I will present recent results on a one-dimensional wire with strong Rashba
and Dresselhaus spin-orbit coupling (SOC), which supports Majorana
fermions when subject to a Zeeman magnetic field and in proximity of a
superconductor. By introducing  a new local quantity dubbed as "Majorana
polarization", I will argue that it can be used as a local order
parameter to characterize the topological transition between a trivial
system and a system exhibiting Majorana bound modes. We have found that
the local "Majorana polarization" is correlated to the transverse spin
polarization, and we propose to test the presence of Majorana fermions in
a 1D system by a spin-polarized density of states measurement [1]. Such
quantity turns out to be very useful to analyze one-dimensional
topological SN and SNS long junctions obtained by placing a topological
insulating nanowire in the proximity of either one or two superconducting
finite-size leads: for a finite-size SN junction the Andeev Bound States
(ABS) spectrum exhibits a zero-energy extended state which carries a full
Majorana fermion, while the ABS of long SNS junctions with phase
difference pi transform into two zero-energy states carrying two
Majorana fermions [2].

[1] D. Sticlet, C. Bena, P. Simon, Phys. Rev. Lett. 108, 096802 (2012).
[2] D. Chevallier,  D. Sticlet, P. Simon, C. Bena, arXiv:1203.2643.