Explore what happens when we change one the important knobs of the nanowire model, the external magnetic field. We studied what happens when $B$ is pointing along the $z$ direction. However, what happens when the magnetic field is tilted?

Generalize the Hamiltonian of the nanowire to the case of a magnetic field with three components $B_x, B_y, B_z$. How do the new terms look like?

Go into the nanowire notebook. Modify the nanowire_chain function to include the magnetic field pointing in general direction.
Plot the band structure for different field directions, and compare to the original case of having only $B_z$. What changes?

Compare your results with what you find over here:

Effects of tilting the magnetic field in 1D Majorana nanowires

We investigate the effects that a tilting of the magnetic field from the
parallel direction has on the states of a 1D Majorana nanowire. Particularly,
we focus on the conditions for the existence of Majorana zero modes, uncovering
an analytical relation (the sine rule) between the field orientation relative
to the wire, its magnitude and the superconducting parameter of the material.
The study is then extended to junctions of nanowires, treated as magnetically
inhomogeneous straight nanowires composed of two homogeneous arms. It is shown
that their spectrum can be explained in terms of the spectra of two independent
arms. Finally, we investigate how the localization of the Majorana mode is
transferred from the magnetic interface at the corner of the junction to the
end of the nanowire when increasing the arm length.

Now let's switch to the signatures of Majoranas. The code for these is in the signatures notebook.

How does the $4\pi$-periodic Josephson effect disapper? We argued that we cannot just remove a single crossing. Also periodicity isn't a continuous variable and cannot just change. So what is happening?

Study the spectrum of a superconducting ring as a function of magnetic field, as you make a transition between the trivial and the topological regimes.

What do you see? Compare your results with the paper below.

Signatures of topological phase transitions in mesoscopic
superconducting rings

Falko Pientka, Alessandro Romito, Mathias Duckheim, Yuval Oreg, Felix von Oppen

We investigate Josephson currents in mesoscopic rings with a weak link which
are in or near a topological superconducting phase. As a paradigmatic example,
we consider the Kitaev model of a spinless p-wave superconductor in one
dimension, emphasizing how this model emerges from more realistic settings
based on semiconductor nanowires. We show that the flux periodicity of the
Josephson current provides signatures of the topological phase transition and
the emergence of Majorana fermions situated on both sides of the weak link even
when fermion parity is not a good quantum number. In large rings, the Majorana
fermions hybridize only across the weak link. In this case, the Josephson
current is h/e periodic in the flux threading the loop when fermion parity is a
good quantum number but reverts to the more conventional h/2e periodicity in
the presence of fermion-parity changing relaxation processes. In mesoscopic
rings, the Majorana fermions also hybridize through their overlap in the
interior of the superconducting ring. We find that in the topological
superconducting phase, this gives rise to an h/e-periodic contribution even
when fermion parity is not conserved and that this contribution exhibits a peak
near the topological phase transition. This signature of the topological phase
transition is robust to the effects of disorder. As a byproduct, we find that
close to the topological phase transition, disorder drives the system deeper
into the topological phase. This is in stark contrast to the known behavior far
from the phase transition, where disorder tends to suppress the topological
phase.

MoocSelfAssessment description

In the live version of the course, you would need to share your solution and grade yourself.

Discussion entitled 'Majorana nanowire' is available in the online version of the course.

V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers, L. P. Kouwenhoven

Majorana fermions are particles identical to their own antiparticles. They
have been theoretically predicted to exist in topological superconductors. We
report electrical measurements on InSb nanowires contacted with one normal (Au)
and one superconducting electrode (NbTiN). Gate voltages vary electron density
and define a tunnel barrier between normal and superconducting contacts. In the
presence of magnetic fields of order 100 mT we observe bound, mid-gap states at
zero bias voltage. These bound states remain fixed to zero bias even when
magnetic fields and gate voltages are changed over considerable ranges. Our
observations support the hypothesis of Majorana fermions in nanowires coupled
to superconductors.

Hint: Welcome to the real world.

Quantum point contact as a probe of a topological superconductor

M. Wimmer, A. R. Akhmerov, J. P. Dahlhaus, C. W. J. Beenakker

We calculate the conductance of a ballistic point contact to a
superconducting wire, produced by the s-wave proximity effect in a
semiconductor with spin-orbit coupling in a parallel magnetic field. The
conductance G as a function of contact width or Fermi energy shows plateaus at
half-integer multiples of 4e^2/h if the superconductor is in a topologically
nontrivial phase. In contrast, the plateaus are at the usual integer multiples
in the topologically trivial phase. Disorder destroys all plateaus except the
first, which remains precisely quantized, consistent with previous results for
a tunnel contact. The advantage of a ballistic contact over a tunnel contact as
a probe of the topological phase is the strongly reduced sensitivity to finite
voltage or temperature.

Hint: Majorana conductance with many modes.

Non-Abelian statistics and topological quantum information processing in
1D wire networks

Jason Alicea, Yuval Oreg, Gil Refael, Felix von Oppen, Matthew P. A. Fisher

Topological quantum computation provides an elegant way around decoherence,
as one encodes quantum information in a non-local fashion that the environment
finds difficult to corrupt. Here we establish that one of the key
operations---braiding of non-Abelian anyons---can be implemented in
one-dimensional semiconductor wire networks. Previous work [Lutchyn et al.,
arXiv:1002.4033 and Oreg et al., arXiv:1003.1145] provided a recipe for driving
semiconducting wires into a topological phase supporting long-sought particles
known as Majorana fermions that can store topologically protected quantum
information. Majorana fermions in this setting can be transported, created, and
fused by applying locally tunable gates to the wire. More importantly, we show
that networks of such wires allow braiding of Majorana fermions and that they
exhibit non-Abelian statistics like vortices in a p+ip superconductor. We
propose experimental setups that enable the Majorana fusion rules to be probed,
along with networks that allow for efficient exchange of arbitrary numbers of
Majorana fermions. This work paves a new path forward in topological quantum
computation that benefits from physical transparency and experimental realism.

Hint: To play a nice melody, you just need a keyboard. This paper first showed how Majoranas in wire networks can be moved around

Search for Majorana fermions in multiband semiconducting nanowires

We study multiband semiconducting nanowires proximity-coupled with an s-wave
superconductor. We show that when odd number of subbands are occupied the
system realizes non-trivial topological state supporting Majorana modes
localized at the ends. We study the topological quantum phase transition in
this system and analytically calculate the phase diagram as a function of the
chemical potential and magnetic field. Our key finding is that multiband
occupancy not only lifts the stringent constraint of one-dimensionality but
also allows to have higher carrier density in the nanowire and as such
multisubband nanowires are better-suited for observing the Majorana particle.
We study the robustness of the topological phase by including the effects of
the short- and long-range disorder. We show that in the limit of strong
interband mixing there is an optimal regime in the phase diagram ("sweet spot")
where the topological state is to a large extent insensitive to the presence of
disorder.

Do you know of another paper that fits into the topics of this week, and you think is good?
Then you can get bonus points by reviewing that paper instead!

Read one of the above papers and see how it is
related to the current topic.

In the live version of the course, you would need to write a summary which is then assessed by your peers.

Discussion entitled 'Majoranas' is available in the online version of the course.