90 research outputs found
An Introduction to Topological Insulators
Electronic bands in crystals are described by an ensemble of Bloch wave
functions indexed by momenta defined in the first Brillouin Zone, and their
associated energies. In an insulator, an energy gap around the chemical
potential separates valence bands from conduction bands. The ensemble of
valence bands is then a well defined object, which can possess non-trivial or
twisted topological properties. In the case of a twisted topology, the
insulator is called a topological insulator. We introduce this notion of
topological order in insulators as an obstruction to define the Bloch wave
functions over the whole Brillouin Zone using a single phase convention.
Several simple historical models displaying a topological order in dimension
two are considered. Various expressions of the corresponding topological index
are finally discussed.Comment: 46 pages, 29 figures. This papers aims to be a pedagogical review on
topological insulators. It was written for the topical issue of "Comptes
Rendus de l'Acad\'emie des Sciences - Physique" devoted to topological
insulators and Dirac matte
Parallel Transport and Band Theory in Crystals
We show that different conventions for Bloch Hamiltonians on non-Bravais
lattices correspond to different natural definitions of parallel transport of
Bloch eigenstates. Generically the Berry curvatures associated with these
parallel transports differ, while physical quantities are naturally related to
a canonical choice of the parallel transport.Comment: 5 pages, 1 figure ; minor updat
Dualities and non-Abelian mechanics
Dualities are mathematical mappings that reveal unexpected links between
apparently unrelated systems or quantities in virtually every branch of
physics. Systems that are mapped onto themselves by a duality transformation
are called self-dual and they often exhibit remarkable properties, as
exemplified by an Ising magnet at the critical point. In this Letter, we unveil
the role of dualities in mechanics by considering a family of so-called twisted
Kagome lattices. These are reconfigurable structures that can change shape
thanks to a collapse mechanism easily illustrated using LEGO. Surprisingly,
pairs of distinct configurations along the mechanism exhibit the same spectrum
of vibrational modes. We show that this puzzling property arises from the
existence of a duality transformation between pairs of configurations on either
side of a mechanical critical point. This critical point corresponds to a
self-dual structure whose vibrational spectrum is two-fold degenerate over the
entire Brillouin zone. The two-fold degeneracy originates from a general
version of Kramers theorem that applies to classical waves in addition to
quantum systems with fermionic time-reversal invariance. We show that the
vibrational modes of the self-dual mechanical systems exhibit non-Abelian
geometric phases that affect the semi-classical propagation of wave packets.
Our results apply to linear systems beyond mechanics and illustrate how
dualities can be harnessed to design metamaterials with anomalous symmetries
and non-commuting responses.Comment: See http://home.uchicago.edu/~vitelli/videos.html for Supplementary
Movi
Probing (topological) Floquet states through DC transport
We consider the differential conductance of a periodically driven system
connected to infinite electrodes. We focus on the situation where the
dissipation occurs predominantly in these electrodes. Using analytical
arguments and a detailed numerical study we relate the differential
conductances of such a system in two and three terminal geometries to the
spectrum of quasi-energies of the Floquet operator. Moreover these differential
conductances are found to provide an accurate probe of the existence of gaps in
this quasi-energy spectrum, being quantized when topological edge states occur
within these gaps. Our analysis opens the perspective to describe the
intermediate time dynamics of driven mesoscopic conductors as topological
Floquet filters.Comment: 8 pages, 6 figures, invited contribution to the special issue of
Physica E on "Frontiers in quantum electronic transport" in memory of Markus
Buttike
Computation of topological phase diagram of disordered PbSnTe using the kernel polynomial method
We present an algorithm to determine topological invariants of inhomogeneous
systems, such as alloys, disordered crystals, or amorphous systems. Based on
the kernel polynomial method, our algorithm allows us to study samples with
more than degrees of freedom. Our method enables the study of large
complex compounds, where disorder is inherent to the system. We use it to
analyse PbSnTe and tighten the critical concentration for the
phase transition.Comment: 4 pages + supplemental materia
Soft self-assembly of Weyl materials for light and sound
Soft materials can self-assemble into highly structured phases which
replicate at the mesoscopic scale the symmetry of atomic crystals. As such,
they offer an unparalleled platform to design mesostructured materials for
light and sound. Here, we present a bottom-up approach based on self-assembly
to engineer three-dimensional photonic and phononic crystals with topologically
protected Weyl points. In addition to angular and frequency selectivity of
their bulk optical response, Weyl materials are endowed with topological
surface states, which allows for the existence of one-way channels even in the
presence of time-reversal invariance. Using a combination of group-theoretical
methods and numerical simulations, we identify the general symmetry constraints
that a self-assembled structure has to satisfy in order to host Weyl points,
and describe how to achieve such constraints using a symmetry-driven pipeline
for self-assembled material design and discovery. We illustrate our general
approach using block copolymer self-assembly as a model system.Comment: published version, SI are available as ancillary files, code and data
are available on Zenodo at https://doi.org/10.5281/zenodo.1182581, PNAS
(2018
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