Orbital magnetization is known empirically to play an important role in several magnetic phenomena, such as permanent magnetism and ferromagnetic superconductivity. Within the recently developed ``modern theory of orbital magnetization,'' theoretical insight has been gained into the nature of this often neglected contribution to magnetism but is based on an underlying mean-field approximation. From this theory, a few treatments have emerged which also take into account correlations beyond the mean-field approximation. Here, we apply the scheme developed in a previous work [F. Aryasetiawan et al., Phys. Rev. B 93, 161104(R) (2016)] to the spin-$\frac{1}{2}$ Haldane-Hubbard model to investigate the effect of charge fluctuations on the orbital magnetization within the $GW$ approximation. Qualitatively, we are led to distinguish between two quite different situations: (i) When the lattice potential is larger than the nearest-neighbor hopping, the correlations are found to boost the orbital magnetization. (ii) If the nearest-neighbor hopping is instead larger than the lattice potential, the correlations reduce the magnetization. The boost and reduction are identified to stem from interband and intraband correlations, respectively, and the relative importance of the two varies with the strength of the lattice potential. We finally study graphene with parameters obtained from first principles.
Orbital magnetization is known empirically to play an important role in several magnetic phenomena, such as permanent magnetism and ferromagnetic superconductivity. Within the recently developed ``modern theory of orbital magnetization,'' theoretical insight has been gained into the nature of this often neglected contribution to magnetism but is based on an underlying mean-field approximation. From this theory, a few treatments have emerged which also take into account correlations beyond the mean-field approximation. Here, we apply the scheme developed in a previous work [F. Aryasetiawan et al., Phys. Rev. B 93, 161104(R) (2016)] to the spin-21 Haldane-Hubbard model to investigate the effect of charge fluctuations on the orbital magnetization within the GW approximation. Qualitatively, we are led to distinguish between two quite different situations: (i) When the lattice potential is larger than the nearest-neighbor hopping, the correlations are found to boost the orbital magnetization. (ii) If the nearest-neighbor hopping is instead larger than the lattice potential, the correlations reduce the magnetization. The boost and reduction are identified to stem from interband and intraband correlations, respectively, and the relative importance of the two varies with the strength of the lattice potential. We finally study graphene with parameters obtained from first principles.
This work deals with the calculation of magnetic properties for strongly correlated electron systems. One of most important model in this field of physics is the Hubbard model. It describes the situation of, for example, transition metal oxides. This particular class of materials is known for magnetic order at low temperatures. The reason for this and at the same time the difficulty for theoretically describing them are strong electron-electron-interactions. For calculating the magnetic properties I used the dynamical mean field theory (DMFT), which relates the lattice model to an impurity model. Even this simplified model is highly non trivial. For solving it I used two different renormalization group approaches. I used the numerical renormalization group (NRG) and the density matrix renormalization group (DMRG) and also compared both methods with each other. These modern non-perturbative techniques are able to solve the impurity model, which then can be used for calculating the magnetic phases of the lattice models. In this work I mainly focused on the frustrated one-orbital Hubbard model and the two-orbital Hubbard model. Frustration represents a situation where the aimed magnetic solution cannot be stabilized for the used lattice. Frustration can, for example, be generated by long range hopping of the electrons. This can lead to a situation where new magnetic phases are stable. For systematically analyzing the effects of frustration, I performed calculations for a Bethe lattice with nearest and next nearest neighbor hopping, for which I varied the strength of the interaction and the next nearest neighbor hopping. Another very important aspect for describing transition metal oxides are orbital degrees of freedom. For analyzing the influence of these degrees of freedom I used the two-orbital Hubbard model, which shows new effects.
We investigated the honeycomb lattice using a two-orbital (α and β) Hubbard model and many-body exact calculations. The spin gap (Δ s ), charge gap (Δ c ), intrasite ( γ i i α β = ⟨ S → i α · S → i β ⟩ ), intersite ( γ i j α β = ⟨ S → i α · S → j β ⟩ ), spin–spin correlations, and orbital magnetic moments μ iα are analyzed for different band-filling ( n ), which denotes the number of electrons per site. Special attention was set in one-quarter-filled ( n = 1) and half-filled ( n = 2) bands. The results are presented by varying Coulomb repulsion ( U), Hund’s coupling ( J H ), external magnetic field ( B ), and orbital energy splitting (Δϵ). For the one-quarter-filled band, an intrasite magnetic disorder ( γ i i α β ≈ 0 ) begins to form for large values of U / t, which is interpreted as a possible signature of a resonating valence bond state. Full orbital polarization ( γ i i α β = 0.25 ) with intersite antiferromagnetic (AFM) order ( γ i j α β = − 0.25 ) was observed in the half-filled band for U / t ≫ 1, indicating that the two-orbital Hubbard model can be mapped to an AFM Heisenberg model with S = 1. The increase of J H is conductive to insulating and metallic phases for half- and one-quarter-filled bands, respectively. For a fractional n, the honeycomb lattice can exhibit spin-glass-like behavior, predominating as a disordered magnetic system characterized by random values of γ i j α β . The magnetic field induced a reduction in Δ c, which became zero at the critical external magnetic field. Twisted bilayer graphene (TBG) exhibits approximately flat electronic bands at magic angles and interesting correlated phenomena, such as unconventional superconductivity and correlated insulating states around n = 1 and n = 2, respectively. The dominance of electron–electron interactions in TBG makes the Hubbard model, in various versions (extended or multi-orbital), offer a practical, structure-scale account for investigating many-body effects in large moiré unit cells, where first-principles density-functional theory calculations are computationally challenging.
