  In an attempt to theoretically investigate the quantum phase transition and
criticality in topological models, we study Kitaev chain with longer-range
couplings (finite number of neighbors) as well as truly long-range couplings
(infinite number of neighbors). We carry out an extensive topological
characterization of the momentum space to explore the possibility of obtaining
higher order winding numbers and analyze the nature of their stability in the
model. The occurrences of phase transitions from even-to-even and odd-to-odd
winding numbers are observed with decreasing longer-rangeness in the system. We
derive topological quantum critical lines and study them to understand the
behavior of criticality. A suppression of higher order winding numbers is
observed with decreasing longer-rangeness in the model. We show that the
mechanism behind such phenomena is due to the superposition and vanishing of
the topological quantum critical lines associated with the higher winding
number. Through the study of Berry connection we show the possible different
behaviors of critical lines when they undergo superposition along with the
corresponding critical exponents. We analyze the behavior of the long-range
models through the momentum space characterization. We also provide exact
solution for the problem and discuss the experimental aspects of the work.
