Properties of spin–orbit-coupled Bose–Einstein condensates

The experimental and theoretical research of spin–orbit-coupled ultracold atomic gases has advanced and expanded rapidly in recent years. Here, we review some of the progress that either was pioneered by our own work, has helped to lay the foundation, or has developed new and relevant techniques. After examining the experimental accessibility of all relevant spin–orbit coupling parameters, we discuss the fundamental properties and general applications of spin–orbit-coupled Bose–Einstein condensates (BECs) over a wide range of physical situations. For the harmonically trapped case, we show that the ground state phase transition is a Dicke-type process and that spin–orbit-coupled BECs provide a unique platform to simulate and study the Dicke model and Dicke phase transitions. For a homogeneous BEC, we discuss the collective excitations, which have been observed experimentally using Bragg spectroscopy. They feature a roton-like minimum, the softening of which provides a potential mechanism to understand the ground state phase transition. On the other hand, if the collective dynamics are excited by a sudden quenching of the spin–orbit coupling parameters, we show that the resulting collective dynamics can be related to the famous Zitterbewegung in the relativistic realm. Finally, we discuss the case of a BEC loaded into a periodic optical potential. Here, the spin–orbit coupling generates isolated flat bands within the lowest Bloch bands whereas the nonlinearity of the system leads to dynamical instabilities of these Bloch waves. The experimental verification of this instability illustrates the lack of Galilean invariance in the system.     

Homogeneous Einstein–Randers metrics on Aloff–Wallach spaces

Aloff–Wallach spaces are important in the study of positively curved homogeneous Riemannian manifolds. In this paper, we find some homogeneous Einstein–Randers metrics on such spaces.     

A new characterization of the n-dimensional Einstein static spacetime

The Einstein static spacetime is characterized as the unique geodesically complete and simply connected Lorentzian manifold such that the geodesic flow acts by isometries of the Sasaki metric on any null congruence associated to a conformal timelike vector field.     

Cohomological uniqueness of the Cauchy problem solutions for the Einstein–Maxwell equation

In this paper we analyze the Cauchy problem for the Einstein–Maxwell equation in the case of non-characteristic initial hypersurface. To find the correct notions of characteristic and Cauchy data we introduce a complex, which we call the Einstein–Maxwell complex. Then the Cauchy problem acquires correctness in terms of an associated spectral sequence. We define a Cauchy data in such way that they allow us to reconstruct a cohomologously unique formal solution.     

On Yau’s theorem for effective orbifolds

In 1978 Yau (Yau, 1978) confirmed a conjecture due to Calabi (1954) stating the existence of Kähler metrics with prescribed Ricci forms on compact Kähler manifolds. A version of this statement for effective orbifolds can be found in the literature (Joyce, 2000; Boyer and Galicki, 2008; Demailly and Kollár, 2001). In this expository article, we provide details for a proof of this orbifold version of the statement by adapting Yau’s original continuity method to the setting of effective orbifolds in order to solve a Monge–Ampère equation. We then outline how to obtain Kähler–Einstein metrics on orbifolds with negative first Chern class by solving a slightly different Monge–Ampère equation. We conclude by listing some explicit examples of Calabi–Yau orbifolds, which consequently admit Ricci flat metrics by Yau’s theorem for effective orbifolds.     

迷向表示分为6个不可约直和的旗流形上不变爱因斯坦度量

众所周知,计算广义旗流形G/K上不变爱因斯坦度量存在两个困难:(1)如何计算旗流形的非零结构常数;(2)如何计算旗流形爱因斯坦方程组的Grobner基.在这篇文章中用定理2.1来计算旗流形的非零结构常数,用Maple软件来计算旗流形爱因斯坦方程组的Gr?bnexr基.最后得到旗流形F_4/U~2(1)×SU(3),E_6/U~2(1)×SU(3)×SU(3),E_7/U~2(1)×SU(2)×SU(5),E_7/U~2(1)×SU(6),E_7/U~2(1)×SU(2)×SO(8)与E_8/U~2(1)×E_6上爱因斯坦度量.     

Permutations,power operations,and the center of the category of racks

Racks and quandles are rich algebraic structures that are strong enough to classify knots. Here we develop several fundamental categorical aspects of the theories of racks and quandles and their relation to the theory of permutations. In particular, we compute the centers of the categories and describe power operations on them, thereby revealing free extra structure that is not apparent from the definitions. This also leads to precise characterizations of these theories in the form of universal properties.