# Simulations¶

As usual, start by grabbing the notebooks of this week (w10_extensions). They are once again over here.

## Weyl semimetal with time-reversal symmetry.¶

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.

# Review assignment¶

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

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'

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

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

Hoi Chun Po, Yasaman Bahri, Ashvin Vishwanath

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

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.