Moving and merging of Dirac points on a square lattice and hidden symmetry protection

First, we study a square fermionic lattice that supports the existence of massless Dirac fermions, where the Dirac points are protected by a hidden symmetry. We also consider two modified models with a staggered potential and the diagonal hopping terms, respectively. In the modified model with a staggered potential, the Dirac points exist in some range of magnitude of the staggered potential, and move with the variation of the staggered potential. When the magnitude of the staggered potential reaches a critical value, the two Dirac points merge. In the modified model with the diagonal hopping terms, the Dirac points always exist and just move with the variation of amplitude of the diagonal hopping. We develop a mapping method to find hidden symmetries evolving with the parameters. In the two modified models, the Dirac points are protected by this kind of hidden symmetry, and their moving and merging process can be explained by the evolution of the hidden symmetry along with the variation of the corresponding parameter

Quantum Phases of Ultracold Bosonic Atoms in a Two-Dimensional Optical Superlattice

We study quantum phases of ultracold bosonic atoms in a two-dimensional optical superlattice. The extended Bose-Hubbard model derived from the system of ultracold bosonic atoms in an optical superlattice is solved numerically with Gutzwiller approach. We find that the modulated superfluid(MS), Mott-insulator (MI) and density-wave(DW) phases appear in some regimes of parameters. The experimental detection of the first order correlations and the second order correlations of different quantum phases with time-of-flight and noise-correlation techniques is proposed

Hidden-Symmetry-Protected Topological Semimetals on a Square Lattice

We study a two-dimensional fermionic square lattice, which supports the existence of two-dimensional Weyl semimetal, quantum anomalous Hall effect, and $2\pi$-flux topological semimetal in different parameter ranges. We show that the band degenerate points of the two-dimensional Weyl semimetal and $2\pi$-flux topological semimetal are protected by two distinct novel hidden symmetries, which both corresponds to antiunitary composite operations. When these hidden symmetries are broken, a gap opens between the conduction and valence bands, turning the system into a insulator. With appropriate parameters, a quantum anomalous Hall effect emerges. The degenerate point at the boundary between the quantum anomalous Hall insulator and trivial band insulator is also protected by the hidden symmetry

A possible new path to proving the Riemann Hypothesis

In the past 100 years, the research of Riemann Hypothesis meets many difficulties. Such situation may be caused by that people used to study Zeta function only regarding it as a complex function. Generally, complex functions are far more complex than real functions, and are hard to graph. So, people cannot grasp the nature of them easily. Therefore, it may be a promising way to try to correspond Zeta function to real function so that we can return to the real domain to study RH. In fact, under Laplace transform, the whole picture of Zeta function is very clear and simple, and the problem can be greatly simplified. And by Laplace transform, most integral and convolution operations can be converted into algebraic operations, which greatly simplifies calculating and analysis.Comment: 14 pages, 6 figure

A new insight into neutrino energy loss by electron capture of iron group nuclei in magnetars surface

Based on the relativistic mean-field effective interactions theory, and Lai dong model \citep{b37, b38, b39}, we discuss the influences of superstrong magnetic fields (SMFs) on electron Fermi energy, nuclear blinding energy, and single-particle level structure in magnetars surface. By using the method of Shell-Model Monte Carlo (SMMC), and the Random Phase Approximation (RPA) theory, we detailed analyze the neutrino energy loss rates(NELRs) by electron capture (EC) for iron group nuclei in SMFs.Comment: 22 pages, 8 figures, accepted for publication in ApJS. arXiv admin note: text overlap with arXiv:astro-ph/0606674, arXiv:nucl-th/9707052, arXiv:nucl-th/9801012, arXiv:1505.07304 by other author

Hidden symmetry and protection of Dirac points on the honeycomb lattice

The honeycomb lattice possesses a novel energy band structure, which is characterized by two distinct Dirac points in the Brillouin zone, dominating most of the physical properties of the honeycomb structure materials. However, up till now, the origin of the Dirac points is unclear yet. Here, we discover a hidden symmetry on the honeycomb lattice and prove that the existence of Dirac points is exactly protected by such hidden symmetry. Furthermore, the moving and merging of the Dirac points and a quantum phase transition, which have been theoretically predicted and experimentally observed on the honeycomb lattice, can also be perfectly explained by the parameter dependent evolution of the hidden symmetry.Comment: 5 pages, 2 figures, +6 pages of supplementary information. Welcome any comments

Weyl semimetals in optical lattices: moving and merging of Weyl points, and hidden symmetry at Weyl points

We propose to realize Weyl semimetals in a cubic optical lattice. We find that there exist three distinct Weyl semimetal phases in the cubic optical lattice for different parameter ranges. One of them has two pairs of Weyl points and the other two have one pair of Weyl points in the Brillouin zone. For a slab geometry with (010) surfaces, the Fermi arcs connecting the projections of Weyl points with opposite topological charges on the surface Brillouin zone is presented. By adjusting the parameters, the Weyl points can move in the Brillouin zone. Interestingly, for two pairs of Weyl points, as one pair of them meet and annihilate, the originial two Fermi arcs coneect into one. As the remaining Weyl points annihilate further, the Fermi arc vanishes and a gap is opened. Furthermore, we find that there always exists a hidden symmetry at Weyl points, regardless of anywhere they located in the Brillouin zone. The hidden symmetry has an antiunitary operator with its square being $-1$.Comment: 10 pages, 5 figure

A New Kind of Deformed Hermite Polynomials and Its Applications

A new kind of deformed calculus was introduced recently in studying of parabosonic coordinate representation. Based on this deformed calculus, a new deformation of Hermite polynomials is proposed, its some properties such as generating function, orthonormality, differential and integral representaions, and recursion relations are also discussed in this paper. As its applications, we calculate explicit forms of parabose squeezed number states, derive a particularly simple subset of minimum uncertainty states for parabose amplitude-squared squeezing, and discuss their basic squeezing behaviours.Comment: 18 pages, LaTe

Graded Lie Algebra Generating of Parastatistical Algebraic Structure

A new kind of graded Lie algebra (we call it $Z_{2,2}$ graded Lie algebra) is introduced as a framework for formulating parasupersymmetric theories. By choosing suitable bose subspace of the $Z_{2,2}$ graded Lie algebra and using relevant generalized Jacobi identities, we generate the whole algebraic structure of parastatistics.Comment: 8 pages, LaTe

Space Structure for the Simplest Parasupersymmetric System

Structure of the state-vector space for a system consisting of one mode parabose and one mode parafermi degree of freedom with the same parastatistics order $p$ is studied and a complete, orthonormal set of basis vectors in this space is constructed. There is an intrinsic double degeneracy for state vectors with $m$ parabosons and $n$ parafermions, where $m \not= 0$, $n \not= 0$, and $n \not= p$. It is also shown that the degeneracy plays a key role in realization of exact supersymmetry for such a system