Time-reversal symmetry has a very similar effect on Weyl semimetals as it has on gapless superconductors: it keeps the value of the Chern number around the Weyl point the same, and leads to appearance of quadruplets of Weyl points.

Your task is to construct a Weyl semimetal with time reversal symmetry. As we discussed, 4 Weyl points are needed.

If you don't know where to start, here's a hint: you're not the first one who wants to construct a Weyl semimetal with time reversal, search on arxiv.

Graphene, just like $d$-wave superconductors has edge states. They only exist when the Dirac points are not located at coinciding momenta parallel to the boundary.

Define a graphene ribbon supporting edge states. For that you'll need to figure out which orientation to choose.

Then try to add a term to the boundary that breaks the sublattice symmetry and moves the edge states from zero energy. What happens?

What if you add the next-nearest neighbor hopping in the bulk. What do you see now?

Try to remove the edge states completely by tweaking the sublattice symmetry breaking term at the edge. Did you succeed? How?

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 'Extensions' is available in the online version of the course.

A topological insulator is characterized by a dichotomy between the interior
and the edge of a finite system: While the bulk has a non-zero energy gap, the
edges are forced to sustain excitations traversing these gaps. Originally
proposed for electrons governed by quantum mechanics, it has remained an
important open question if the same physics can be observed for systems obeying
Newton's equations of motion. Here, we report on measurements that characterize
the collective behavior of mechanical oscillators exhibiting the phenomenology
of the quantum spin hall effect. The phononic edge modes are shown to be
helical and we demonstrate their topological protection via the stability
against imperfections. Our results open the door to the design of topological
acoustic meta-materials that can capitalize on the stability of the surfaces
phonons as reliable wave guides.

Hint: Different mechanical TI

Quantum phase transitions of a disordered antiferromagnetic topological
insulator

P. Baireuther, J. M. Edge, I. C. Fulga, C. W. J. Beenakker, J. Tworzydło

We study the effect of electrostatic disorder on the conductivity of a
three-dimensional antiferromagnetic insulator (a stack of quantum anomalous
Hall layers with staggered magnetization). The phase diagram contains regions
where the increase of disorder first causes the appearance of surface
conduction (via a topological phase transition), followed by the appearance of
bulk conduction (via a metal-insulator transition). The conducting surface
states are stabilized by an effective time-reversal symmetry that is broken
locally by the disorder but restored on long length scales. A simple
self-consistent Born approximation reliably locates the boundaries of this
socalled "statistical" topological phase.

Topological band structures in electronic systems like topological insulators
and semimetals give rise to highly unusual physical properties. Analogous
topological effects have also been discussed in bosonic systems, but the novel
phenomena typically occur only when the system is excited by finite-frequency
probes. A mapping recently proposed by Kane and Lubensky [Nat. Phys. 10, 39
(2014)], however, establishes a closer correspondence. It relates the
zero-frequency excitations of mechanical systems to topological zero modes of
fermions that appear at the edges of an otherwise gapped system. Here we
generalize the mapping to systems with an intrinsically gapless bulk. In
particular, we construct mechanical counterparts of topological semimetals. The
resulting gapless bulk modes are physically distinct from the usual acoustic
Goldstone phonons, and appear even in the absence of continuous translation
invariance. Moreover, the zero-frequency phonon modes feature adjustable
momenta and are topologically protected as long as the lattice coordination is
unchanged. Such protected soft modes with tunable wavevector may be useful in
designing mechanical structures with fault-tolerant properties.

Hint: The best of both worlds

Topologically stable gapless phases of time-reversal invariant
superconductors

We show that time-reversal invariant superconductors in d=2 (d=3) dimensions
can support topologically stable Fermi points (lines), characterized by an
integer topological charge. Combining this with the momentum space symmetries
present, we prove analogs of the fermion doubling theorem: for d=2 lattice
models admitting a spin X electron-hole structure, the number of Fermi points
is a multiple of four, while for d=3, Fermi lines come in pairs. We show two
implications of our findings for topological superconductors in d=3: first, we
relate the bulk topological invariant to a topological number for the surface
Fermi points in the form of an index theorem. Second, we show that the
existence of topologically stable Fermi lines results in extended gapless
regions in a generic topological superconductor phase diagram.

Hint: A general approach to gapless superconductors.

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 'Extensions' is available in the online version of the course.