Topological quantum phase transitions in a Majorana chain with spatial modulation

TL;DR

This paper investigates how spatially modulated chemical potentials affect quantum phase transitions and Majorana zero modes in a one-dimensional Kitaev model, revealing topological phase changes and stability conditions.

Contribution

It introduces a detailed numerical analysis of topological phase transitions in a modulated Majorana chain, highlighting the effects of modulation phase and amplitude on Majorana modes and topological invariants.

Findings

Number of Majorana zero modes varies with modulation phase.

Quantum phase transitions occur with changes in modulation amplitude.

Certain phases exhibit stable degeneracy of entanglement spectrum despite modulation.

Abstract

We numerically study the quantum phase transitions and the stability of Majorana zero modes in a generalized Kitaev model in one dimension when the chemical potential is periodically modulating in space. By using the exact diagonalization method for open boundary condition, we investigate the ground-state phases in terms of the non-local properties such as the entanglement spectrum (ES) and the string correlation functions. When we vary the phase of the modulation, the number of the Majorana zero modes changes, which manifests itself in the degeneracy of the lowest level of the ES. Next, we study the quantum phase transitions driven by the change in the amplitude of the modulation. In particular, for certain values of the wave number and the phase of the modulation, we observe a quantum phase transition from one topological phase into another where the string correlation function oscillates in space. We also show a case where the degeneracy of the ES does not change even for large enough amplitude of the modulation. Finally, we characterize the phases of the system with periodic boundary condition by the topological invariant, which reflects the number of the zero-energy excitations.