Knotted pictures of the GHZ states on the surface of a trivial torus

By means of the torus knot theory method, this paper presents the complete process of obtaining the knotted pictures of eight GHZ states on the surface of a trivial torus from the knotted pictures of eight basic three-qubit states on the surface of a trivial torus. Thus, we obtain eight knotted pictures 121 linkage on the ordinary plane.     

Knotted pictures of the Bell bases on the surface of a trivial torus

By means of the method of torus knot theory, this paper gives the complete processes of obtaining the knotted pictures of four Bell bases from the knotted pictures of four basic two qubit states.     

On the correspondence between three nodes W states in quantum network theory and the oriented links in knot theory

The GHZ states and W states are two fundamental types of three qubits quantum entangled states. For finding the knotted pictures of three nodes W states, on the one side, we empty any one node, thus obtaining three degenerated twonode W states, then we find the nonzero submatrix of the corresponding covariance correlation tensor in quantum network theory. On the other side, excepting the linkage 41 corresponding to Bell bases, we conjecture that the another one possible oriented link(which is composed of two-component knots entangled with each other and has four crossings) would be the required knotted pictures of the two nodes W states, thence obtain the nonzero submatrix of the Alexander relation matrix in the theory of knot crystals for these knotted pictures. The equality of the two nonzero submatrices of different kinds thus verify the exactness of our conjecture. The superposition of three knotted pictures of two-node W states from different choices of the emptied node gives the knotted pictures of three-node W states, thus shows the correspondence between three-node W states in quantum network theory and the oriented links in knot theory. Finally we point out that there is an intimate and simple relationship between the knotted pictures of GHZ states and W states.     

Knotted picture of the whole complete quantum measurement process of quantum teleportation

<正>Based on the previous work about the knotted pictures of quantum states,quantum logic gates and unitary transformations,this paper further gives the whole complete quantum measurement process of quantum teleportation from the viewpoint of knot theory.     

Homology of Fortuin–Kasteleyn Clusters of Potts Models on the Torus

Topological properties of Fortuin–Kasteleyn clusters are studied on the torus. Namely, the probability that their topology yields a given subgroup of the first homology group of the torus is computed for Q=1, 2, 3 and 4. The expressions generalize those obtained by Pinson for percolation (Q=1). Numerical results are also presented for three tori of different moduli. They agree with the theoretical predictions for Q=1, 2 and 3. For Q=4 agreement is not ruled out but logarithmic corrections are probably present and they make it harder to decide.     

Closed-form solutions of the Schrödinger equation for a particle on the torus

Two results on the Schrödinger equation for a particle on the surface of a torus are obtained: In the first part of this note closed-form zero-energy solutions for a free particle are given. In the second part we show for which potentials depending only on the polar angle the Schrodinger equation becomes exactly solvable.     

On Quasi-Exact Solvability of the Schrödinger Equation for a Free Particle on the Surface of a Spindle Torus

We show that the Schrödinger equation for a free particle on the surface of a spindle torus is quasi-exactly solvable. Our result complements former ones in an interesting way: it is known that the Schrödinger equation for a free particle on a ring torus is non-solvable, whereas it is exactly solvable for a particle on a horn torus.     

The Landau Problem on the θ-Deformed Two-Torus

We study the Landau problem on the -deformed two-torus and use well-known projective modules to obtain perturbed energy spectra. For a strong magnetic field B, the problem can be restricted to one particular Landau level. First we represent generators of the algebra of the noncommutative torus T ² as finite-dimensional matrices. A second approach leads to a reducible representation with a -dependent center. For a simple periodic potential, the rational part of the Hofstadter butterfly spectrum is obtained.     

Stability of a Special Class of $$q_{ij}$$ -CCR and Extensions of Higher-Dimensional Noncommutative Tori

The -algebras A_{{q i}}, generated by generalised quon commutation relations are considered. The nuclearity of these algebras is proved. It is shown that A_{{q i}}, is isomorphic to the extension of a higher-dimensional noncommutative torus. Irreducible representations of A_{{q i}}, are considered. It is shown that the Fock representation is faithful.     

Disorder and the theory of the fractionally quantized Hall effect

In a recent article we showed that weak disorder does not change the ground states of a system of interacting electrons on a torus in a strong magnetic field at a filling factorf ₀, if the denominator off ₀ is odd and small. At filling factors near but not equalf ₀ disorder becomes important. We show that the ground states and the low excited states of such a system may be constructed from the ground states atf ₀ through adding localized electrons or holes. These states have the same degeneracy as the ground states atf ₀ and lead to the same value of the Hall conductance.     

Knotted periodic orbits in suspensions of smale's horseshoe: Period multiplying and cabled knots

Following earlier work on knotted periodic orbits in a suspension of Smale's [1] horseshoe diffeomorphism (Holmes and Williams [2]), we define a notion of iterated horseshoe knots. We show that approximately half the horseshoe knots are cabled and that, among these cablings, are infinitely many, non-isotopic, iterated torus knots. We obtain uniqueness results for certain families of resonant torus knots, related to cascades of period-doubling and period-multiplying bifurcations which occur as a horseshoe is created by passing through a family of diffeomorphisms such as the Hénon maps. We give formulae from which linking numbers can be computed and provide closed form expressions for the type numbers of certain resonant iterated torus knots.     

