Quantum phase transitions in the Kitaev--Heisenberg model on a single hexagon

We present a detailed analysis of the Kitaev--Heisenberg model on a single hexagon. The energy spectra and spin--spin correlations obtained using exact diagonalisation indicate quantum phase transitions between antiferromagnetic and anisotropic spin correlations when the Kitaev interactions increase. In cluster mean-field approach frustrated nearest neighbor exchange stabilizes the stripe phase in between the N\'eel phase and frustrated one which evolves towards the Kitaev spin liquid.Comment: 3 pages, 3 figures, Euroconference Physics of Magnetism, Pozna\'n 201

Evolution of Spin-Orbital Entanglement with Increasing Ising Spin-Orbit Coupling

Several realistic spin-orbital models for transition metal oxides go beyond the classical expectations and could be understood only by employing the quantum entanglement. Experiments on these materials confirm that spin-orbital entanglement has measurable consequences. Here, we capture the essential features of spin-orbital entanglement in complex quantum matter utilizing 1D spin-orbital model which accommodates SU(2)xSU(2) symmetric Kugel-Khomskii superexchange as well as the Ising on-site spin-orbit coupling. Building on the results obtained for full and effective models in the regime of strong spin-orbit coupling, we address the question whether the entanglement found on superexchange bonds always increases when the Ising spin-orbit coupling is added. We show that (i) quantum entanglement is amplified by strong spin-orbit coupling and, surprisingly, (ii) almost classical disentangled states are possible. We complete the latter case by analyzing how the entanglement existing for intermediate values of spin-orbit coupling can disappear for higher values of this coupling.Comment: Proceedings paper [Quantum Complex Matter Conference, June 2020, Frascati] reporting on the research first presented in Physical Review Research 2, 013353 (2020) [arXiv:1911.12180

How spin-orbital entanglement depends on the spin-orbit coupling in a Mott insulator

The concept of the entanglement between spin and orbital degrees of freedom plays a crucial role in understanding various phases and exotic ground states in a broad class of materials, including orbitally ordered materials and spin liquids. We investigate how the spin-orbital entanglement in a Mott insulator depends on the value of the spin-orbit coupling of the relativistic origin. To this end, we numerically diagonalize a 1D spin-orbital model with the 'Kugel-Khomskii' exchange interactions between spins and orbitals on different sites supplemented by the on-site spin-orbit coupling. In the regime of small spin-orbit coupling w.r.t. the spin-orbital exchange, the ground state to a large extent resembles the one obtained in the limit of vanishing spin-orbit coupling. On the other hand, for large spin-orbit coupling the ground state can, depending on the model parameters, either still show negligible spin-orbital entanglement, or can evolve to a highly spin-orbitally entangled phase with completely distinct properties that are described by an effective XXZ model. The presented results suggest that: (i) the spin-orbital entanglement may be induced by large on-site spin-orbit coupling, as found in the 5d transition metal oxides, such as the iridates; (ii) for Mott insulators with weak spin-orbit coupling of Ising-type, such as e.g. the alkali hyperoxides, the effects of the spin-orbit coupling on the ground state can, in the first order of perturbation theory, be neglected.Comment: 16 pages, 8 figures; accepted in Physical Review Researc

Origin of monoclinic distortion and its impact on the electronic properties in KO2_2

We use the density functional theory and lattice dynamics calculations to investigate the properties of potassium superoxide KO$_2$ in which spin, orbital, and lattice degrees of freedom are interrelated and determine the low-temperature phase. After calculating phonon dispersion relations in the high-temperature tetragonal $I4/mmm$ structure, we identify a soft phonon mode leading to the monoclinic $C2/c$ symmetry and optimize the crystal geometry resulting from this mode. Thus we reveal a displacive character of the structural transition with the group-subgroup relation between the tetragonal and monoclinic phases. We compare the electronic structure of KO$_2$ with antiferromagnetic spin order in the tetragonal and monoclinic phases. We emphasize that realistic treatment of the electronic structure requires including the local Coulomb interaction $U$ in the valence orbitals of the O$^-_2$ ions. The presence of the `Hubbard' $U$ leads to the gap opening at the Fermi energy in the tetragonal structure without orbital order but with weak spin-orbit interaction. We remark that the gap opening in the tetragonal phase could also be obtained when the orbital order is initiated in the calculations with a realistic value of $U$. Finally, we show that the local Coulomb interactions and the finite lattice distortion, which together lead to the orbital order via the Jahn-Teller effect, are responsible for the enhanced insulating gap in the monoclinic structure.Comment: accepted by Physical Review

Spin states and correlations for frustrated magnetic interactions on the honeycomb lattice.

