Sonja Gombar

We discuss the geometry behind the classical Heisenberg model at the level suitable for third- or fourth-year students who did not have the opportunity to take a course on differential geometry. The arguments presented here rely solely on elementary algebraic concepts such as vectors, dual vectors and tensors, as well as Hamiltonian equations and Poisson brackets in their simplest form. We derive Poisson brackets for classical spins, along with the corresponding equations of motion for the classical Heisenberg model, starting from the two-sphere geometry, thereby demonstrating the relevance of standard canonical procedures in the case of the Heisenberg model.

In this paper certain classes of infinite sums involving special functions are evaluated analytically by application of basic quantum mechanical principles to simple models of half harmonic oscillator and a particle trapped inside an infinite potential well. The infinite sums ∑ n = 0 ∞ 2 2 n ( 2 n + 1 ) ! Γ 2 ( n + 3 2 ) [   2 F 1 ( − n , ν + 2 2 ; 3 2 ; 1 2 ) ] 2 , ∑ n = 0 ∞ [ L ν 2 n + 1 − ν ( b 2 2 ) ] 2 b 4 n 2 2 n ( 2 n + 1 ) ! and ∑ n = 1 ∞ [ J ν + 1 ( n π ) ] 2 n 2 ν , where 2 F 1 ( − n , ν + 2 2 ; 3 2 ; 1 2 ) is generalized hypergeometric function, L ν 2 n + 1 − ν ( b 2 2 ) associated Laguerre polynomial and J ν + 1 ( n π ) Bessel function of the first kind, are calculated for integer ν. It is also demonstrated that the same procedure can be generalized by application to some classes of functions which are not regular wave functions leading to additional infinite sums, as a consequence of which the series ∑ n = 1 ∞ [ H ν ( n π ) ] 2 n 2 ν containing Struve functions of the first kind H ν ( n π ) are evaluated. Convergence of the evaluated series, additionally verified by the application of different convergence tests, is secured by the properties of the corresponding Hilbert space.

We demonstrate that certain classes of Schlömilch-like infinite series and series that include generalized hypergeometric functions can be calculated in closed form starting from a simple quantum model of a particle trapped inside an infinite potential well and using principles of quantum mechanics. We provide a general framework based on the Hilbert space eigenproblem that can be applied to different exactly solvable quantum models. Obtaining series from normalization conditions in well-defined quantum problems secures their convergence.

Dynamics of the driven Frenkel-Kontorova model with asymmetric deformable substrate potential is examined by analyzing response function, the largest Lyapunov exponent, and Poincaré sections for two neighboring particles. The obtained results show that the largest Lyapunov exponent, besides being used for investigating integral quantities, can be used for detecting microchanges in chain configuration of both damped Frenkel-Kontorova model with inertial term and its strictly overdamped limit. Slight changes in relative positions of the particles are registered through jumps of the largest Lyapunov exponent in the pinning regime. The occurrence of such jumps is highly dependent on type of commensurate structure and deformation of substrate potential. The obtained results also show that the minimal force required to initiate collective motion of the chain is not dependent on the number of Lyapunov exponent jumps in the pinning regime. These jumps are also registered in the sliding regime, where they are a consequence of a more complex structure of largest Lyapunov exponent on the step.

Recently, there has been an increased interest in studying quantum entanglement and quantum coherence. Since both of these properties are attributed to the existence of quantum superposition, it would be useful to determine whether some type of correlation between them exists. Hence, the purpose of this paper is to explore the type of the correlation in several systems with different types of anisotropy. The focus will be on the XY spin chains with the Dzyaloshinskii–Moriya interaction and the type of the mentioned bond will be explored using the quantum renormalization group method.

Recently, there has been an increased interest in studying quantum entanglement and quantum coherence. Since both of these properties are attributed to the existence of quantum superposition, it would be useful to determine if some type of correlation between them exists. Hence, the purpose of this paper is to explore the type of the correlation in several systems with different types of anisotropy. The focus will be on the XY spin chains with the Dzyaloshinskii-Moriya interaction and the type of the mentioned bond will be explored using the quantum renormalization group method.

The dispersion relation for noninteracting excitons and the influence of perturbative corrections are examined in the case of pentacene structure. The values of exchange integrals are determined by nonlinear fits to the experimental dispersion data, obtained by the inelastic electron scattering reported in recent experiments. We obtain theoretical dispersion curves along four different directions in the Brillouin zone which possess the same periodicity as the experimental data. We also show that perturbative corrections are negligible since the exciton gap in the dispersion relation is huge in comparison to the exchange integrals.