As usual, start by grabbing the notebooks of this week (w11_extensions2). They are once again over here.

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.

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 'Floquet and crystalline' is available in the online version of the course.

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

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

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

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?

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 'Floquet and crystalline' is available in the online version of the course.