  In this general article, we map the one-dimensional transverse field quantum
Ising model of ferromagnetism to Kitaev's one-dimensional p-wave
superconductor, which has its application in fault-tolerant topological quantum
computing. Mapping Pauli's spin operators of transverse Ising chain to spinless
fermionic creation and annihilation operators by Inverse Jordan-Wigner
transformation leads to a Hamiltonian form closely related to Kitaev Chain,
which exhibits topological phase transition where phases are characterized by
different topological invariant that changes discontinuously at the transition
point. Kitaev Chain supports two Majorana zero modes (MZMs) in the non-trivial
topological phase, while none is in the trivial phase. The doubly degenerate
ground state of the transverse Ising in ferromagnetic phase corresponds to
non-local free fermion degree made from MZMs. The quasi-particle excitations of
Ising chain, viz., domain wall formation in the ferromagnetic phase and
spin-flip in paramagnetic phase maps to Bogoliubon excitations. The mapping
suggests that a non-local order parameter can be defined for Kitaev Chain to
work with the usual paradigm of Landau's theory.
