150 research outputs found
Interplay between electronic topology and crystal symmetry: Dislocation-line modes in topological band-insulators
We elucidate the general rule governing the response of dislocation lines in
three-dimensional topological band insulators. According to this rule, the lattice topology, represented by
dislocation lines oriented in direction with Burgers vector , combines with the electronic-band topology, characterized by the
band-inversion momentum , to produce gapless propagating
modes when the plane orthogonal to the dislocation line features a band
inversion with a nontrivial ensuing flux . Although it has already been discovered by Y. Ran
{\it et al.}, Nature Phys. {\bf 5}, 298 (2009), that dislocation lines host
propagating modes, the exact mechanism of their appearance in conjunction with
the crystal symmetries of a topological state is provided by the rule . Finally, we discuss possible
experimentally consequential examples in which the modes are oblivious for the
direction of propagation, such as the recently proposed
topologically-insulating state in electron-doped BaBiO.Comment: Main text + supplementary material, published versio
Impurity Bound States and Greens Function Zeroes as Local Signatures of Topology
We show that the local in-gap Greens function of a band insulator
, with
the position perpendicular to a codimension-1 or -2
impurity, reveals the topological nature of the phase. For a topological
insulator, the eigenvalues of this Greens function attain zeros in the gap,
whereas for a trivial insulator the eigenvalues remain nonzero. This
topological classification is related to the existence of in-gap bound states
along codimension-1 and -2 impurities. Whereas codimension-1 impurities can be
viewed as 'soft edges', the result for codimension-2 impurities is nontrivial
and allows for a direct experimental measurement of the topological nature of
2d insulators.Comment: 11 pages, 8 figure
Self-organized pseudo-graphene on grain boundaries in topological band insulators
Semi-metals are characterized by nodal band structures that give rise to
exotic electronic properties. The stability of Dirac semi-metals, such as
graphene in two spatial dimensions (2D), requires the presence of lattice
symmetries, while akin to the surface states of topological band insulators,
Weyl semi-metals in three spatial dimensions (3D) are protected by band
topology. Here we show that in the bulk of topological band insulators,
self-organized topologically protected semi-metals can emerge along a grain
boundary, a ubiquitous extended lattice defect in any crystalline material. In
addition to experimentally accessible electronic transport measurements, these
states exhibit valley anomaly in 2D influencing edge spin transport, whereas in
3D they appear as graphene-like states that may exhibit an odd-integer quantum
Hall effect. The general mechanism underlying these novel semi-metals -- the
hybridization of spinon modes bound to the grain boundary -- suggests that
topological semi-metals can emerge in any topological material where lattice
dislocations bind localized topological modes.Comment: 14 pages, 6 figures. Improved discussion compared to the earlier
versio
Hopf characterization of two-dimensional Floquet topological insulators
We present a topological characterization of time-periodically driven
two-band models in 2+1 dimensions as Hopf insulators. The intrinsic periodicity
of the Floquet system with respect to both time and the underlying
two-dimensional momentum space constitutes a map from a three dimensional torus
to the Bloch sphere. As a result, we find that the driven system can be
understood by appealing to a Hopf map that is directly constructed from the
micromotion of the drive. Previously found winding numbers are shown to
correspond to Hopf invariants, which are associated with linking numbers
describing the topology of knots in three dimensions. Moreover, after being
cast as a Hopf insulator, not only the Chern numbers, but also the winding
numbers of the Floquet topological insulator become accessible in experiments
as linking numbers. We exploit this description to propose a feasible scheme
for measuring the complete set of their Floquet topological invariants in
optical lattices.Comment: 6 pages, 3 figures + 2 pages, 1 figure supplementar
Generalized liquid crystals: giant fluctuations and the vestigial chiral order of , and matter
The physics of nematic liquid crystals has been subject of intensive research
since the late 19th century. However, because of the limitations of chemistry
the focus has been centered around uni- and biaxial nematics associated with
constituents bearing a or symmetry respectively. In
view of general symmetries, however, these are singularly special since nematic
order can in principle involve any point group symmetry. Given the progress in
tailoring nano particles with particular shapes and interactions, this vast
family of "generalized nematics" might become accessible in the laboratory.
Little is known since the order parameter theories associated with the highly
symmetric point groups are remarkably complicated, involving tensor order
parameters of high rank. Here we show that the generic features of the
statistical physics of such systems can be studied in a highly flexible and
efficient fashion using a mathematical tool borrowed from high energy physics:
discrete non-Abelian gauge theory. Explicitly, we construct a family of lattice
gauge models encapsulating nematic ordering of general three dimensional point
group symmetries. We find that the most symmetrical "generalized nematics" are
subjected to thermal fluctuations of unprecedented severity. As a result, novel
forms of fluctuation phenomena become possible. In particular, we demonstrate
that a vestigial phase carrying no more than chiral order becomes ubiquitous
departing from high point group symmetry chiral building blocks, such as ,
and symmetric matter.Comment: 14 pages, 5 figures; published versio
Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time reversal symmetry
We present a general methodology towards the systematic characterization of
crystalline topological insulating phases with time reversal symmetry (TRS).~In
particular, taking the two-dimensional spinful hexagonal lattice as a proof of
principle we study windings of Wilson loop spectra over cuts in the Brillouin
zone that are dictated by the underlying lattice symmetries.~Our approach finds
a prominent use in elucidating and quantifying the recently proposed
``topological quantum chemistry" (TQC) concept.~Namely, we prove that the split
of an elementary band representation (EBR) by a band gap must lead to a
topological phase.~For this we first show that in addition to the Fu-Kane-Mele
classification, there is -symmetry protected
classification of two-band subspaces that is obstructed by the
other crystalline symmetries, i.e.~forbidding the trivial phase. This accounts
for all nontrivial Wilson loop windings of split EBRs \textit{that are
independent of the parameterization of the flow of Wilson loops}.~Then, we show
that while Wilson loop winding of split EBRs can unwind when embedded in
higher-dimensional band space, two-band subspaces that remain separated by a
band gap from the other bands conserve their Wilson loop winding, hence
revealing that split EBRs are at least "stably trivial", i.e. necessarily
non-trivial in the non-stable (few-band) limit but possibly trivial in the
stable (many-band) limit.~This clarifies the nature of \textit{fragile}
topology that has appeared very recently.~We then argue that in the many-band
limit the stable Wilson loop winding is only determined by the Fu-Kane-Mele
invariant implying that further stable topological phases must
belong to the class of higher-order topological insulators.Comment: 27 pages, 13 figures, v2: minor corrections, new references included,
v3: metastable topology of split EBRs emphasized, v4: prepared for
publicatio
Wannier representation of Floquet topological states
A universal feature of topological insulators is that they cannot be
adiabatically connected to an atomic limit, where individual lattice sites are
completely decoupled. This property is intimately related to a topological
obstruction to constructing a localized Wannier function from Bloch states of
an insulator. Here we generalize this characterization of topological phases
toward periodically driven systems. We show that nontrivial connectivity of
hybrid Wannier centers in momentum space and time can characterize various
types of topology in periodically driven systems, which include Floquet
topological insulators, anomalous Floquet topological insulators with
micromotion-induced boundary states, and gapless Floquet states realized with
topological Floquet operators. In particular, nontrivial time dependence of
hybrid Wannier centers indicates impossibility of continuous deformation of a
driven system into an undriven insulator, and a topological Floquet operator
implies an obstruction to constructing a generalized Wannier function which is
localized in real and frequency spaces. Our results pave a way to a unified
understanding of topological states in periodically driven systems as a
topological obstruction in Floquet states.Comment: 17 pages, 5 figure
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