Non-Bloch topological phases in a Hermitian system

The search for novel topological states of matter remains to be a research focus in the past several decades. While a topology theory based on Bloch bands is thoroughly investigated in systems with finite-range hopping, mostly in the context of condensed matter physics, here we study a generalized one-dimensional Su-SchriefferHeeger model with semi-infinite long-range hopping, and demonstrate another type of topological phase referred to as a Hermitian non-Bloch topological phase. This phase presents a pair of symmetry-protected edge modes, and can be characterized by a topological invariant defined upon real-space wave functions. Interestingly, we also find a large number of localized bulk modes near the band edges, residing at specific positions determined by the ratio between hopping range and system size. The proposed phenomena of a Hermitian non-Bloch topological phase can be realized in metamaterials such as topolectrical circuits and mechanical oscillator lattices.