43,896 research outputs found
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
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
Decomposing labeled interval orders as pairs of permutations
We introduce ballot matrices, a signed combinatorial structure whose
definition naturally follows from the generating function for labeled interval
orders. A sign reversing involution on ballot matrices is defined. We show that
matrices fixed under this involution are in bijection with labeled interval
orders and that they decompose to a pair consisting of a permutation and an
inversion table. To fully classify such pairs, results pertaining to the
enumeration of permutations having a given set of ascent bottoms are given.
This allows for a new formula for the number of labeled interval orders
Matrix representations and independencies in directed acyclic graphs
For a directed acyclic graph, there are two known criteria to decide whether
any specific conditional independence statement is implied for all
distributions factorized according to the given graph. Both criteria are based
on special types of path in graphs. They are called separation criteria because
independence holds whenever the conditioning set is a separating set in a graph
theoretical sense. We introduce and discuss an alternative approach using
binary matrix representations of graphs in which zeros indicate independence
statements. A matrix condition is shown to give a new path criterion for
separation and to be equivalent to each of the previous two path criteria.Comment: Published in at http://dx.doi.org/10.1214/08-AOS594 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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