New gap theorem on complete Riemannian manifolds

In this short note, we find a new gap phenomena on Riemannian manifolds, which says that for any complete noncompact Riemannian manifold with nonnegative Ricci curvature, if the scalar curvature decays faster than quadratically, then it is Ricci flat.Comment: This paper has been withdrawn by the author due to some crucial error

A double inequality for bounding Toader mean by the centroidal mean

In the paper, the authors find the best numbers $\alpha$ and $\beta$ such that $$\overline{C}\bigl(\alpha a+(1-\alpha)b,\alpha b+(1-\alpha)a\bigr)<T(a,b) <\overline{C}\bigl(\beta a+(1-\beta)b,\beta b+(1-\beta)a\bigr)$$ for all $a,b>0$ with $a\ne b$, where $\overline{C}(a,b)={2\bigl(a^2+ab+b^2\bigr)}{3(a+b)}$ and $T(a,b)=\frac{2}{\pi}\int_{0}^{{\pi}/{2}}\sqrt{a^2{\cos^2{\theta}}+b^2{\sin^2{\theta}}}\,d\theta$ denote respectively the centroidal mean and Toader mean of two positive numbers $a$ and $b$.Comment: 5 page

Exact holographic mapping in free fermion systems

In this paper, we shall perform a detailed analysis of the Exact Holographic Mapping first introduced in arXiv:1309.6282, which was proposed as an explicit example of holographic duality between quantum many-body systems and gravitational theories. We obtain analytic results for free-fermion systems that not only confirm previous numerical results, but also elucidate the exact relationships between the various physical properties of the bulk and boundary systems. Our analytic results allow us to study the asymptotic properties that are difficult to probe numerically, such as the near horizon regime of the black hole geometry. We shall also explore a few interesting but hitherto unexplored bulk geometries, such as that corresponding to a boundary critical fermion with nontrivial dynamic critical exponent. Our analytic framework also allows us to study the holographic mapping of some of these boundary theories in dimensions 2+1 or higher.Comment: 32 pages, 7 figure

On the M-eigenvalues of elasticity tensor and the strong ellipticity condition

Strong ellipticity is an important property in the elasticity theory. In 2009, M-eigenvalues were introduced for the elasticity tensor. It was shown that M-eigenvalues are invariant under coordinate system choices, and the strong ellipticity condition holds if and only if all the M-eigenvalues of the elasticity tensor are positive. Thus, M-eigenvalues are some intrinsic parameters of the elasticity tensor. In this paper, we show that the M-eigenvalues of the elasticity tensor are closely related with some elastic moduli, such as the bulk modulus, the shear modulus, Lam\'e's first parameter, the P-wave modulus, etc, and the positiveness of the M-eigenvalues are corresponding to some existing conditions for strong ellipticity in some special cases, such as the isotropic case, the cubic case, the polar anisotropic case and the tetragonal case. We also present new sufficient conditions for the strong ellipticity of the orthotropic case. These, in a certain sense, further reveal the physical meanings of M-eigenvalues

Lattice construction of pseudopotential Hamiltonians for Fractional Chern Insulators

Fractional Chern insulators are new realizations of fractional quantum Hall states in lattice systems without orbital magnetic field. These states can be mapped onto conventional fractional quantum Hall states through the Wannier state representation (Phys. Rev. Lett. 107, 126803 (2011)). In this paper, we use the Wannier state representation to construct the pseudopotential Hamiltonians for fractional Chern insulators, which are interaction Hamiltonians with certain ideal model wavefunctions as exact ground states. We show that these pseudopotential Hamiltonians can be approximated by short-ranged interactions in fractional Chern insulators, and that their range will be minimized by an optimal gauge choice for the Wannier states. As illustrative examples, we explicitly write down the form of the lowest pseudopotential for several fractional Chern insulator models including the lattice Dirac model and the checkerboard model with Chern number 1, and the d-wave model and the triangular lattice model with Chern number 2. The proposed pseudopotential Hamiltonians have the 1/3 Laughlin state as their groundstate when the Chern number C=1, and a topological nematic (330) state as their groundstate when C=2. Also included are the results of an interpolation between the d-wave model and two decoupled layers of lattice Dirac models, which explicitly demonstrate the relation between C=2 fractional Chern insulators and bilayer fractional quantum Hall states. The proposed states can be verified by future numerical works, and in particular provide a model Hamiltonian for the topological nematic states that have not been realized numerically.Comment: 16 pages, 9 figure

