Elementary derivation of the stacking rules of invertible fermionic topological phases in one dimension

Abstract

Invertible fermionic topological (IFT) phases are gapped phases of matter with nondegenerate ground states on any closed spatial manifold. When open boundary conditions are imposed, nontrivial IFT phases support gapless boundary degrees of freedom. Distinct IFT phases in one-dimensional space with an internal symmetry group $G_f^{}$ have been characterized by a triplet of indices $([(\nu ,\rho )],[\mu ])$. Our main result is an elementary derivation of the fermionic stacking rules of one-dimensional IFT phases for any given internal symmetry group $G_f^{}$ from the perspective of the boundary, i.e., we give an explicit operational definition for the boundary representation $([({\nu }_{\wedge }^{},{\rho }_{\wedge }^{})],[{\mu }_{\wedge }^{}])$ obtained from stacking two IFT phases characterized by the triplets of boundary indices $([({\nu }_1^{},{\rho }_1^{})],[{\mu }_1^{}])$ and $([({\nu }_2^{},{\rho }_2^{})],[{\mu }_2^{}])$, respectively.

27 pages + 17 pages of supplemental material, 1 figure

References in corpus (6)

• One-Dimensional Symmetry Protected Topological Phases and their Transitions
• Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge
• Fermionic Matrix Product States and One-Dimensional Topological Phases
• Lieb-Schultz-Mattis type theorems for Majorana models with discrete symmetries
• Boundary supersymmetry of 1+1 d fermionic SPT phases
• Supersymmetric boundaries of one-dimensional phases of fermions beyond symmetry-protected topological states

Cited by in corpus (7)

• Lieb-Schultz-Mattis anomalies and web of dualities induced by gauging in quantum spin chains
• Non-perturbative constraints from symmetry and chirality on Majorana zero modes and defect quantum numbers in (2+1)D
• Duality and Stacking of Bosonic and Fermionic SPT Phases
• Self-$G$-ality in 1+1 dimensions
• Incommensurate gapless ferromagnetism connecting competing symmetry-enriched deconfined quantum phase transitions
• Stacking Group Structure of Fermionic Symmetry-Protected Topological Phases
• Interacting Crystalline Topological Insulators in two-dimensions with Time-Reversal Symmetry