Let us study the spectrum of a Josephson junction on a quantum spin Hall edge in more detail. As in the lecture, we can add a magnet in the middle of the junction, which adds a Zeeman energy term to the Hamiltonian.

First, make such a junction. The code from week 2 for making a Josephson junction may be useful.

We are interested in the spectrum below the gap. There are two interesting parameters to vary: the Zeeman energy and the length of the junction. What happens to the energy levels as you increase the length of the junction. In particular, what happens when the junction is very long? What if you turn off the magnet?

Compare your results to the following paper, particularly Fig. 2.

Josephson Current and Noise at a Superconductor-Quantum Spin Hall
Insulator-Superconductor Junction

Following Taylor Hughes suggestion from the summary of the lecture about crystalline defects, create an edge dislocation carrying a Majorana mode in an array of weakly coupled Kitaev chains.

Then try to split the dislocation into two disclinations. What happens to the Majorana mode?

Note that Kwant only supports regular lattices, so crystallographic defects can be implemented by altering some hoppings, as was done in the simulations in the lecture.

MoocSelfAssessment description

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

Now share your results:

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

We study the proximity effect between an s-wave superconductor and the
surface states of a strong topological insulator. The resulting two dimensional
state resembles a spinless p_x+ip_y superconductor, but does not break time
reversal symmetry. This state supports Majorana bound states at vortices. We
show that linear junctions between superconductors mediated by the topological
insulator form a non chiral 1 dimensional wire for Majorana fermions, and that
circuits formed from these junctions provide a method for creating,
manipulating and fusing Majorana bound states.

Hint: In detail, how to create and manipulate Majoranas on the 3D TI surface.

Josephson supercurrent through a topological insulator surface state

M. Veldhorst, M. Snelder, M. Hoek, T. Gang, X. L. Wang, V. K. Guduru, U. Zeitler, W. G. v. d. Wiel, A. A. Golubov, H. Hilgenkamp, A. Brinkman

Topological insulators are characterized by an insulating bulk with a finite
band gap and conducting edge or surface states, where charge carriers are
protected against backscattering. These states give rise to the quantum spin
Hall effect without an external magnetic field, where electrons with opposite
spins have opposite momentum at a given edge. The surface energy spectrum of a
threedimensional topological insulator is made up by an odd number of Dirac
cones with the spin locked to the momentum. The long-sought yet elusive
Majorana fermion is predicted to arise from a combination of a superconductor
and a topological insulator. An essential step in the hunt for this emergent
particle is the unequivocal observation of supercurrent in a topological phase.
Here, we present the first measurement of a Josephson supercurrent through a
topological insulator. Direct evidence for Josephson supercurrents in
superconductor (Nb) - topological insulator (Bi2Te3) - superconductor e-beam
fabricated junctions is provided by the observation of clear Shapiro steps
under microwave irradiation, and a Fraunhofer-type dependence of the critical
current on magnetic field. The dependence of the critical current on
temperature and length shows that the junctions are in the ballistic limit.
Shubnikov-de Haas oscillations in magnetic fields up to 30 T reveal a
topologically non-trivial two-dimensional surface state. We argue that the
ballistic Josephson current is hosted by this surface state despite the fact
that the normal state transport is dominated by diffusive bulk conductivity.
The lateral Nb-Bi2Te3-Nb junctions hence provide prospects for the realization
of devices supporting Majorana fermions.

Hint: The Josephson effect on a 3D TI, in real life.

Majorana Fermions and Disclinations in Topological Crystalline
Superconductors

We prove a topological criterion for the existence of zero-energy Majorana
bound-state on a disclination, a rotation symmetry breaking point defect, in
4-fold symmetric topological crystalline superconductors (TCS). We first
establish a complete topological classification of TCS using the Chern
invariant and three integral rotation invariants. By analytically and
numerically studying disclinations, we algebraically deduce a Z_2-index that
identifies the parity of the number of Majorana zero-modes at a disclination.
Surprisingly, we also find weakly-protected Majorana fermions bound at the
corners of superconductors with trivial Chern and weak invariants.

Three-dimensional topological insulators are classified into "strong" (STI)
and "weak" (WTI) according to the nature of their surface states. While the
surface states of the STI are topologically protected from localization, this
does not hold for the WTI. In this work we show that the surface states of the
WTI are actually protected from any random perturbation that does not break
time-reversal symmetry, and does not close the bulk energy gap. Consequently,
the conductivity of metallic surfaces in the clean system remains finite even
in the presence of strong disorder of this type. In the weak disorder limit the
surfaces are found to be perfect metals, and strong surface disorder only acts
to push the metallic surfaces inwards. We find that the WTI differs from the
STI primarily in its anisotropy, and that the anisotropy is not a sign of its
weakness but rather of its richness.

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.

Do you have questions about what you read? Would you like to suggest other papers? Tell us:

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