417 research outputs found
Pseudo-peakons and Cauchy analysis for an integrable fifth-order equation of Camassa-Holm type
In this paper we discuss integrable higher order equations {\em of
Camassa-Holm (CH) type}. Our higher order CH-type equations are "geometrically
integrable", that is, they describe one-parametric families of pseudo-spherical
surfaces, in a sense explained in Section 1, and they are integrable in the
sense of zero curvature formulation ( Lax pair) with infinitely many
local conservation laws. The major focus of the present paper is on a specific
fifth order CH-type equation admitting {\em pseudo-peakons} solutions, that is,
weak bounded solutions with differentiable first derivative and continuous and
bounded second derivative, but such that any higher order derivative blows up.
Furthermore, we investigate the Cauchy problem of this fifth order CH-type
equation on the real line and prove local well-posedness under the initial
conditions , . In addition, we study
conditions for global well-posedness in as well as conditions
causing local solutions to blow up in a finite time. We conclude our paper with
some comments on the geometric content of the high order CH-type equations.Comment: 6 figures; 32 page
Band Connectivity for Topological Quantum Chemistry: Band Structures As A Graph Theory Problem
The conventional theory of solids is well suited to describing band
structures locally near isolated points in momentum space, but struggles to
capture the full, global picture necessary for understanding topological
phenomena. In part of a recent paper [B. Bradlyn et al., Nature 547, 298
(2017)], we have introduced the way to overcome this difficulty by formulating
the problem of sewing together many disconnected local "k-dot-p" band
structures across the Brillouin zone in terms of graph theory. In the current
manuscript we give the details of our full theoretical construction. We show
that crystal symmetries strongly constrain the allowed connectivities of energy
bands, and we employ graph-theoretic techniques such as graph connectivity to
enumerate all the solutions to these constraints. The tools of graph theory
allow us to identify disconnected groups of bands in these solutions, and so
identify topologically distinct insulating phases.Comment: 19 pages. Companion paper to arXiv:1703.02050 and arXiv:1706.08529
v2: Accepted version, minor typos corrected and references added. Now
19+epsilon page
Graph Theory Data for Topological Quantum Chemistry
Topological phases of noninteracting particles are distinguished by global
properties of their band structure and eigenfunctions in momentum space. On the
other hand, group theory as conventionally applied to solid-state physics
focuses only on properties which are local (at high symmetry points, lines, and
planes) in the Brillouin zone. To bridge this gap, we have previously [B.
Bradlyn et al., Nature 547, 298--305 (2017)] mapped the problem of constructing
global band structures out of local data to a graph construction problem. In
this paper, we provide the explicit data and formulate the necessary algorithms
to produce all topologically distinct graphs. Furthermore, we show how to apply
these algorithms to certain "elementary" band structures highlighted in the
aforementioned reference, and so identified and tabulated all orbital types and
lattices that can give rise to topologically disconnected band structures.
Finally, we show how to use the newly developed BANDREP program on the Bilbao
Crystallographic Server to access the results of our computation.Comment: v1: 29 Pages, 13 Figures. Explains how to access the data presented
in arXiv:1703.02050 v2: Accepted version. References updated, figures
improve
Higher-Order Topological Insulators
Three-dimensional topological (crystalline) insulators are materials with an
insulating bulk, but conducting surface states which are topologically
protected by time-reversal (or spatial) symmetries. Here, we extend the notion
of three-dimensional topological insulators to systems that host no gapless
surface states, but exhibit topologically protected gapless hinge states. Their
topological character is protected by spatio-temporal symmetries, of which we
present two cases: (1) Chiral higher-order topological insulators protected by
the combination of time-reversal and a four-fold rotation symmetry. Their hinge
states are chiral modes and the bulk topology is -classified. (2)
Helical higher-order topological insulators protected by time-reversal and
mirror symmetries. Their hinge states come in Kramers pairs and the bulk
topology is -classified. We provide the topological invariants for
both cases. Furthermore we show that SnTe as well as surface-modified
BiTeI, BiSe, and BiTe are helical higher-order topological insulators and
propose a realistic experimental setup to detect the hinge states.Comment: 8 pages (4 figures) and 16 pages supplemental material (7 figures
Building Blocks of Topological Quantum Chemistry: Elementary Band Representations
The link between chemical orbitals described by local degrees of freedom and
band theory, which is defined in momentum space, was proposed by Zak several
decades ago for spinless systems with and without time-reversal in his theory
of "elementary" band representations. In Nature 547, 298-305 (2017), we
introduced the generalization of this theory to the experimentally relevant
situation of spin-orbit coupled systems with time-reversal symmetry and proved
that all bands that do not transform as band representations are topological.
