Ground-state properties of the attractive Hubbard model in one dimension are studied by means of both the exact Bethe-ansatz formalism and the self-consistent field (SCF) approach with renormalized chemical potential for general band fillings n and a wide range of coupling strength $U/t.\mathrm{}$ The energy, the concentration of double occupied sites, the kinetic energy and the chemical potential of the ground state are in a good numerical agreement with the exact results over a wide range of parameters $U/t$ and n. The concentration of local pairs or double occupied sites in the Bethe-ansatz solution serves as a suitable parameter measuring the electron pairing correlations. The SCF theory provides a simple analytical relationship between the concentration of double occupied sites, the band filling and the BCS order parameter, valid for arbitrary $U/t$ and n. The calculated energy gap, the BCS order parameter, the phase diagram and the compressibility are also discussed. The SCF theory in one dimension distinguishes the order parameter from the excitation gap and suggests a smooth crossover away from half-filling (at $n\ne 1)$ from the BCS pairing to the Bose condensation regime under the variation of $U/t$ and n.

Rare-earth iron nitrides are emerging as an important class of magnetic materials. In certain rare-earth iron compounds, the insertion of small atoms such as nitrogen and boron has resulted in significant changes in the magnetic properties in the form of higher Curie temperatures, enhanced magnetic moments, and stronger anisotropies. In an attempt to understand some of the above, we have focused on two nitride phases of Fe, namely ${\mathrm{Fe}}_4\mathrm{N}$ (cubic) and ${\mathrm{Fe}}_{16}{\mathrm{N}}_2$ (tetragonal). For the ${\mathrm{Fe}}_{16}{\mathrm{N}}_2$ phase, the average Fe moment reported by different experimental groups varies over a wide range of values, from $2.3{\mu }_B$ to $3.5{\mu }_B.$ We will discuss some of the recent experiments and examine some related theoretical questions with regard to Fe having such an unusually large moment in a metallic environment. Employing a Hubbard-Stoner-like model in addition to local-density results, it is shown that an unusually large on-site Coulomb repulsion is necessary if one is to obtain a moment as large as $3.5{\mu }_B.$

We present the basic regularities that govern the time evolution dynamics of earthquakes in the seismic system (SS) defined as a unique active volume of lithosphere responsible for preparation of strong earthquakes above a certain threshold magnitude. This concept was articulated previously within phenomenological approach of seismic entropy which utilizes real-time monitoring of strong earthquakes. Seismic parameters such as cumulative energy, number of seismic system states and entropy are calculated to reliably describe in real space–time the non-equilibrium dynamics of the active volume of the Earth's lithosphere. The behaviour of seismic parameters also shows the direction of tectonic processes which preserve a long-term memory of all previously released energies of preceding earthquakes as well as the time when they occurred. The obtained power-law relationship between the information entropy and cumulative energy within a cycle implies that seismically ideal homogeneous system over time tends to the certain critical conditions (attractor). The well-defined SS forms an ensemble of strong earthquakes with critical instabilities which periodically restore the equilibrium state of the system. The finite size of SS and minimal unit of microquake leads to selectivity and discreteness of the earthquake magnitude. It is found that indicator earthquakes above certain threshold magnitudes at which the Gutenberg–Richter law is often violated, play a crucial role in preparation of strong earthquakes. The calculated trajectory diagrams describing the dynamic evolutions of lithosphere allow construct the attractor for real SSs by solving the inverse problem.