# Topics for self-study¶

## Simulation: powers combined¶

Are you tired yet of all the different kinds of topology? If no, this assignment is for you :-)

By now you should have a feel for how to make new topological phases. Your task now is to combine the two systems you’ve learned about and to create a Floquet crystalline topological insulator. If you want even more challenge, create also a gapless Floquet topological material.

Take care however: if you take a topologically nontrivial system and just apply rapid driving, you’ll still get a topological one. This cheating way is prohibited: at any moment during the driving cycle the Hamiltonian of your system should remain gapped.

**Now share your results:**

## Review assignment¶

### Topological characterization of periodically-driven quantum systems (arXiv:1010.6126)¶

Takuya Kitagawa, Erez Berg, Mark Rudner, Eugene Demler

Topological properties of physical systems can lead to robust behaviors that are insensitive to microscopic details. Such topologically robust phenomena are not limited to static systems but can also appear in driven quantum systems. In this paper, we show that the Floquet operators of periodically driven systems can be divided into topologically distinct (homotopy) classes, and give a simple physical interpretation of this classification in terms of the spectra of Floquet operators. Using this picture, we provide an intuitive understanding of the well-known phenomenon of quantized adiabatic pumping. Systems whose Floquet operators belong to the trivial class simulate the dynamics generated by time-independent Hamiltonians, which can be topologically classified according to the schemes developed for static systems. We demonstrate these principles through an example of a periodically driven two–dimensional hexagonal lattice model which exhibits several topological phases. Remarkably, one of these phases supports chiral edge modes even though the bulk is topologically trivial.

**Hint:** Computes topological edge states from Floquet Hamiltonian.

### Anomalous edge states and the bulk-edge correspondence for periodically-driven two dimensional systems (arXiv:1212.3324)¶

Mark S. Rudner, Netanel H. Lindner, Erez Berg, Michael Levin

Recently, several authors have investigated topological phenomena in periodically-driven systems of non-interacting particles. These phenomena are identified through analogies between the Floquet spectra of driven systems and the band structures of static Hamiltonians. Intriguingly, these works have revealed that the topological characterization of driven systems is richer than that of static systems. In particular, in driven systems in two dimensions (2D), robust chiral edge states can appear even though the Chern numbers of all the bulk Floquet bands are zero. Here we elucidate the crucial distinctions between static and driven 2D systems, and construct a new topological invariant that yields the correct edge state structure in the driven case. We provide formulations in both the time and frequency domains, which afford additional insight into the origins of the “anomalous” spectra which arise in driven systems. Possible realizations of these phenomena in solid state and cold atomic systems are discussed.

**Hint:** Points out and explains why the floquet Hamiltonian in momentum space does not capture the presence of Floquet edge states.

### Topological Crystalline Insulators in the SnTe Material Class (arXiv:1202.1003)¶

Timothy H. Hsieh, Hsin Lin, Junwei Liu, Wenhui Duan, Arun Bansil, Liang Fu

Topological crystalline insulators are new states of matter in which the topological nature of electronic structures arises from crystal symmetries. Here we predict the first material realization of topological crystalline insulator in the semiconductor SnTe, by identifying its nonzero topological index. We predict that as a manifestation of this nontrivial topology, SnTe has metallic surface states with an even number of Dirac cones on high-symmetry crystal surfaces such as {001}, {110} and {111}. These surface states form a new type of high-mobility chiral electron gas, which is robust against disorder and topologically protected by reflection symmetry of the crystal with respect to {110} mirror plane. Breaking this mirror symmetry via elastic strain engineering or applying an in-plane magnetic field can open up a continuously tunable band gap on the surface, which may lead to wide-ranging applications in thermoelectrics, infrared detection, and tunable electronics. Closely related semiconductors PbTe and PbSe also become topological crystalline insulators after band inversion by pressure, strain and alloying.

**Hint:** Theoretical prediction of topological crystalline insulator.

### Statistical Topological Insulators (arXiv:1212.6191)¶

I. C. Fulga, B. van Heck, J. M. Edge, A. R. Akhmerov

We define a class of insulators with gapless surface states protected from localization due to the statistical properties of a disordered ensemble, namely due to the ensemble’s invariance under a certain symmetry. We show that these insulators are topological, and are protected by a \(\mathbb{Z}_2\) invariant. Finally, we prove that every topological insulator gives rise to an infinite number of classes of statistical topological insulators in higher dimensions. Our conclusions are confirmed by numerical simulations.

**Hint:** Are topological crystalline surface states stable against disorder?

### 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!