Niniejsza praca magisterska opisuje układ sześciu spinów 1/2 umieszczonych na wierzchołkachsześciokąta foremnego. Jest to najprostszy klaster sieci heksagonalnej.Do obliczeń numerycznych dotyczących powyższego układu zostały zaimplementowane założenia modeli: Heisenberga, Isinga - Kitaeva oraz Kitaeva - Heisenbrga.W pierwszym rozdziale przeanalizowano stany spinowe i funkcje korelacji otrzymane numerycznie dla następujacych modeli Heisenberga, uwzględniających odpowiednio pierwszych,pierwszych i drugich a także pierwszych, drugich i trzecich sąsiadów.W drugim rozdziale opisano pokrótce model Kitaeva oraz pokazano wyniki obliczeń dla modeli Isinga - Kitaeva i Kitaeva - Heisenberga (dla ostatniego z modeli również rozważono trzy wersje zależnie od typów uwzględnianych sąsiadów). W trzecim rozdziale zaprezentowano rezultaty obliczeń w samouzgodnionym średnim polu na zredukowanym klastrze obejmującym pojedynczy heksagon wraz z zewnętrznymi wiązaniami.Następnie podsumowano na jważniejsze wyniki i zasugerowano kierunek dalszych badań. Na końcu pracy umieszczono dodatek matematycznyprezentujący szczegółowo obliczanie komutatora [Sz , H] dla dwóch spinów 1/2.The master thesis presented here describes microscopic system which consists of six spins 1/2.The said spins are placed on vertices of a single hexagon.It is the simplest claster of hexagonal lattice.For numerical calculations purpose we implemented Heisenberg, Ising - Kitaev and Kitaev - Heisenberg models assumptions.First chapter consists of spin states and correlation functions analysis.The quantities mentioned above were obtained in a numerical way, for Heisenberg models which include first, first and second as well as first, second and third neighbors.Second chapter presents shortly Kitaev model. Furthermore it shows the results of Ising - Kitaev and Kitaev - Heisenberg models calculations.Third chapter presents the results of self consistant mean field calculations.These calculations were conducted on a reduced claster which included single hexagon with outher bonds.Lastly we summarized the main results and suggested future research directions.Appendix A shows detailed calculations of [Sz,H] commutator for two spins 1.2

Phase diagram and spin correlations of the Kitaev-Heisenberg model : importance of quantum effects

We explore the phase diagram of the Kitaev-Heisenberg model with nearest neighbor interactions on the honeycomb lattice using the exact diagonalization of finite systems combined with the cluster mean field approximation, and supplemented by the insights from analytic approaches: the linear spin-wave and second-order perturbation theories. This study confirms that by varying the balance between the Heisenberg and Kitaev term, frustrated exchange interactions stabilize in this model either one of four phases with magnetic long range order: Néel phase, ferromagnetic phase, and two other phases with coexisting antiferromagnetic and ferromagnetic bonds, zigzag and stripy phase, or one of two distinct spin-liquid phases. Out of these latter disordered phases, the one with ferromagnetic Kitaev interactions has a substantially broader range of stability as the neighboring competing ordered phases, ferromagnetic and stripy, have very weak quantum fluctuations. Focusing on the quantum spin-liquid phases, we study spatial spin correlations and dynamic spin structure factor of the model by the exact diagonalization technique, and discuss the evolution of gapped low-energy spin response across the quantum phase transitions between the disordered spin liquid and phases with long range magnetic order