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

TL;DR

This paper provides a straightforward derivation of the stacking rules for one-dimensional invertible fermionic topological phases, focusing on boundary representations and their composition from stacked phases.

Contribution

It offers an explicit, elementary derivation of the boundary stacking rules for 1D IFT phases based on boundary indices, enhancing understanding of their compositional structure.

Findings

Derived explicit boundary stacking rules for 1D IFT phases.

Connected boundary indices to the physical stacking process.

Provided operational definitions for boundary representations.

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.