---
abstract: 'The possibility to observe and manipulate Majorana fermions as end states of one-dimensional topological superconductors has been actively discussed recently. In a quantum wire with strong spin-orbit coupling placed in proximity to a bulk superconductor, a topological superconductor has been expected to be realized when the band energy is split by the application of a magnetic field. When a periodic lattice modulation is applied multiple topological superconductor phases appear in the phase diagram. Some of them occur for higher filling factors compared to the case without the modulation. We study the effects of phase jumps and argue that the topologically nontrivial state of the whole system is retained even if they are present. We also study the effect of the spatial modulation in the hopping parameter.'
author:
- Masaki
- Norio
title: Reentrant topological transitions with Majorana end states in 1D superconductors by lattice modulation
---

Introduction
============

Much work has been devoted to realize topologically nontrivial states of matter with topologically protected surface states. Majorana surface states are expected to be formed at the boundaries and vortex cores of topological superconductors (TSs).[@Read2000; @Kitaev2001; @Ivanov2001; @Fu2008; @Fujimoto2008; @Sato2009; @Linder2010] Such states have been expected in a one-dimensional (1D) 