A relation between chiral central charge and ground state degeneracy in 2+1-dimensional topological orders

A bosonic topological order on $d$-dimensional closed space $\Sigma^d$ may have degenerate ground states. The space $\Sigma^d$ with different shapes (different metrics) form a moduli space ${\cal M}_{\Sigma^d}$. Thus the degenerate ground states on every point in the moduli space ${\cal M}_{\Sigma^d}$ form a complex vector bundle over ${\cal M}_{\Sigma^d}$. It was suggested that the collection of such vector bundles for $d$-dimensional closed spaces of all topologies completely characterizes the topological order. Using such a point of view, we propose a direct relation between two seemingly unrelated properties of 2+1-dimensional topological orders: (1) the chiral central charge $c$ that describes the many-body density of states for edge excitations (or more precisely the thermal Hall conductance of the edge), (2) the ground state degeneracy $D_g$ on closed genus $g$ surface. We show that $c D_g/2 \in \mathbb{Z},\ g\geq 3$ for bosonic topological orders. We explicitly checked the validity of this relation for over 140 simple topological orders. For fermionic topological orders, let $D_{g,\sigma}^{e}$ ($D_{g,\sigma}^{o}$) be the degeneracy with even (odd) number of fermions for genus-$g$ surface with spin structure $\sigma$. Then we have $2c D_{g,\sigma}^{e} \in \mathbb{Z}$ and $2c D_{g,\sigma}^{o} \in \mathbb{Z}$ for $g\geq 3$.Comment: 8 pages. This paper supersedes Section XIV of an unpublished work arXiv:1405.5858. We add new results on fermionic topological orders and some numerical check

A classification of 3+1D bosonic topological orders (I): the case when point-like excitations are all bosons

Topological orders are new phases of matter beyond Landau symmetry breaking. They correspond to patterns of long-range entanglement. In recent years, it was shown that in 1+1D bosonic systems there is no nontrivial topological order, while in 2+1D bosonic systems the topological orders are classified by a pair: a modular tensor category and a chiral central charge. In this paper, we propose a partial classification of topological orders for 3+1D bosonic systems: If all the point-like excitations are bosons, then such topological orders are classified by unitary pointed fusion 2-categories, which are one-to-one labeled by a finite group $G$ and its group 4-cocycle $\omega_4 \in \mathcal H^4[G;U(1)]$ up to group automorphisms. Furthermore, all such 3+1D topological orders can be realized by Dijkgraaf-Witten gauge theories.Comment: An important new result "Untwisted sector of dimension reduction is the Drinfeld center of E" is added in Sec. IIIC; other minor refinements and improvements; 23 pages, 10 figure

A theory of 2+1D fermionic topological orders and fermionic/bosonic topological orders with symmetries

We propose that, up to invertible topological orders, 2+1D fermionic topological orders without symmetry and 2+1D fermionic/bosonic topological orders with symmetry $G$ are classified by non-degenerate unitary braided fusion categories (UBFC) over a symmetric fusion category (SFC); the SFC describes a fermionic product state without symmetry or a fermionic/bosonic product state with symmetry $G$, and the UBFC has a modular extension. We developed a simplified theory of non-degenerate UBFC over a SFC based on the fusion coefficients $N^{ij}_k$ and spins $s_i$. This allows us to obtain a list that contains all 2+1D fermionic topological orders (without symmetry). We find explicit realizations for all the fermionic topological orders in the table. For example, we find that, up to invertible $p+\hspace{1pt}\mathrm{i}\hspace{1pt} p$ fermionic topological orders, there are only four fermionic topological orders with one non-trivial topological excitation: (1) the $K={\scriptsize \begin{pmatrix} -1&0\\0&2\end{pmatrix}}$ fractional quantum Hall state, (2) a Fibonacci bosonic topological order $2^B_{14/5}$ stacking with a fermionic product state, (3) the time-reversal conjugate of the previous one, (4) a primitive fermionic topological order that has a chiral central charge $c=\frac14$, whose only topological excitation has a non-abelian statistics with a spin $s=\frac14$ and a quantum dimension $d=1+\sqrt{2}$. We also proposed a categorical way to classify 2+1D invertible fermionic topological orders using modular extensions.Comment: 23 pages, 8 table

Transport in three-dimensional topological insulators: theory and experiment

This article reviews recent theoretical and experimental work on transport due to the surface states of three-dimensional topological insulators. The theoretical focus is on longitudinal transport in the presence of an electric field, including Boltzmann transport, quantum corrections and weak localization, as well as longitudinal and Hall transport in the presence of both electric and magnetic fields and/or magnetizations. Special attention is paid to transport at finite doping, to the $\pi$-Berry phase, which leads to the absence of backscattering, Klein tunneling and half-quantized Hall response. Signatures of surface states in ordinary transport and magnetotransport are clearly identified. The review also covers transport experiments of the past years, reviewing the initial obscuring of surface transport by bulk transport, and the way transport due to the surface states has increasingly been identified experimentally. Current and likely future experimental challenges are given prominence and the current status of the field is assessed.Comment: Review article to appear in Physica

Copper-containing mesoporous bioactive glass promotes angiogenesis in an in vivo zebrafish model

The osteogenic and angiogenic responses of organisms to the ionic products of degradation of bioactive glasses (BGs) are being intensively investigated. The promotion of angiogenesis by copper (Cu) has been known for more than three decades. This element can be incorporated to delivery carriers, such as BGs, and the materials used in biological assays. In this work, Cu-containing mesoporous bioactive glass (MBG) in the SiO2-CaO-P2O5 compositional system was prepared incorporating 5% mol Cu (MBG-5Cu) by replacement of the corresponding amount of Ca. The biological effects of the ionic products of MBG biodegradation were evaluated on a well-known endothelial cell line, the bovine aorta endothelial cells (BAEC), as well as in an in vivo zebrafish (Danio rerio) embryo assay. The results suggest that ionic products of both MBG (Cu free) and MBG-5Cu materials promote angiogenesis. In vitro cell cultures show that the ionic dissolution products of these materials are not toxic and promote BAEC viability and migration. In addition, the in vivo assay indicates that both exposition and microinjection of zebrafish embryos with Cu free MBG material increase vessel number and thickness of the subintestinal venous plexus (SIVP), whereas assays using MBG-5Cu enhance this effect.The authors gratefully acknowledge the financial support provided by the Andalusian Ministry of Economy, Science and Innovation (Proyectos Excelencia Grants no. P10-CTS-6681 and no. P12-CTS-1507) and Spanish Ministry of Economy and Competitivity (BIO2014-56092-R). LBRS acknowledges the CONACYT-Mexico Fellowship PhD Program

Rejoinder: Quantifying the Fraction of Missing Information for Hypothesis Testing in Statistical and Genetic Studies

Rejoinder to "Quantifying the Fraction of Missing Information for Hypothesis Testing in Statistical and Genetic Studies" [arXiv:1102.2774]Comment: Published in at http://dx.doi.org/10.1214/08-STS244REJ the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org