Here, we give the full details of this construction. We prove that elementary
band representations are either connected as bands in the Brillouin zone and
are described by localized Wannier orbitals respecting the symmetries of the
lattice (including time-reversal when applicable), or, if disconnected,
describe topological insulators. We then show how to generate a band
representation from a particular Wyckoff position and determine which Wyckoff
positions generate elementary band representations for all space groups. This
theory applies to spinful and spinless systems, in all dimensions, with and
without time reversal. We introduce a homotopic notion of equivalence and show
that it results in a finer classification of topological phases than approaches
based only on the symmetry of wavefunctions at special points in the Brillouin
zone. Utilizing a mapping of the band connectivity into a graph theory problem,
which we introduced in Nature 547, 298-305 (2017), we show in companion papers
which Wyckoff positions can generate disconnected elementary band
representations, furnishing a natural avenue for a systematic materials search.Comment: 15+9 pages, 4 figures; v2: minor corrections; v3: updated references
(published version
Double crystallographic groups and their representations on the Bilbao Crystallographic Server
A new section of databases and programs devoted to double crystallographic
groups (point and space groups) has been implemented in the Bilbao
Crystallographic Server (http://www.cryst.ehu.es). The double crystallographic
groups are required in the study of physical systems whose Hamiltonian includes
spin-dependent terms. In the symmetry analysis of such systems, instead of the
irreducible representations of the space groups, it is necessary to consider
the single- and double-valued irreducible representations of the double space
groups. The new section includes databases of symmetry operations (DGENPOS) and
of irreducible representations of the double (point and space) groups
(REPRESENTATIONS DPG and REPRESENTATIONS DSG). The tool DCOMPATIBILITY
RELATIONS provides compatibility relations between the irreducible
representations of double space groups at different k-vectors of the Brillouin
zone when there is a group-subgroup relation between the corresponding little
groups. The program DSITESYM implements the so-called site-symmetry approach,
which establishes symmetry relations between localized and extended crystal
states, using representations of the double groups. As an application of this
approach, the program BANDREP calculates the band representations and the
elementary band representations induced from any Wyckoff position of any of the
230 double space groups, giving information about the properties of these
bands. Recently, the results of BANDREP have been extensively applied in the
description and the search of topological insulators.Comment: 32 pages, 20 figures. Two extra figures and minor typo mistakes
fixed. Published versio
Generalized r-matrix structure and algebro-geometric solution for integrable systems
The purpose of this paper is to construct a generalized r-matrix structure of
finite dimensional systems and an approach to obtain the algebro-geometric
solutions of integrable nonlinear evolution equations (NLEEs). Our starting
point is a generalized Lax matrix instead of usual Lax pair. The generalized
r-matrix structure and Hamiltonian functions are presented on the basis of
fundamental Poisson bracket. It can be clearly seen that various nonlinear
constrained (c-) and restricted (r-) systems, such as the c-AKNS, c-MKdV,
c-Toda, r-Toda, c-Levi, etc, are derived from the reduction of this structure.
All these nonlinear systems have {\it r}-matrices, and are completely
integrable in Liouville's sense. Furthermore, our generalized structure is
developed to become an approach to obtain the algebro-geometric solutions of
integrable NLEEs. Finally, the two typical examples are considered to
illustrate this approach: the infinite or periodic Toda lattice equation and
the AKNS equation with the condition of decay at infinity or periodic boundary.Comment: 41 pages, 0 figure
Andreev Reflection without Fermi surface alignment in High T-Topological heterostructures
We address the controversy over the proximity effect between topological
materials and high T superconductors. Junctions are produced between
BiSrCaCuO and materials with different Fermi
surfaces (BiTe \& graphite). Both cases reveal tunneling spectra
consistent with Andreev reflection. This is confirmed by magnetic field that
shifts features via the Doppler effect. This is modeled with a single parameter
that accounts for tunneling into a screening supercurrent. Thus the tunneling
involves Cooper pairs crossing the heterostructure, showing the Fermi surface
mis-match does not hinder the ability to form transparent interfaces, which is
accounted for by the extended Brillouin zone and different lattice symmetries
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