Quantized polarization and Majorana fermions beyond tenfold classification

Abstract Exploration of topology is central in condensed matter physics and applications to fault-tolerant quantum information. The bulk-boundary correspondence and tenfold classification determine the topological state compared to a vacuum. Contrary to this belief, we demonstrate that topological zero-energy domain-wall states can emerge for all forbidden 1D classes of the tenfold classification table. The guiding principle is that the difference in the topological quantities of two trivial domains can be quantized, and hence, a topologically protected state can emerge at the domain wall. Such nontrivial domain-wall states are demonstrated using generalized Su-Schrieffeer-Heeger and generalized Kitaev models, which manifest quantized polarization and Majorana fermions, respectively. The quantized Berry phase difference between the domains protects the non-trivial nature of the domain-wall states, extending the bulk-boundary correspondence, also confirmed by the tight-binding and Jackiw-Rebbi methods. Furthermore, we show that the seemingly trivial electronic and superconducting models can be transformed into their topological counterparts in the framework of the topological Fermi-liquid theory. Finally, we propose potential systems where our results may be realized, spanning from electronic and superconducting to optical systems.