From Ising model to Kitaev Chain -- An introduction to topological phase transitions

Abstract

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.