Non-Bloch band theory of sub-symmetry-protected topological phases

A generic feature of symmetry-protected topological (SPT) phases of matter is the bulkboundary correspondence (BBC) which connects the concept of bulk topology to the emergence of robust boundary states. In recent years, non-Hermitian systems have shown unconventional properties and phenomena such as exceptional points, non-Hermitian skin effect, and many more in different research fields without Hermitian analog. Therefore, the topological Bloch band theory with the notion of the Brillouin zone (BZ) has been extended to the non-Bloch band theory with the notion of the generalized Brillouin zone (GBZ) defined by generalized momenta which can take complex values. The non-Bloch band theory has successfully proven that non-Hermitian systems show two types of modified BBC: (i) complex eigenvalue topology of the bulk leads to non-Hermitian skin effect, where all bulk states localize at one boundary of the system, and (ii) the eigenstate topology in the GBZ leads to the conventional topological boundary modes.

In our recent work [1], we reported a new type of BBC in non-Hermitian systems, which originates from the intrinsic topology of the GBZ. Topologically non-trivial GBZ appears due to general boundary conditions that break the translation symmetry of the system locally. In our case, the topological phase transition is characterized by the generalized momentum touching of GBZ, which accompanies the emergence of exceptional points.

In this presentation, I will discuss a simple extension of our work to Hermitian systems. Firstly, we found that the intrinsic topology of GBZ successfully characterizes the SPT phases. In this case, the topological phase transition of GBZ is characterized by the generalized momenta touching of GBZ, which accompanies the emergence of band touching points. Moreover, the non-Bloch band theory provides a natural topological invariant characterizing the sub-symmetry-protected topological phases that a conventional Bloch topological invariant (winding number) fails to characterize.

Reference: [1] Sonu Verma and Moon Jip Park, arXiv:2305.08584.