  In this study, a spin-1/2 anisotropic extended XY chain has been introduced
in which both time reversal and SU(2) symmetries are broken but $Z_2$ symmetry
is preserved. System exhibits the faithful coexistence of magnetic and
topological superconducting phases even in the presence of transverse magnetic
field. Location of those phases in the parameter space has been determined
precisely. Quantum phase transition is noted at zero magnetic field, as well as
magnetic long range order is found to withstand magnetic field of any strength.
Exact analytic results for spin-spin correlation functions have been obtained
in terms of Jordan Wigner fermionization. Existence of long range magnetic
order has been investigated numerically by finding correlation functions as
well as the Binder cumulant in the ground state. Dispersion relation, ground
state energy, and energy gap are obtained analytically. In order to find the
topologically nontrivial phase, sign of Pfaffian invariant and value of winding
number have been evaluated. Both magnetic and topological phases are robust
against the magnetic field and found to move coercively in the parameter space
with the variation of its strength. Long range orders along two orthogonal
directions and two different topological phases are found and their one-to-one
correspondence has been established. Finally casting the spinless fermions onto
Majorana fermions, properties of zero energy edge states are studied. Three
different kinds of Majorana pairings are noted. In the trivial phase,
next-nearest-neighbor Majorana pairing is found, whereas two different types of
nearest-neighbor Majorana pairings are identified in the topological
superconducting phase.

Mesoscale and Nanoscale Physics