  The long-range interaction can fundamentally alter properties in gapped
topological phases such as emergent massive edge modes. However, recent
research has shifted attention to topological nontrivial critical points or
phases, and it is natural to explore how long-range interaction influences
them. In this work, we investigate the topological behavior and phase
transition of extended Kitaev chains with long-range interactions, which can be
derived from the critical Ising model via the Jordan-Wigner transformation in
the short-range limit. Specifically, we analytically find the critical edge
modes at the critical point remain stable against long-range interaction. More
importantly, we observe these critical edge modes remain massless even when
long-range interactions become substantially strong. As a byproduct, we
numerically find that the critical behavior of the long-range model belongs to
the free Majorana fermion universality class, which is entirely different from
the long-range universality class in usual long-range spin models. Our work
could shed new light on the interplay between long-range interactions
(frustrated) and the gapless topological phases of matter.
