Projection operator approach to spin diffusion in the anisotropic Heisenberg chain at high temperatures

We investigate spin transport in the anisotropic Heisenberg chain in the limit of high temperatures ({\beta} \to 0). We particularly focus on diffusion and the quantitative evaluation of diffusion constants from current autocorrelations as a function of the anisotropy parameter {\Delta} and the spin quantum number s. Our approach is essentially based on an application of the time-convolutionless (TCL) projection operator technique. Within this perturbative approach the projection onto the current yields the decay of autocorrelations to lowest order of {\Delta}. The resulting diffusion constants scale as 1/{\Delta}^2 in the Markovian regime {\Delta}<<1 (s=1/2) and as 1/{\Delta} in the highly non-Markovian regime above {\Delta} \sim 1 (arbitrary s). In the latter regime the dependence on s appears approximately as an overall scaling factor \sqrt{s(s+1)} only. These results are in remarkably good agreement with diffusion constants for {\Delta}>1 which are obtained directly from the exact diagonalization of autocorrelations or have been obtained from non-equilibrium bath scenarios.Comment: 4 pages, 3 figure

Calculating the energy spectra of magnetic molecules: application of real- and spin-space symmetries

The determination of the energy spectra of small spin systems as for instance given by magnetic molecules is a demanding numerical problem. In this work we review numerical approaches to diagonalize the Heisenberg Hamiltonian that employ symmetries; in particular we focus on the spin-rotational symmetry SU(2) in combination with point-group symmetries. With these methods one is able to block-diagonalize the Hamiltonian and thus to treat spin systems of unprecedented size. In addition it provides a spectroscopic labeling by irreducible representations that is helpful when interpreting transitions induced by Electron Paramagnetic Resonance (EPR), Nuclear Magnetic Resonance (NMR) or Inelastic Neutron Scattering (INS). It is our aim to provide the reader with detailed knowledge on how to set up such a diagonalization scheme.Comment: 29 pages, many figure

Symmetry assisted exact and approximate determination of the energy spectra of magnetic molecules using irreducible tensor operators

In this work a numerical approach for the determination of the energy spectra and the calculation of thermodynamic properties of magnetic molecules is presented. The work is focused on the treatment of spin systems which exhibit point-group symmetries. Ring-like and archimedean-type structures are discussed as prominent examples. In each case the underlying spin quantum system is modeled by an isotropic Heisenberg Hamiltonian. Its energy spectrum is calculated either by numerical exact diagonalization or by an approximate diagonalization method introduced here. In order to implement full spin-rotational symmetry the numerical approach at hand is based on the use of irreducible tensor operators. Furthermore, it is shown how an unrestricted use of point-group symmetries in combination with the use of irreducible tensor operators leads to a reduction of the dimensionalities as well as to additional information about the physics of the systems. By exemplarily demonstrating how the theoretical foundations of the irreducible tensor operator technique can be realized within small spin systems the technical aspect of this work is covered. These considerations form the basis of the computational realization that was implemented and used in order to get insight into the investigated systems

Numerically exact and approximate determination of energy eigenvalues for antiferromagnetic molecules using irreducible tensor operators and general point-group symmetries

Schnalle R, Schnack J. Numerically exact and approximate determination of energy eigenvalues for antiferromagnetic molecules using irreducible tensor operators and general point-group symmetries. Physical Review B. 2009;79(10): 104419.For small-enough quantum systems numerical exact and complete diagonalization of the Hamiltonian enables one to evaluate and discuss all static, dynamic, and thermodynamic properties. In this article we consider Heisenberg spin systems and extend the range of applicability of the exact diagonalization method by showing how the irreducible tensor operator technique can be combined with an unrestricted use of general point-group symmetries. We also present ideas on how to use spin-rotational and point-group symmetries in order to obtain approximate spectra

Predicting User Behavior in e-Commerce Using Machine Learning

Ketipov R, Angelova V, Doukovska L, Schnalle R. Predicting User Behavior in e-Commerce Using Machine Learning. Cybernetics and Information Technologies . 2023;23(3):89-101.Each person's unique traits hold valuable insights into their consumer behavior, allowing scholars and industry experts to develop innovative marketing strategies, personalized solutions, and enhanced user experiences. This study presents a conceptual framework that explores the connection between fundamental personality dimensions and users' online shopping styles. By employing the TIPI test, a reliable and validated alternative to the Five-Factor model, individual consumer profiles are established. The results reveal a significant relationship between key personality traits and specific online shopping functionalities. To accurately forecast customers' needs, expectations, and preferences on the Internet, we propose the implementation of two Machine Learning models, namely Decision Trees and Random Forest. According to the applied evaluation metrics, both models demonstrate fine predictions of consumer behavior based on their personality

Predicting User Behavior in e-Commerce Using Machine Learning

Each person’s unique traits hold valuable insights into their consumer behavior, allowing scholars and industry experts to develop innovative marketing strategies, personalized solutions, and enhanced user experiences. This study presents a conceptual framework that explores the connection between fundamental personality dimensions and users’ online shopping styles. By employing the TIPI test, a reliable and validated alternative to the Five-Factor model, individual consumer profiles are established. The results reveal a significant relationship between key personality traits and specific online shopping functionalities. To accurately forecast customers’ needs, expectations, and preferences on the Internet, we propose the implementation of two Machine Learning models, namely Decision Trees and Random Forest. According to the applied evaluation metrics, both models demonstrate fine predictions of consumer behavior based on their personality