Covariant string amplitudes on exotic topologies to one loop

Rules are provided for the calculation of string amplitudes on all exotic topologies to one loop. These topologies include the real projective plane, the disk, the annulus, the Möbius strip, the torus and the Klein bottle. The rules are valid for bosonic strings, superstrings, and heterotic strings either in their critical dimensions or toroidally compactified, and they are applicable to amplitudes with arbitrary external bosonic and fermionic states, including mixed open- and closed-string transitions. The modular invariance of the torus amplitudes is discussed.     

Dimer statistics on the Möbius strip and the Klein bottle

Closed-form expressions are obtained for the generating function of close-packed dimers on a 2M×2N simple quartic lattice embedded on a Möbius strip and a Klein bottle. Finite-size corrections are also analyzed and compared with those under cylindrical and free boundary conditions. Particularly, it is found that, for large lattices of the same size and with a square symmetry, the number of dimer configurations on a Möbius strip is 70.2% of that on a cylinder. We also establish two identities relating dimer generating functions for Möbius strips and cylinders.     

Winding Clusters in Percolation on the Torus and the Möbius Strip

Using a simulation technique introduced recently, we study winding clusters in percolation on the torus and the Möbius strip for different aspect ratios. The asynchronous parallelization of the simulation makes very large system and sample sizes possible. Our high accuracy results are fully consistent with predictions from conformal field theory. The numerical results for the Möbius strip and the number distribution of winding clusters on the torus await theoretical explanation. To our knowledge, this study is the first of its kind.     

One loop partition functions in a compactified open bosonic string theory

We calculate the one loop partition functions of a toroidally compactified open bosonic string (fromD=26 toD=10) in the path integral formalism. By a certain combination of the compactified sectors it is shown that the partition functions of a compactified open bosonic string vanish in both annulus and Möbius strip topologies as in the open superstring case. We also discuss the consistency requirement for the equivalence between the compactified open bosonic string and the open superstring with the gauge groups (SO(8192) andSO(32), respectively.     

The Klein-Gordon Operator on Möbius Strip Domains and the Klein Bottle in ℝ n

In this paper we present explicit formulas for the fundamental solution to the Klein-Gordon operator on some higher dimensional generalizations of the Möbius strip and the Klein bottle with values in distinct pinor bundles. The fundamental solution is described in terms of generalizations of the Weierstraß $\wp $ -function that are adapted to the context of these geometries. The explicit formulas for the kernel then allow us to express all solutions to the homogeneous and inhomogeneous Klein-Gordon problem with given boundary data in the context of these manifolds. In the case of the Klein bottle we are able to describe all null solutions of the Klein-Gordon equation in terms of finite linear combinations of the fundamental solution and its partial derivatives.     

Dimer statistics of honeycomb lattices on Klein bottle,Möbius strip and cylinder

Dimer statistics is a central problem in statistical physics. In this paper the enumerations of close-packed dimers of honeycomb lattices on Klein bottle, Möbius strip and cylinder are considered. By establishing a Pfaffian orientation or a crossing orientation, and then computing the determinants of the skew-symmetric matrices of the resulting orientation graphs, we obtain explicit expressions of the number of close-packed dimers of the Klein-bottle polyhex, the Möbius polyhex and the cylindrical polyhex.     

Electronic, energy, and thermal properties of the Möbius strip and related ring nanostructures of NbS3

The electronic structure and energy characteristics of the Möbius strip and related ring nanostructures of niobium trisulfide NbS₃ are investigated using the density functional tight-binding (DFTB) method. The results obtained are analyzed and compared with those obtained for the crystal and planar ribbon structures of NbS₃. The formation of NbS₃ ring structures leads to a considerable increase in the density of Nb 4d states at the Fermi level. The simulation of the behavior of the ring structures in the temperature range T = 0–700 K demonstrates that the Möbius strip has a lower thermal stability as compared to the NbS₃ ring.     

Plasmonic microcavity formed by the Möbius strip

We propose a Möbius-strip-type plasmonic cavity with a silver Möbius strip sandwiched between dielectric layers. By brief theoretical and simulation analyses, we obtain that the Q factor of the cavity remains about 40 and the mode volume is ultrasmall (less than 1 μm³) which is more compact than that of the cylindric cavity. This Möbius-strip-type plasmonic cavity supporting the propagation of surface plasmon polaritons owns some unusual properties such as more effective volume and the spatial separation. More potential applications based on this cavity remain to be explored in future nanophotonics.     

Dimer statistics on the Klein bottle

The problem of enumerating close-packed dimers, or perfect matchings, on a quadratic lattice embedded on the Klein bottle is considered. Thomassen [C. Thomassen, Tilings of the torus and the Klein Bottle and vertex-transitive graphs on a fixed surface, Trans. Amer. Math. Soc. 323(2) (1991) 605-635] characterized that there are six quadrilateral lattices embedded on the Klein bottle. Lu and Wu [W. T. Lu and F. Y. Wu, Close-packed dimers on nonorientable surfaces, Phys. Lett. A 293 (2002) 235-246] had obtained a expression for the number of close-packed dimers on one of them. In this paper we investigate four other embeddings and obtain explicit expressions of the numbers of close-packed dimers and free energy per dimer by enumerating Pfaffians.     

Coherent states for a quantum particle on a Möbius strip

The coherent states for a quantum particle on a Möbius strip are constructed and the relation with the natural phase space for fermionic fields is shown. The explicit comparison of the obtained states with previous works where the cylinder quantization was used and the spin 1/2 was introduced by hand is given.     

