  Intriguing issues in one-dimensional non-reciprocal topological systems
include the breakdown of usual bulk-edge correspondence and the occurrence of
half-integer topological invariants. In order to understand these unusual
topological properties, we investigate the topological phase diagrams and the
zero-mode edge states of a generalized non-reciprocal Su-Schrieffer-Heeger
model, based on some analytical results. Meanwhile, we provide a concise
geometrical interpretation of the bulk topological invariants in terms of two
independent winding numbers and also give an alternative interpretation related
to the linking properties of curves in three-dimensional space. For the system
under the open boundary condition, we construct analytically the wavefunctions
of zero-mode edge states by properly considering a hidden symmetry of the
system and the normalization condition with the use of biorthogonal
eigenvectors. Our analytical results directly give the phase boundary for the
existence of zero-mode edge states and unveil clearly the evolution behavior of
edge states. In comparison with results via exact diagonalization of
finite-size systems, we find our analytical results agree with the numerical
results very well.
