ИСТИНА

Studying the magnetic characteristics of the two dimensional (2D) lattice (hexagonal, graphene-like [1] and the square one [2]) by solving numerically the exact system of discrete equations, which fully describes the broadening of Landau levels, we have found that the energy spectrum of the Landau level at the energy of the saddle point (which is also known as a Van Hove peak) is principally continuous, so that even in a very small magnetic field this Landau level is always broadened in a miniband. In contrast to that, at other energies in a weak magnetic the spectrum of Landau levels is discrete. At the energy of the saddle point of the Brillouin zone the corresponding density of states 𝑁(𝐸𝐹) is formally infinite (𝑁(𝐸𝐹) → +∞). We then consider the 2D square lattice [2], used as a prototype electron system in which the Fermi level lies exactly at the Van Hove peak. According to the electron band treatment this could lead to a formally divergent paramagnetic susceptibility and an infinite electron contribution to the specific heat at zero temperature. Our accurate analysis shows that both values remain finite. Taking into account the electron spin polarization and the obtained numerical solution, we reproduce the temperature dependence of the induced magnetic moment proportional to the magnetic susceptibility, and the electron contribution to the specific heat. Both plots demonstrate unusual dependencies reflecting the “metal”-like or “insulator”-like structure of the Landau minibands in the neighborhood of the Fermi energy. At low temperatures all quantities display oscillatory behavior. We also prove rigorously that the fully occupied electron band has no contribution to the diamagnetic susceptibility and specific heat [1,2]. [1] A. V. Nikolaev, Phys. Rev. B, 104, 035419 (2021); Phys. Rev. B, 105, 039902 (2022). [2] A.V. Nikolaev, M.Ye. Zhuravlev, Journal of Magnetism and Magnetic Materials, 560, 169674 (2022).