10,785 research outputs found
Projective Ribbon Permutation Statistics: a Remnant of non-Abelian Braiding in Higher Dimensions
In a recent paper, Teo and Kane proposed a 3D model in which the defects
support Majorana fermion zero modes. They argued that exchanging and twisting
these defects would implement a set R of unitary transformations on the zero
mode Hilbert space which is a 'ghostly' recollection of the action of the braid
group on Ising anyons in 2D. In this paper, we find the group T_{2n} which
governs the statistics of these defects by analyzing the topology of the space
K_{2n} of configurations of 2n defects in a slowly spatially-varying gapped
free fermion Hamiltonian: T_{2n}\equiv {\pi_1}(K_{2n})$. We find that the group
T_{2n}= Z \times T^r_{2n}, where the 'ribbon permutation group' T^r_{2n} is a
mild enhancement of the permutation group S_{2n}: T^r_{2n} \equiv \Z_2 \times
E((Z_2)^{2n}\rtimes S_{2n}). Here, E((Z_2)^{2n}\rtimes S_{2n}) is the 'even
part' of (Z_2)^{2n} \rtimes S_{2n}, namely those elements for which the total
parity of the element in (Z_2)^{2n} added to the parity of the permutation is
even. Surprisingly, R is only a projective representation of T_{2n}, a
possibility proposed by Wilczek. Thus, Teo and Kane's defects realize
`Projective Ribbon Permutation Statistics', which we show to be consistent with
locality. We extend this phenomenon to other dimensions, co-dimensions, and
symmetry classes. Since it is an essential input for our calculation, we review
the topological classification of gapped free fermion systems and its relation
to Bott periodicity.Comment: Missing figures added. Fixed some typos. Added a paragraph to the
conclusio
Lecture Notes on Topological Crystalline Insulators
We give an introduction to topological crystalline insulators, that is,
gapped ground states of quantum matter that are not adiabatically connected to
an atomic limit without breaking symmetries that include spatial
transformations, like mirror or rotational symmetries. To deduce the
topological properties, we use non-Abelian Wilson loops. We also discuss in
detail higher-order topological insulators with hinge and corner states, and in
particular present interacting bosonic models for the latter class of systems.Comment: Lectures given at the San Sebasti\'an Topological Matter School 2017,
published in "Topological Matter. Springer Series in Solid-State Sciences,
vol 190. Springer, Cham
Loop-closure principles in protein folding
Simple theoretical concepts and models have been helpful to understand the
folding rates and routes of single-domain proteins. As reviewed in this
article, a physical principle that appears to underly these models is loop
closure.Comment: 27 pages, 5 figures; to appear in Archives of Biochemistry and
Biophysic
On the Closing Lemma problem for vector fields of bounded type on the torus
We investigate the open Closing Lemma problem for vector fields on the
2-dimensional torus. Under the assumption of bounded type rotation number, the
Closing Lemma is verified for smooth vector fields that are
area-preserving at all saddle points. Namely, given such a vector field
, , with a non-trivially recurrent point , there exists a vector
field arbitrarily near to in the topology and obtained from
by a twist perturbation, such that is a periodic point of .
The proof relies on a new result in 1-dimensional dynamics on the
non-existence of semi-wandering intervals of smooth maps of the circle.Comment: 11 pages, 1 figur
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
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