Boundary topological orders of (4+1)d fermionic ${\mathbb{Z}}_{2N}^{\mathrm{F}}$ SPT states

Abstract

We investigate (3+1)d topological orders in fermionic systems with an anomalous ${\mathbb{Z}}_{2N}^{\mathrm{F}}$ symmetry, where its ${\mathbb{Z}}_2^{\mathrm{F}}$ subgroup is the fermion parity. Such an anomalous symmetry arises as the discrete subgroup of the chiral U(1) symmetry of $\nu$ copies of Weyl fermions of the same chirality. Inspired by the crystalline correspondence principle, we deform the anomalous ${\mathbb{Z}}_{2N}^{\mathrm{F}}$ symmetry of (3+1)d Weyl fermion to the anomalous $C_N\times {\mathbb{Z}}_2^{\mathrm{F}}$ symmetry. Then we microscopically construct symmetry-preserving gapped boundary states of the closely-related (4+1)d $C_N\times {\mathbb{Z}}_2^{\mathrm{F}}$ symmetry-protected topological (SPT) state (with $C_N$ being the $N$-fold rotation), whenever it is possible. In particular, for $\nu =N$, we show that the (3+1)d symmetric gapped state admits a topological ${\mathbb{Z}}_4$ gauge theory description at low energy, and propose that a similar theory saturates the corresponding ${\mathbb{Z}}_{2N}^{\mathrm{F}}$ anomaly. For $N\nmid \nu$, our construction admits no topological quantum field theory (TQFT) symmetric gapped state; while for $\nu =N/2$, we find a non-TQFT symmetric gapped state via stacking lower-dimensional (2+1)d non-discrete-gauge-theory topological order inhomogeneously. For other values of $\nu$, no symmetric gapped state is possible within our construction, which is consistent with the no-go theorem by Cordova-Ohmori.

arXiv comment16 pages, 3 figures

References in corpus (33)

• Electronic Structure of Pyrochlore Iridates: From Topological Dirac Metal to Mott Insulator
• Interacting fermionic topological insulators/superconductors in three dimensions
• Reflection positivity and invertible topological phases
• The "Parity" Anomaly On An Unorientable Manifold
• Topological phases protected by point group symmetry
• On the classification of topological orders
• Gauging spatial symmetries and the classification of topological crystalline phases
• Dai-Freed anomalies in particle physics
• Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions
• A classification of 3+1D bosonic topological orders (I): the case when point-like excitations are all bosons
• Why are fractional charges of orientifolds compatible with Dirac quantization?
• Fermionic Finite-Group Gauge Theories and Interacting Symmetric/Crystalline Orders via Cobordisms
• A classification of 3+1D bosonic topological orders (II): the case when some point-like excitations are fermions
• Construction and classification of symmetry protected topological phases in interacting fermion systems
• Classification of Symmetry-Protected Phases for Interacting Fermions in Two Dimensions
• Beyond Standard Models and Grand Unifications: Anomalies, Topological Terms, and Dynamical Constraints via Cobordisms
• Anomaly Constraints on Gapped Phases with Discrete Chiral Symmetry
• Anomaly indicators for time-reversal symmetric topological orders
• Interacting fermionic symmetry-protected topological phases in two dimensions
• Relative Anomalies in (2+1)D Symmetry Enriched Topological States
• Construction and classification of point group symmetry protected topological phases in 2D interacting fermionic systems
• Rotation Symmetry-Protected Topological Phases of Fermions
• Microscopic Theory of Surface Topological Order for Topological Crystalline Superconductors
• Cobordism and Deformation Class of the Standard Model
• 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
• Ultra Unification
• Categorical Symmetry of the Standard Model from Gravitational Anomaly
• Anomaly cascade in (2+1)D fermionic topological phases
• Proton Stability: From the Standard Model to Beyond Grand Unification
• A no-go result for implementing chiral symmetries by locality-preserving unitaries in a 3 dimensional Hamiltonian lattice model of fermions
• Mirror anomaly in fermionic topological orders
• Gapped boundary of (4+1)d beyond-cohomology bosonic SPT phase

Cited by in corpus (1)

• Anomaly of 4d Weyl fermions with discrete symmetries