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Few-layer CrI$_3$ is the most known example among two-dimensional (2D) ferromagnets, which have attracted growing interest in recent years. Despite considerable efforts and progress in understanding the properties of 2D magnets both from theory and experiment, the mechanism behind the formation of in-plane magnetic ordering in chromium halides is still under debate. Here, we propose a microscopic orbitally-resolved description of ferromagnetism in monolayer CrI$_3$. Starting from first-principles calculations, we construct a low-energy model for the isotropic Heisenberg exchange interactions. We find that there are two competing contributions to the long-range magnetic ordering in CrI$_3$: (i) Antiferromagnetic Anderson's superexchange between half-filled $t_{2g}$ orbitals of Cr atoms; and (ii) Ferromagnetic exchange governed by the Kugel-Khomskii mechanism, involving the transitions between half-filled $t_{2g}$ and empty $e_g$ orbitals. Using numerical calculations, we estimate the exchange interactions in momentum-space, which allows us to restore the spin-wave spectrum, as well as estimate the Curie temperature. Contrary to the nearest-neighbor effective models, our calculations suggest the presence of sharp resonances in the spin-wave spectrum at 5--7 meV, depending on the vertical bias voltage. Our estimation of the Curie temperature in monolayer CrI$_3$ yields 55--65 K, which is in good agreement with experimental data.

Rutile CrO$_2$ is an important half-metallic ferromagnetic material, which is also widely used in magnetic recording. In an attempt to find the conditions, which lead to the increase the Curie temperature ($T_{\rm C}$), we study theoretically the band-filling dependence of interatomic exchange interactions in the rutile compounds. For these purposes, we use the effective low-energy model for the magnetic $t_{2g}$ bands, derived from the first-principles electronic structure calculations in the Wannier basis, which is solved by means of dynamical mean-field theory. After the solution, we calculate the interatomic exchange interactions, by using the theory of infinitesimal spin rotations, and evaluate $T_{\rm C}$. We argue that, as far as the Curie temperature is concerned, the band filling realized in CrO$_2$ is far from being the optimal one and much higher $T_{\rm C}$ can be obtained by decreasing the number of $t_{2g}$ electrons ($n$) via the hole doping. We find that the optimal $n$ is close to $1$, which should correspond to the case of VO$_2$, provided that it is crystallized in the rutile structure. This finding was confirmed by using the experimental rutile structure for both CrO$_2$ and VO$_2$ and reflects the general tendency towards ferromagnetism for the narrow-band compounds at the beginning of the band filling. In particular, our results suggest that the strong ferromagnetism can be achieved in the thin films of VO$_2$, whose crystal structure is controlled by the substrate., Comment: 13 pages, 6 figures

We propose a distinct numerical approach to effectively solve the problem of partial diagonalization of the super-large-scale quantum electronic Hamiltonian matrices. The key ingredients of our scheme are the new method for arranging the basis vectors in the computer's RAM and the algorithm allowing not to store a matrix in RAM, but to regenerate it on-the-fly during diagonalization procedure. This scheme was implemented in the program, solving the Anderson impurity model in the framework of dynamical mean-field theory (DMFT). The DMFT equations for electronic Hamiltonian with 18 effective orbitals that corresponds to the matrix with the dimension of 2.4 * 10^9 were solved on the distributed memory computational cluster., Comment: 4 pages, 6 figures, 4 tables

Chromium dioxide (CrO2) offers a rare example of metallic ferromagnetism among stoichiometric transition-metal oxides. What makes it even more remarkable is the half-metallic electronic structure. Today, CrO2 is widely used in magnetorecording and regarded as a promising spintronic material. Nevertheless, the key question "Why is it ferromagnetic?" remains largely unanswered, despite general interest to the problem and practical importance of CrO2. In the present work we challenge this question by combining first-principles electronic structure calculations with the model Hamiltonian approach and modern many-body methods for treating electron correlations. Our analysis demonstrates that the problem is indeed highly nontrivial: at the first glance, the ferromagnetism in CrO2 can be easily explained by Hund's rule related exchange processes in the narrow t2g band. However, the electron correlations, rigorously treated in the frameworks of dynamical mean-field theory, tend to destabilize this state. The ferromagnetism reemerges if, besides conventional kinetic energy changes in the t2g band, to consider other mechanism, involving direct exchange and magnetic polarization of the oxygen band. We show how all these contributions can be evaluated using first-principles electronic structure calculations. Our results explain the overall stability of the ferromagnetic state and provide the firm microscopic basis for understanding the magnetism of CrO2., Comment: 37 pages, 11 figure, 5 tables

We study the role of static and dynamical Coulomb correlation effects on the electronic and magnetic properties of individual Mn, Fe and Co adatoms deposited on the CuN surface. For these purposes, we construct a realistic Anderson model, solve it by using finite-temperature exact diagonalization method and compare the calculated one-particle spectral functions with the LDA+$U$ densities of states. In contrast to Mn/CuN and Fe/CuN, the cobalt system tends to form the electronic excitations at the Fermi level. Based on the calculated magnetic response functions, the relative relaxation times for the magnetic moments of impurity orbitals are estimated. To study the effect of the dynamical correlations on the exchange interaction in nanoclusters, we solve the two-impurity Anderson model for the Mn dimer on the CuN surface. It is found that the experimental exchange interaction can be well reproduced by employing $U$=3 eV, which is two times smaller than the value used in static mean-field LDA+$U$ calculations. This suggests on important role of dynamical correlations in the interaction between adatoms on a surface., Comment: 8 pages, 8 figures