As usual, start by grabbing the notebooks of this week (`w10_extensions`

). They are once again over here.

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 edge states.¶

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.**

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

### Discovery of a Weyl Fermion semimetal state in NbAs

http://arxiv.org/abs/1504.01350

Su-Yang Xu, Nasser Alidoust, Ilya Belopolski, Chenglong Zhang, Guang Bian, Tay-Rong Chang, Hao Zheng, Vladimir Strokov, Daniel S. Sanchez, Guoqing Chang, Zhujun Yuan, Daixiang Mou, Yun Wu, Lunan Huang, Chi-Cheng Lee, Shin-Ming Huang, BaoKai Wang, Arun Bansil, Horng-Tay Jeng, Titus Neupert, Adam Kaminski, Hsin Lin, Shuang Jia, M. Zahid Hasan

We report the discovery of Weyl semimetal NbAs featuring topological Fermi
arc surface states.

### Observation of phononic helical edge states in a mechanical 'topological
insulator'

http://arxiv.org/abs/1503.06808

Roman Süsstrunk, Sebastian D. Huber

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

http://arxiv.org/abs/1309.5846

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.

**Hint:** Weyl + disorder

### Phonon analogue of topological nodal semimetals

http://arxiv.org/abs/1410.1320

Hoi Chun Po, Yasaman Bahri, Ashvin Vishwanath

Recently, Kane and Lubensky proposed a mapping between bosonic phonon
problems on isostatic lattices to chiral fermion systems based on factorization
of the dynamical matrix [Nat. Phys. 10, 39 (2014)]. The existence of
topologically protected zero modes in such mechanical problems is related to
their presence in the fermionic system and is dictated by a local index
theorem. Here we adopt the proposed mapping to construct a two-dimensional
mechanical analogue of a fermionic topological nodal semimetal that hosts a
robust bulk node in its linearized phonon spectrum. Such topologically
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

http://arxiv.org/abs/0909.5680

B. Béri

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.

### 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!

** 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.