  This work comprehensively investigates the non-Hermitian skin effect (NHSE)
in a spinless Bernevig- Hughes-Zhang (BHZ)-like model in one dimension. It is
generally believed that a system with non-reciprocal hopping amplitudes
demonstrates NHSE. However, we show that there are exceptions, and more
in-depth analyses are required to decode the presence of NHSE or its variants
in a system. The fascinating aspects of our findings, depending on the
inclusion of non-reciprocity in the inter-orbital hopping terms, concede the
existence of conventional NHSE or NHSE at both edges and even a surprising
absence of NHSE. The topological properties and the (bi-orthogonal)
bulk-boundary correspondence, enumerated via computation of the (complex) Berry
phase and spatial localization of the edge modes, highlight the topological
phase transitions occurring therein. Further, to facilitate a structured
discussion of the non-Hermitian model, we split the results into PT symmetric
and non-PT symmetric cases with a view to comparing the two.
