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

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. In this talk, I will focus on the IFT phases of matter in one-dimensional space. I will start by a set of necessary conditions for certain Hamiltonians to realize the IFT phases dictated by the Lieb-Schultz-Mattis (LSM) type theorems [1-2]. Motivated by these necessary conditions, I will discuss how the one-dimensional IFT phases can be classified by their boundary projective representations characterized by a triplet of topological indices $([(\nu,\rho)],[\mu])$ [3-6]. I will present 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., I will give an explicit operational derivation for the boundary representation obtained from stacking two IFT phases[4-6]. [1] M. Cheng, Phys. Rev. B 99, 075143 (2019). [2] Ö. M. Aksoy, A. Tiwari, and C. Mudry, Phys. Rev. B 104, 075146 (2021). [3] L. Fidkowski and A. Kitaev, Phys. Rev. B 83, 075103 (2011). [4] A. Turzillo and M. You, Phys. Rev. B 99, 035103 (2019). [5] C. Bourne and Y. Ogata, Forum of Mathematics, Sigma 9, e25 (2021). [6] Ö. M. Aksoy and C. Mudry, arXiv:2204.10333.