  Bulk-boundary correspondence (BBC) of symmetry-protected topological (SPT)
phases relates the non-trivial topological invariant of the bulk to the number
of topologically protected boundary states. Recently, a finer classification of
SPT phases has been discovered, known as sub-symmetry- protected topological
(sub-SPT) phases. In sub- SPT phases, a fraction of the boundary states is
protected by the sub-symmetry of the system, even when the full symmetry is
broken. While the conventional topological invariant derived from the Bloch
band is not applicable to describe the BBC in these systems, we propose to use
the non-Bloch topological band theory to describe the BBC of sub-SPT phases.
Using the concept of the generalized Brillouin zone (GBZ), where Bloch momenta
are generalized to take complex values, we show that the non-Bloch band theory
naturally gives rise to a non-Bloch topological invariant, establishing the BBC
in both SPT and sub-SPT phases. In a one-dimensional system, we define the
winding number, whose physical meaning corresponds to the reflection amplitude
in the scattering matrix. Furthermore, the non-Bloch topological invariant
characterizes the hidden intrinsic topology of the GBZ under translation
symmetry-breaking boundary conditions. The topological phase transitions are
characterized by the generalized momenta touching the GBZ, which accompanies
the emergence of diabolic or band-touching points. Additionally, we discuss the
BBCs in the presence of local or global full-symmetry or sub-symmetry-breaking
deformations.