Uncovering the community structure associated with the diffusion dynamics of networks

As two main focuses of the study of complex networks, the community structure and the dynamics on networks have both attracted much attention in various scientific fields. However, it is still an open question how the community structure is associated with the dynamics on complex networks. In this paper, through investigating the diffusion process taking place on networks, we demonstrate that the intrinsic community structure of networks can be revealed by the stable local equilibrium states of the diffusion process. Furthermore, we show that such community structure can be directly identified through the optimization of the conductance of network, which measures how easily the diffusion occurs among different communities. Tests on benchmark networks indicate that the conductance optimization method significantly outperforms the modularity optimization methods at identifying the community structure of networks. Applications on real world networks also demonstrate the effectiveness of the conductance optimization method. This work provides insights into the multiple topological scales of complex networks, and the obtained community structure can naturally reflect the diffusion capability of the underlying network.Comment: 10 pages, 5 figure

Breather-to-soliton and rogue wave-to-soliton transitions in a resonant erbium-doped fiber system with higher-order effects

Under investigation in this paper is the higherorder nonlinear Schrodinger and Maxwell-Bloch (HNLSMB) system which describes the wave propagation in an erbium-doped nonlinear fiber with higher-order effects including the fourth-order dispersion and quintic nonKerr nonlinearity. The breather and rogue wave (RW) solutions are shown that they can be converted into various soliton solutions including the multipeak soliton, periodic wave, antidark soliton, M-shaped soliton, and W-shaped soliton. In addition, under different values of higher-order effect, the locus of the eigenvalues on the complex plane which converts breathers or RWs into solitons is calculated

Quantum Anomalous Hall Effect in a Perovskite and Inverse-Perovskite Sandwich Structure

Based on first-principles calculations, we propose a sandwich structure composed of a G-type anti-ferromagnetic (AFM) Mott insulator LaCrO$_3$ grown along the [001] direction with one atomic layer replaced by an inverse-perovskite material Sr$_3$PbO. We show that the system is in a topologically nontrivial phase characterized by simultaneous nonzero charge and spin Chern numbers, which can support a spin-polarized and dissipationless edge current in a finite system. Since these two materials are stable in bulk and match each other with only small lattice distortions, the composite material is expected easy to synthesize.Comment: 4 pages, 4 figure

Electrically Tunable Topological State in [111] Perovskite Materials with Antiferromagnetic Exchange Field

A topological state with simultaneous nonzero Chern number and spin Chern number is possible for electrons on honeycomb lattice based on band engineering by staggered electric potential and antiferromagnetic exchange field in presence of intrinsic spin-orbit coupling. With first principles calculation we confirm that the scheme can be realized by material modification in perovskite G-type antiferromagnetic insulators grown along [111] direction, where d electrons hop on a single buckled honeycomb lattice. This material is ideal for spintronics applications, since it provides a spin-polarized quantized edge current, robust to both nonmagnetic and magnetic defects, with the spin polarization tunable by inverting electric field.Comment: 5 pages, 5 figure

Resonance interaction of two dipoles in optically active surroundings

We study the resonance interaction between two quantum electric dipoles immersed in optically active surroundings. Quantum electrodynamics is employed to deal with dipole-vacuum interaction. Our results show that the optical activity of surroundings will not change the single atom behaviors while it can change the collective behaviors of the two dipoles, as well as greatly affect the dipole-dipole resonance interaction. Especially, if the orientations of two dipoles are orthogonal and respectively perpendicular to the interdipole axis, the interdipole resonance interaction can be established with the help of optically active surroundings while there is no resonance interaction in vacuum.Comment: 13 pages, 3 figure