzebruch spectral sequence in crystalline symmetry protected topological phenomena
• Excitation basis for (3+1)d topological phases
• Dimensional transmutation from non-Hermiticity
• Codimension-2 defects and higher symmetries in (3+1)D topological phases
• Tunneling Topological Vacua via Extended Operators: (Spin-)TQFT Spectra and Boundary Deconfinement in Various Dimensions
• Symmetry-protected sign problem and magic in quantum phases of matter
• Real-space construction of crystalline topological superconductors and insulators in 2D interacting fermionic systems
• Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions
• Construction and classification of point group symmetry protected topological phases in 2D interacting fermionic systems
• Full Commuting Projector Hamiltonians of Interacting Symmetry-Protected Topological Phases of Fermions
• Generalized Charges, Part II: Non-Invertible Symmetries and the Symmetry TFT
• Non-invertible symmetry-protected topological order in a group-based cluster state
• Disentangling supercohomology symmetry-protected topological phases in three spatial dimensions
• Non-perturbative constraints from symmetry and chirality on Majorana zero modes and defect quantum numbers in (2+1)D
• Intrinsically Interacting Higher-Order Topological Superconductors
• Fractionalization and Anomalies in Symmetry-Enriched U(1) Gauge Theories
• Strongly correlated crystalline higher-order topological phases in two-dimensional systems: A coupled-wire study
• Computing classification of interacting fermionic symmetry-protected topological phases using topological invariants
• $Z_3$ and $(\times Z_3{)}^3$ symmetry protected topological paramagnets
• Non-Abelian Three-Loop Braiding Statistics for 3D Fermionic Topological Phases
• Towards a complete classification of non-chiral topological phases in 2D fermion systems
• Stacking Group Structure of Fermionic Symmetry-Protected Topological Phases
• Generalized Statistics on Lattices
• Spacetime symmetry-enriched SymTFT: from LSM anomalies to modulated symmetries and beyond
• Systematic Construction of Interfaces and Anomalous Boundaries for Fermionic Symmetry-Protected Topological Phases
• Two-dimensional topological paramagnets protected by ${\mathbb{Z}}_3$ symmetry: Properties of the boundary Hamiltonian
• Tensor network representations of fermionic crystalline topological phases on two-dimensional lattices
• Detection of 2D SPT Order with Partial Symmetries