On the origin of two-dimensional electron gas states at the surface of topological insulators

The ab initio calculations of the electronic structure in the bulk and at the (0001) surface of narrow-band Bi₂Se₃, Sb₂Te₃, Sb₂STe₃, and Sb₂SeTe₂ semiconductors have been performed. It has been shown that ternary compounds Sb₂STe₂ and Sb₂SeTe₂, as well as the previously known compounds Bi₂Se₃ and Sb₂Te₃, are three-dimensional topological insulators. The influence of the subsurface van der Waals gap expansion on the surface electronic structure of these compounds has been analyzed. It has been shown that this expansion leads to the formation of new (trivial) surface states, namely a parabolic state in the conduction band and an M-shaped state in the valence band. These results explain the phenomena discovered recently in photoemission experiments and reveal the nature of new states that are caused by the adsorption of atoms on the surfaces of the layered topological insulators.     

Topological phase transition in the trirutile-type MgBi2O6

Based on the first-principle calculations and k⋅p effective model analysis, we predicted a new topological semimetal (TSM) MgBi₂O₆. Without spin-orbit-coupling (SOC) and under the generalized-gradient-approximation (GGA), MgBi₂O₆ is a nodal-line semimetal. When the exchange-correlation energy was changed to HSE06, MgBi₂O₆ was trivial insulator in the equilibrium volume, but it became TSM under 7% hydrostatic tensile strain. MgBi₂O₆ might be an important platform to study the topological properties because of the two following advantages for measurements: (1) The nodal line, drumhead-liked surface state and Fermi Arc are very closely to the Fermi level; (2) The band structure is very “clean” (no other bulk bands except the related inverted conduction and valence bands around the Fermi level), which avoids the surface states been embedded into the bulk states.     

一类多折叠环面混沌吸引子

在双折叠环面混沌吸引子方程中构造一种具有多个分段线性化的奇函数，基于递归算法，导出可产生一类多折叠环面混沌吸引子的递归公式.通过适当选取各个分段线性区间的斜率值，利用所得的递归公式计算出分段线性化函数中各个平衡点和转折点之值，最终可产生一类多折叠环面混沌吸引子.给出了产生这类混沌吸引子的计算机数值模拟结果. 关键词： 双折叠环面混沌吸引子 多折叠环面混沌吸引子 递归算法     

Critical percolation on the torus

We compute the various crossing probabilities defined by R. Langlands, P. Pouliot, and Y. Saint-Aubin for the critical percolation on the torus.     

铁磁绝缘体间的极薄Bi2Se3薄膜的相变研究

研究一个极薄三维拓扑绝缘体Bi₂Se₃薄膜处于两个铁磁绝缘体层之间, 其铁磁层的磁化方向都处于竖直平面, 系统拓扑性质随磁化方向夹角的变化. 从表面态电子低能有效哈密顿量出发计算系统的Chern 数, 和运用一个具有Armchair边界的单层六角晶格带的紧束缚模型模拟系统的体能带和边缘态, 来确定系统所处的拓扑相. 发现两个铁磁层的磁化方式从平行转到反平行的某一临界角度, 系统经历从反常量子霍尔相到普通绝缘相的转变.     

Non-Existence of Liouvillian Solutions for a Free Quantum Particle on a Torus Surface,Part 1: Polar States

We consider a quantum particle constrained to the surface of a torus that we parametrize by its azimuthal and polar angle. We show that the corresponding Schrödinger equation does not have closed-form solutions (in the sense of Liouvillian functions) that depend on the polar angle only. It follows that if there are any wavefunctions in closed form, they must contain nondegenerate, special functions.     

CO2激光干涉检验轮胎面面形及双曲率半径

轮胎面属于高次非球面,不同于二次回转球面,在光学系统中具有独特重要的用途,但至今无有效的面形检验方法.本文用自行研制的移相式CO₂激光(λ=10.6μm)球波面干涉仪,检验双曲率半径之差为亚毫米量级的轮胎面面形;文中分析了测试原理,给出了测试实例的移相干涉图、双曲率半径测量值及面形波差三维立体图,并分析了测试误差.     

Phase behaviour of polyethylene knotted ring chains

The phase behaviour of polyethylene knotted ring chains is investigated by using molecular dynamics simulations. In this paper, we focus on the collapse of the polyethylene knotted ring chain, and also present the results of linear and ring chains for comparison. At high temperatures, a fully extensive knot structure is observed. The mean-square radius of gyration per bond ² / (Nb²) and the shape factor <δ^*> depend on not only the chain length but also the knot type. With temperature decreasing, chain collapse is observed, and the collapse temperature decreases with the chain length increasing. The actual collapse transition can be determined by the specific heat capacity C_v, and the knotted ring chain undergoes gas-liquid-solid-like transition directly. The phase transition of a knotted ring chain is only one-stage collapse, which is different from the polyethylene linear and ring chains. This investigation can provide some insights into the statistical properties of knotted polymer chains.     

Witten–Reshetikhin–Turaev Invariants of¶Seifert Manifolds

For Seifert homology spheres, we derive a holomorphic function of K whose value at integer K is the sl ₂ Witten–Reshetikhin–Turaev invariant, Z _K , at q= exp 2πi/K. This function is expressed as a sum of terms, which can be naturally corresponded to the contributions of flat connections in the stationary phase expansion of the Witten–Chern–Simons path integral. The trivial connection contribution is found to have an asymptotic expansion in powers of K ⁻¹ which, for K an odd prime power, converges K-adically to the exact total value of the invariant Z _K at that root of unity. Evaluations at rational $K$ are also discussed. Using similar techniques, an expression for the coloured Jones polynomial of a torus knot is obtained, providing a trivial connection contribution which is an analytic function of the colour. This demonstrates that the stationary phase expansion of the Chern–Simons–Witten theory is exact for Seifert manifolds and for torus knots in S ³. The possibility of generalising such results is also discussed. Received: 26 October 1998 / Accepted: 1 March 1999