Topological quantum matter with cold atoms

This is an introductory review of the physics of topological quantum matter with cold atoms. Topological quantum phases, originally discovered and investigated in condensed matter physics, have recently been explored in a range of different systems, which produced both fascinating physics findings and exciting opportunities for applications. Among the physical systems that have been considered to realize and probe these intriguing phases, ultracold atoms become promising platforms due to their high flexibility and controllability. Quantum simulation of topological phases with cold atomic gases is a rapidly evolving field, and recent theoretical and experimental developments reveal that some toy models originally proposed in condensed matter physics have been realized with this artificial quantum system. The purpose of this article is to introduce these developments. The article begins with a tutorial review of topological invariants and the methods to control parameters in the Hamiltonians of neutral atoms. Next, topological quantum phases in optical lattices are introduced in some detail, especially several celebrated models, such as the Su–Schrieffer–Heeger model, the Hofstadter–Harper model, the Haldane model and the Kane–Mele model. The theoretical proposals and experimental implementations of these models are discussed. Notably, many of these models cannot be directly realized in conventional solid-state experiments. The newly developed methods for probing the intrinsic properties of the topological phases in cold-atom systems are also reviewed. Finally, some topological phases with cold atoms in the continuum and in the presence of interactions are discussed, and an outlook on future work is given.     

A COUNTEREXAMPLE TO THE -CONJECTURE

There was an error in the statement of Corollary 1 in [Comm. Alg. 2000, 28, 3973–3982]. So that corollary, as well as the proof of Theorem 2 in loc. cit., should be replaced.     

Interband excitations in the 1D limit of two-band fractional Chern insulators

We investigate the stability of the one-dimensional limit of $\nu =1/3$ Laughlin-like fractional Chern insulator with respect to the interband interaction. We propose a construction for the excitations in the infinite-interaction case and show that the energy gap remains finite in the thermodynamic limit. Next, by means of exact diagonalization and Density Matrix Renormalization Group approaches, we consider deviations from ideal dimerization and show that they reduce the stability of the FCI-like states. Finally, to show that our approach is not restricted to one model, we identify the dimer structure behind the thin-torus limit of other system – the checkerboard lattice.     

The Chern‐Simons Current in Systems of DNA‐RNA Transcriptions

A Chern‐Simons current, coming from ghost and anti‐ghost fields of supersymmetry theory, can be used to define a spectrum of gene expression in new time series data where a spinor field, as alternative representation of a gene, is adopted instead of using the standard alphabet sequence of bases . After a general discussion on the use of supersymmetry in biological systems, we give examples of the use of supersymmetry for living organism, discuss the codon and anti‐codon ghost fields and develop an algebraic construction for the trash DNA, the DNA area which does not seem active in biological systems. As a general result, all hidden states of codon can be computed by Chern‐Simons 3 forms. Finally, we plot a time series of genetic variations of viral glycoprotein gene and host T‐cell receptor gene by using a gene tensor correlation network related to the Chern‐Simons current. An empirical analysis of genetic shift, in host cell receptor genes with separated cluster of gene and genetic drift in viral gene, is obtained by using a tensor correlation plot over time series data derived as the empirical mode decomposition of Chern‐Simons current.     

布洛赫电子的拓扑与几何

文章回顾了电子的拓扑几何理论发展的初期,大约二十多年的历史。首先介绍拓扑陈数在凝聚态物理中的两个重要应用。其一关于量子霍尔效应,绝缘条件下霍尔电导可以写成一个陈数拓扑不变量,从而解释实验结果的精确量子化。其二关于绝热泵浦,它描述布洛赫能带的绝热电流响应,与电子极化有密切联系。拓扑陈数是布里渊区上贝里曲率的积分,后者本身也有独立的物理意义。接着介绍贝里曲率对电子动力学的影响,包括反常速度和轨道磁化等概念。作者还将这个理论推广到多带情况,使其可以应用到自旋输运等现象。最后,文中展示了再量子化方法,从半经典模型来获得布洛赫电子的有效量子理论。在非相对论极限下,泡利—薛定谔方程可以看作是狄拉克电子在正能谱上的等效量子理论,其中的自旋轨道耦合即是一种几何物理效应。     

Three-dimensional quantum Hall effect in Weyl and double-Weyl semimetals

The well-known quantum Hall effect (QHE) was usually studied in 2D systems. In this work, we investigate the integer QHE in 3D Weyl and double-Weyl semimetals. Based on the lattice models of Weyl and double-Weyl semimetals subjected to a uniform magnetic field, we derive the generalized 3D spinfull Hofstadter Hamiltonians and Harper equations for the two systems, and obtain their corresponding energy spectra. Furthermore, we show that for proper hopping parameters and rational magnetic fluxes, both systems exhibit the 3D QHE when the Fermi level lies in some band gaps. The 3D QHE is topologically characterized by three Chern numbers with one or two nonzero Chern values which are respectively defined for three crystal planes. The possible experimental realization and detection of the 3D QHE are also discussed.     

Qualitative Features of Intra-molecular Dynamics. What can be Learned from Symmetry and Topology?

Mathematical tools which are appropriate for the qualitative analysis of simple molecular quantum systems are briefly introduced and discussed.     

非定域变换的广义Ward恒等式

基于高阶微商奇异拉氏量系统相空间Green函数的生成泛函,导出了该系统在定域和非定域变换下的广义正则Ward恒等式.对规范不变系统,从位形空间生成泛函出发,导出了该系统在定域、非定域和整体变换下的广义Ward恒等式.用于高阶微商非Abel(Chern-Simons CS)理论,无需作出生成泛函中对正则动量的路径积分,即可导出正规顶角的某些关系.此外还给出了BRS变换下的Ward-Takahashi恒等式.     

Classes de Chern Des Ensembles Analytiques

Let V be a compact complex analytic subset of a non-singular holomorphic manifold M. Assume that V has pure complex dimension n. Denote by V₀ its regular part, and by [V] its fundamental class in H_2n(V; ). If V is a locally complete intersection (LCI), it is known that the normal bundle N_{V_0} in M to V₀ in M has a natural extension N_V to all of V, so that we can define its Chern classes c^(*)(N_V) in cohomology, as well as the Chern classes c_vir^(*). If V is a locally complete intersection (LCI), it is known that the normal bundle N_{V_0} in M to V₀ in M has a natural extension N_V to all of V, so that we can define its Chern classes c^(*)(N_V) in cohomology, as well as the Chern classes c_vir^(*) (V) of the virtual tangent bundle T_vir(V):=[TM|_V - N_V] in the K-theory K⁰(V). This has applications