Finite-size scaling for the ising model on the Möbius strip and the klein bottle

We study the finite-size scaling properties of the Ising model on the M?bius strip and the Klein bottle. The results are compared with those of the Ising model under different boundary conditions, that is, the free, cylindrical, and toroidal boundary conditions. The difference in the magnetization distribution function p(m) for various boundary conditions is discussed in terms of the number of the percolating clusters and the cluster size. We also find interesting aspect-ratio dependence of the value of the Binder parameter at T = T(c) for various boundary conditions. We discuss the relation to the finite-size correction calculations for the dimer statistics.     

Finitized Conformal Spectra of the Ising Model on the Klein Bottle and Möbius Strip

We study the conformal spectra of the critical square lattice Ising model on the Klein bottle and Möbius strip using Yang–Baxter techniques and the solution of functional equations. In particular, we obtain expressions for the finitized conformal partition functions in terms of finitized Virasoro characters. This demonstrates that Yang–Baxter techniques and functional equations can be used to study the conformal spectra of more general exactly solvable lattice models in these topologies. The results rely on certain properties of the eigenvalues which are confirmed numerically.     

Phonon dispersion for the bcc metals Mo and Cr

In this paper the phonon dispersion for the bcc metals Mo and Cr is calculated based on the pair potentials obtained from cohesive energies and the Slater-Kirkwood- type three body interaction. In the calculation of the pair potentials the Möbius transform in the number theory is used and the cohesive energy is evaluated by the LMTO method. The results show a good agreement with inelastic neutron scattering data and indicate that the three-body interaction is necessary to account for the phonon dispersion.     

Geometry and Symmetric Coherent States of Three Qubits Systems

In this paper, we first generalize the previous results that relate 1- and 2-qubit geometries to complex and quaternionic Möbius transformations respectively, to the case of 3-qubit states under octonionic Möbius transformations. This completes the correspondence between the qubit geometries and the four normed division algebras. Thereby, new systems of symmetric coherent states with 2 and 3 qubits can be constructed by mapping the spin coherent states to their antipodal symmetric ponits on the generalized Bloch spheres via Möbius transformations in corresponding dimensions. Finally, potential applications of the normed division algebras in physics are discussed.     

Fermionic Casimir effect with helix boundary condition

In this paper, we consider the fermionic Casimir effect under a new type of space-time topology using the concept of quotient topology. The relation between the new topology and that in Feng and Li (Phys. Lett. B 691:167, 2010), Zhai et al. (Mod. Phys. Lett. A 26:669, 2011) is something like that between a Möbius strip and a cylindric. We obtain the exact results of the Casimir energy and force for the massless and massive Dirac fields in the (D+1)-dimensional space-time. For both massless and massive cases, there is a Z ₂ symmetry for the Casimir energy. To see the effect of the mass, we compare the result with that of the massless one and we found that the Casimir force approaches the result of the force in the massless case when the mass tends to zero and vanishes when the mass tends to infinity.     

Möbius invariance and classical solutions of SU(2) gauge theory

We use quaterniomic Möbius transformations to construct classical solutions of pure SU(2) gauge theory which correspond to two generalized merons and to two “excited” merons. We discuss how such transformations can yield distinct gauge invariant quantities and point out the limitations for generating further solutions by these techniques.     

Comprehensive study of the correlations that exist among the members of the [n]cyclacene series and the Möbius[n]cyclacene series

Cyclacenes are the smallest substructures of carbon nanotubes used in modelling studies. The systematics that exists between Hückel molecular orbital eigenvalues and eigenvectors of cyclacenes are delineated. This study of cyclacenes combines the interconnection of concepts of complementarity theorem, characteristic and matching polynomial recursion equations, embedding, greater than twofold symmetry and doubly degenerate eigenvalues, open-shell singlet character, and pairing theorem. Proof that cyclacenes have more open-shell (diradical) character than do polyacenes is also provided by the sum total of this work. Mirror-plane scission of even-ring cyclacenes gives linear polyacene fragments. This shows that the properties of the successor linear polyacene must be contained in the precursor cyclacene. Corresponding Möbius[n]cyclacene isomers display contrasting and unusual comparative properties. A partial list of contrasting properties include alternant (cyclacenes) versus nonalternant (Möbius[n]cyclacene) polyenes, presence of Hamiltonian circuits in Möbius[n]cyclacene and presence of oscillatory electronic properties in cyclacenes.     

Möbius and super-Möbius gauge fixing for the closed string amplitudes on the desk

We systematically study the gauge fixing of the Möbius and the super-Möbius transformations for the N-point closed-string amplitudes on the disk. Using the Faddeev-Popov method, we obtain the explicit formulas for the Koba-Nielsen factors for these amplitudes.     

Mechanisms and Möbius strips: understanding dynamic processes in annulenes

Exploratory computational studies on annulenes with planar, Möbius, and two‐twist topologies have resulted in new mechanisms to explain facile thermal configuration change (cis‐trans isomerization) for medium‐sized annulenes ([12]‐ to [16]annulene). Möbius π‐bond shifting through both aromatic and antiaromatic transition states, two‐twist π‐bond shifting, and planar nondegenerate π‐bond shifting can all be invoked to explain experimental results. Moreover, a simple bond‐shift rule, which is based on the change in number of trans C?C double bonds (Δtrans), was developed that predicts the topology of the transition state(s) necessary to effect the desired cis‐trans isomerization. The bond‐shift rule was also applied to configuration change in dehydro[12]annulene. Finally, extensive investigation of the [14]annulene hypersurface revealed that numerous Möbius minima exist within 10 kcal/mol of the global minimum. Copyright © 2012 John Wiley & Sons, Ltd.