Zero modes of the Kitaev chain with phase-gradients and longer range couplings

TL;DR

This paper analytically solves the spectrum of the Kitaev chain with complex couplings, exploring zero-modes under phase gradients and longer-range interactions, revealing conditions for their presence and non-topological nature.

Contribution

It provides an exact analytical solution for the full spectrum of the Kitaev chain with arbitrary parameters and analyzes the conditions for zero-modes with phase gradients and extended couplings.

Findings

Zero-modes can exist at one end of the chain without topological protection.

Presence of zero-modes depends on fine-tuned parameters.

Zero-modes are not protected by topology or symmetry.

Abstract

We present an analytical solution for the full spectrum of Kitaev's one-dimensional p-wave superconductor with arbitrary hopping, pairing amplitude and chemical potential in the case of an open chain. We also discuss the structure of the zero-modes in the presence of both phase gradients and next nearest neighbor hopping and pairing terms. As observed by Sticlet et al., one feature of such models is that in a part of the phase diagram, zero-modes are present at one end of the system, while there are none on the other side. We explain the presence of this feature analytically, and show that it requires some fine-tuning of the parameters in the model. Thus as expected, these `one-sided' zero-modes are neither protected by topology, nor by symmetry.