Superconducting properties of small metallic grains modelled by highly degenerate two-level spectrum have been studied. We have solved numerically Richardson's exact equations for the system of 2N electrons in two levels. Characterising the size of the grain by the level degeneracy we study the finite size corrections to the thermodynamic limit of the ground and lowest excited state energy. The interparticle distance∝ N^{-1/3} seems to be the expansion parameter. The obtained results have been compared with those of other authors.
It is shown that the magnetic structure of high-T_{c} superconductors is strongly influenced by the next-nearest neighbor hopping parameter t' which distinguishes different families of cuprates. Our investigations indicate that uniform spirals get favored by a large t'/t ratio but are unstable at small doping towards stripes with spin canting. For large |t'/t| spirals can be stabilized under certain conditions in the overdoped regime which may explain the elastic incommensurate magnetic response recently observed in iron-co-doped Bi2201 materials.
The two-level version of the Richardson model presents a unique possibility to calculate numerically exactly thermodynamic properties of the system it describes. The point is that all energies and the degeneracies of the many-body system can be easily calculated. The energies are given by the eigenvalues of the small (of order of N×N, where 2N is a number of electrons in the system) tridiagonal matrices. Here we numerically obtain a complete spectrum of the interacting two-level model and calculate the specific heat and the pairing energy of the small system at finite temperatures.
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