Majorana fermions in the nonuniform Ising-Kitaev chain: exact solution

TL;DR

This paper provides an exact solution for a nonuniform Ising-Kitaev chain model, revealing how Majorana zero modes localize or delocalize at interfaces, challenging the assumption of nearest-neighbor couplings in low-energy descriptions.

Contribution

It presents an exact analytical solution for a minimal nonuniform Ising-Kitaev chain, elucidating the localization and coupling behavior of Majorana zero modes in the system.

Findings

Majorana zero modes are localized at chain interfaces for generic parameters.

Inversion symmetry causes Majorana modes to delocalize between two interfaces.

Couplings between Majorana modes are independent of their separation, contradicting nearest-neighbor assumptions.

Abstract

A quantum computer based on Majorana qubits would contain a large number of zero-energy Majorana states. This system can be modelled as a connected network of the Ising-Kitaev chains alternating the "trivial" and "topological" regions, with the zero-energy Majorana fermions localized at their interfaces. The low-energy sector of the theory describing such a network can be formulated in terms of leading-order couplings between the Majorana zero modes. I consider a minimal model exhibiting effective couplings between four Majorana zero modes - the nonuniform Ising-Kitaev chain, containing two "topological" regions separated by a "trivial" region. Solving the model exactly, I show that for generic values of the model parameters the four zero modes are localized at the four interface points of the chain. In the special case where additional inversion symmetry is present, the Majorana zero modes are "delocalized" between two interface points. In both cases, the low-energy sector of the theory can be formulated in terms of the localized Majorana fermions, but the couplings between some of them are independent of their respective separations: the exact solution does not support the "nearest-neighbor" form of the effective low-energy Hamiltonian.