Five-body choreography on the algebraic lemniscate is a potential motion

In a remarkable paper of 2003 by Fujiwara et al. [1], a figure-eight three-body choreography on the algebraic lemniscate of Bernoulli was discovered. Such a choreography was found to be driven by the action of a pairwise potential $V(r_{ij})$, depending only on the mutual relative distances $r_{ij}$, $i,j=1,2,3$. In the present Letter we show that two different choreographies of five bodies on the same algebraic lemniscate exist and correspond to solutions of ten coupled Newton equations of motion with a pairwise interaction potential. For each choreography the explicit form of the potential is found and ten constants of motion are presented, thus, it is superintegrable.     

Study of the q-Gaussian distribution with the scale index and calculating entropy by normalized inner scalogram

In this paper, we discuss a method based on wavelet analysis for the study of the q-index of the Gaussian distribution. We derive q-index from the scale index, $i_{\mathit{scale}}$, using the expression; $q\equiv 1+2i_{\mathit{scale}}$ where $i_{\mathit{scale}}$ is a wavelet based tool for measuring the degree of aperiodicity of a dynamical system in the range of $0\le i_{\mathit{scale}}\le 1$. We show that this expression gives consistent results with the numerical approach of q-Gaussian distribution which determines the degree of non-extensivity of a dynamical system in the range of $1<q<3$. We also suggest a new entropy calculation method based on the normalized inner scalogram for studying the chaotic characteristics of nonlinear dynamical systems.     

Electronic vacuum charge for an overcritical nucleus

A “cut-off” Coulomb potential taking into account the finite size of the nucleus is finite, and a solution of the Dirac equation can be constructed for any energy, both positive and negative. In the paper we develop an exact solution of the Dirac equation for a fixed value of the total momentum j for the whole spectrum of energies, which allows us to determine the vacuum charge and its spatial distribution. We consider nuclei with different charges Z, both $Z<Z_c$ and $Z>Z_c$, where $Z=Z_c$ is the “critical” charge, at which the energy of the lowest discrete state reaches the boundary of the lower continuum $\epsilon =?m{c}^2$. Polarization of vacuum is determined, and the vacuum charge for several values of Z is found. For an undercritical nuclear charge, $Z<Z_c$, the total vacuum charge appears to be zero, while for $Z>Z_c$, the vacuum gets rearranged, and the total vacuum charge becomes equal to $?2e$. The vacuum charge distribution for $j=1/2$ for both undercritical and overcritical nuclei is calculated.     

Topological states in the Hofstadter model on a honeycomb lattice

We provide a detailed analysis of a topological structure of a fermion spectrum in the Hofstadter model with different hopping integrals along the $x,y,z$-links ($t_x=t,t_y=t_z=1$), defined on a honeycomb lattice. We have shown that the chiral gapless edge modes are described in the framework of the generalized Kitaev chain formalism, which makes it possible to calculate the Hall conductance of subbands for different filling and an arbitrary magnetic flux ϕ. At half-filling the gap in the center of the fermion spectrum opens for $t>t_c=2^{\varphi }$, a quantum phase transition in the 2D-topological insulator state is realized at $t_c$. The phase state is characterized by zero energy Majorana states localized at the boundaries. Taking into account the on-site Coulomb repulsion U (where $U<<1$), the criterion for the stability of a topological insulator state is calculated at $t<<1$, $t\sim U$. Thus, in the case of $U>4\mathrm{\Delta }$, the topological insulator state, which is determined by chiral gapless edge modes in the gap Δ, is destroyed.     

Thermospin effects in a T-shaped spin valves with spin-flip scattering

With the help of the nonequilibrium Green's function technique, we theoretically analyze the thermospin property through a typical T-shaped spin valve with spin-flip scattering in the linear regime. The influences of spin-flip coefficient of interdot λ, spin-flip coefficient of intradot η and interdot hopping coefficient $t+{\delta }_{\sigma }\mathrm{\Delta }t$ on thermospin property are discussed. As interdot hopping coefficient t is equal to energy level ε, the spectrum of $G_s$ shows Fano-like effect with ε variation. Antiresonance position of $G_s$ is almost unchanged and its width becomes narrower with ε increasing. Spin thermopower $S_s$ is close to the maximum of the peak and charge thermopower $S_c$ is equal to zero for $t=\epsilon$. As a result, the pure spin thermopower $S_s$ can be obtained, which means that a pure spin current may be produced by a temperature gradient in our system. It is found that spin figure of merit $Z{T}_s$ can reach a considerable value by adjusting key parameters of the system, such as Δt, β, α, ?. The typical T-shaped spin valve can be treated as a stable thermospin battery which allows to convert the heat energy to spin voltage, thus produces the pure spin current in the device.