Wave packet dynamics in a monolayer graphene

The dynamics of charge particles described by Gaussian wave packet in monolayer graphene is studied analytically and numerically. We demonstrate that the shape of wave packet at arbitrary time depends on correlation between the initial electron amplitudes $\psi_1(\vec r,0)$ and $\psi_2(\vec r,0)$ on the sublattices $A$ and $B$ correspondingly (i.e. pseudospin polarization). For the transverse pseudospin polarization the motion of the center of wave packet occurs in the direction perpendicular to the average momentum ${\vec p_0}=\hbar \vec{k_0}$. Moreover, in this case the initial wave packet splits into two parts moving with opposite velocities along ${\vec p_0}$. If the initial direction of pseudospin coincides with average momentum the splitting is absent and the center of wave packet is displaced at $t>0$ along the same direction. The results of our calculations show that all types of motion experience {\it zitterbewegung}. Besides, depending on initial polarization the velocity of the packet center may have the constant component $v_c=uf(a)$, where $u\approx 10^8 cm/s$ is the Fermi velocity and $f(a)$ is a function of the parameter $a=k_0d$ ($d$ is the initial width of wave packet). As a result, the direction of the packet motion is determined not only by the orientation of the average momentum, but mainly by the phase difference between the up- and low- components of the wave functions. Similar peculiarities of the dynamics of 2D electron wave packet connected with initial spin polarization should take place in the semiconductor quantum well under the influence of the Rashba spin-orbit coupling.Comment: 7 pages, 8 figures, to be published in Phys. Rev.

Wave packet dynamics in hole Luttinger systems

For hole systems with an effective spin 3/2 we analyzed analytically and numerically the evolution of wave packets with the different initial polarizations. The dynamics of such systems is determined by the $4\times 4$ Luttinger Hamiltonian. We work in the space of arbitrary superposition of light- and heavy-hole states of the "one-particle system". For 2D packets we obtained the analytical solution for the components of wave function and analyzed the space-time dependence of probability densities as well as angular momentum densities. Depending on the value of the parameter $a=k_0d$ ($k_0$ is the average momentum vector and $d$ is the packet width) two scenarios of evolution are realized. For $a>>1$ the initial wave packet splits into two parts and the coordinates of packet center experience the transient oscillations or {\it Zitterbewegung} (ZB) as for other two-band systems. In the case when $a0$ remains almost cylindrically symmetric and the ripples arise at the circumference of wave packet. The ZB in this case is absent. We evaluated and visualized for different values of parameter $a$ the space-time dependence of angular momentum densities, which have the multipole structure. It was shown that the average momentum components can precess in the absence of external or effective magnetic fields due to the interference of the light- and heavy hole states. For localized initial states this precession has a transient character.Comment: 9 pages, 8 gigur

Graphene superlattice with periodically modulated Dirac gap

Graphene-based superlattice (SL) formed by a periodic gap modulation is studied theoretically using a Dirac-type Hamiltonian. Analyzing the dispersion relation we have found that new Dirac points arise in the electronic spectrum under certain conditions. As a result, the gap between conduction and valence minibands disappears. The expressions for the position of these Dirac points in ${\bf k}$-space and threshold value of the potential for their emergence were obtained. At some parameters of the system, we have revealed interface states which form the top of the valence miniband.Comment: 5 pages, 4 figures, accepted to Physical Review

Space-time evolution of Dirac wave packets

In this work we study the dynamics of free 3D relativistic Gaussian wave packets with different spin polarization. We analyze the connection between the symmetry of initial state and the dynamical characteristics of moving particle. The corresponding solutions of Dirac equation having different types of symmetry were evaluated analytically and numerically and after that the electron probability densities, as well as, the spin densities were visualized. The average values of velocity of the packet center and the average spin were calculated analytically, and the parameters of transient Zitterbewegung in different directions were obtained. These results can be useful for the interpretation of future experiments with trapped ions.Comment: 10 pages, 7 figure

Geometry-induced reduction of the critical current in superconducting nanowires

Reduction of the critical current in narrow superconducting NbN lines with sharp and rounded bends with respect to the critical current in straight lines was studied at different temperatures. We compare our experimental results with the reduction expected in the framework of the London model and the Ginsburg-Landau model. We have experimentally found that the reduction is significantly less than either model predicts. We also show that in our NbN lines the bends mostly contribute to the reduction of the critical current at temperatures well below the superconducting transition temperature

Proving Craig and Lyndon Interpolation Using Labelled Sequent Calculi

We have recently presented a general method of proving the fundamental logical properties of Craig and Lyndon Interpolation (IPs) by induction on derivations in a wide class of internal sequent calculi, including sequents, hypersequents, and nested sequents. Here we adapt the method to a more general external formalism of labelled sequents and provide sufficient criteria on the Kripke-frame characterization of a logic that guarantee the IPs. In particular, we show that classes of frames definable by quantifier-free Horn formulas correspond to logics with the IPs. These criteria capture the modal cube and the infinite family of transitive Geach logics

Розробка та анімація тривимірної моделі персонажа для комп’ютерної гри

The article shows the process of creating realistic three-dimensional character in the 3D Max environment: from concept art to development of the skeleton and animation of it. In a third-person game, the main character is constantly in the field of a camera view, often occupies most of the screen with sufficient increase. The increased requirements to a study of both the model, and textures are imposed. The importance of optimizing high-polygonal model to improve overall performance of the system resources in the game highlighted in the work. The methods allowing to reduce the number of polygons without essential loss of visual quality of model are considered and analysed.В статье показан процесс создания реалистичного трехмерного персонажа в среде 3D Max: начиная от концепт арта и заканчивая разработкой скелета и анимации для него. Поскольку в играх от третьего лица именно главный герой находится постоянно в поле зрения камеры, нередко занимает большую часть экрана при достаточном приближении, то к качеству выполнения как самой модели, так и текстур для нее предъявляются повышенные требования. В работе подчёркнута важность оптимизации высокополигональной модели для повышения общей производительности использования системных ресурсов в игре. Рассмотрены и проанализированы методы, позволяющие уменьшить количество полигонов без существенной потери визуального качества модели.У статі наведено створеня реалістичного персонажа : починаючи від концепт  розробкою скелету та анімації для нього. Оскільки в іграх від третьої особи саме головний герой знаходиться постійно у полі зору камери, нерідко займає більшу частину екрану при достатньому наближенні, то до якості опрацювання як самої моделі, так і текстур для неї висуваються підвищені вимоги. В роботі підкреслена продуктивностю використання системних ресурсів у грі. Розглянуті та проаналізовані методи, що дозволяють зменшити кількість полігонів без суттєвої втрати візуальної якості моделі