In [1]:
import sys
sys.path.append('../code')
from init_mooc_nb import *
init_notebook()

Populated the namespace with:
np, matplotlib, kwant, holoviews, init_notebook, SimpleNamespace, pprint_matrix, scientific_number, pretty_fmt_complex, plt, pf, display_html
from code/edx_components:
MoocVideo, MoocDiscussion, MoocCheckboxesAssessment, MoocMultipleChoiceAssessment, MoocSelfAssessment
from code/functions:
spectrum, hamiltonian_array, h_k, pauli
Using kwant 1.3.2 and holoviews 1.7.0-x-gf3a9f4f
Executed on 2019-01-31 at 20:44:08.930409.


# Simulations: Majorana defects¶

As usual, start by grabbing the notebooks of this week (w7_defects). They are once again over here.

## Quantum spin Hall junction¶

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 (arXiv:0804.4469) Liang Fu, C. L. Kane

## Majorana in a crystalline defect¶

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.

In [2]:
MoocSelfAssessment()

Out[2]:

MoocSelfAssessment description

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

In [3]:
MoocDiscussion('Labs', 'Topological defects')

Out[3]:

Discussion Topological defects is available in the EdX version of the course.

# Review assignment¶

### Superconducting proximity effect and Majorana fermions at the surface of a topological insulator (arXiv:0707.1692)¶

Liang Fu, C. L. Kane

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 (arXiv:1112.3527)¶

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 (arXiv:1208.6303)¶

Jeffrey C. Y. Teo, Taylor L. Hughes

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.

Hint: Disclinations.

### The strong side of weak topological insulators (arXiv:1105.4351)¶

Zohar Ringel, Yaacov E. Kraus, Ady Stern

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.

Hint: How weak is weak?

### Bonus: Find your own paper to review!¶

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!

In [4]:
MoocSelfAssessment()

Out[4]:

MoocSelfAssessment description

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

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

In [5]:
MoocDiscussion("Reviews", "Topological defects")

Out[5]:

Discussion Topological defects is available in the EdX version of the course.