The Hubbard model has attracted considerable interest due to its prototypical role in describing strongly interacting electronic systems, such as high-critical-temperature superconductors as well as many novel quantum materials. By introducing next-nearest-neighbor (NNN) hoppings to the Hubbard model, the phase diagram becomes richer, and fascinating phenomena arise in both, one-dimensional chains and square lattices, such as: antiferromagnetism, ferromagnetism, superconductivity (SC), as well as charge orders, among others. Moreover, NNN hoppings play a fundamental role in understanding effects of doping on magnetism and pairing orders in strongly interacting regimes. In this article, we review the recent progress in understanding the different competing phases of this model in one and two dimensions from a computational perspective. We comment on the pressing technical challenges, illustrate the controversial results concerning the emergence of the SC phase, and conclude with our perspectives on future explorations.
The Hubbard model has been widely used to describe magnetism due to itinerant electrons. However, except for the one-dimensional case, no exact solution of the model exists. We have studied the two-dimensional Hubbard model on a square lattice using a Monte Carlo simulation technique. Simulations were performed on lattices of up to 8×8 in spatial size. We will discuss results for thermodynamic properties and correlation functions as a function of temperature and interaction strength. For the half-filled band case, our results indicate that the system exhibits long-range antiferromagnetic order in the ground state for any nonzero interaction, although substantially reduced from the classical Néel state due to both charge and spin fluctuations. For the non-half-filled band case, our results suggest that no long-range magnetic order exists. We also don’t find any indication of short-ranged ferromagnetic correlations in the parameter range studied. We have also studied the predictions of various approximate treatments for this model, particularly the static approximation and the coherent potential approximation (CPA). By comparison with the Monte Carlo results, we discuss the validity of these schemes. Finally, we discuss the effect of changing the band structure on the magnetic properties of the system.
We investigate a minimal two-orbital Hubbard model with intra- and inter-orbital nearest-neighbor hopping t and t~, as well as intra- and inter-orbital density-density interactions U and U′ by means of variational ansätze based on Jastrow-Slater wave functions within quantum Monte Carlo techniques. To focus on the electronic mechanisms of superconductivity, we restrict the variational ansätze to uniform nonmagnetic states with an explicit pairing amplitude and compute the pairing correlations as a function of filling and model parameters. At U/t=10, superconducting correlations are highly enhanced by the presence of inter-orbital terms, U′ and t~. For t~=0, a finite value of U′ effectively screens the intra-orbital repulsion U, producing a shift in the superconducting dome. Consequently, inter-orbital repulsion yields a sizable increase in electron pairing compared to the single-orbital baseline. Furthermore, introducing a finite t~ provides an additional boost to superconducting correlations, an effect driven by the simultaneous presence of flat and broad bands in the electronic structure.
To understand correlated insulating and unconventional superconducting states in twisted bilayer graphene, we perform a systematic study of spin and pairing correlations in an effective two-orbital Hubbard model, by using the ground-state constrained-path quantum Monte Carlo method. Our numerical simulations reveal that when only the on-site Hubbard U is considered, the long-range magnetic order develops for $U\ensuremath{\ge}3.2t$ at half filling. Upon doping away from half filling, an analysis based on the pairing correlations and corresponding vertex contributions identifies that pairing with the d+id symmetry is the dominant pairing channel. As the on-site Coulomb interaction is increased, both spin correlation and effective pairing interaction are enhanced simultaneously, indicating the intimate relation between magnetism and superconductivity. An inclusion of nearest neighbor Coulomb interaction V results in a suppression of the d+id superconductivity, but it can survive in the small V regime. Our findings are useful for clarifying the ongoing controversy on the superconducting pairing symmetry in twisted bilayer graphene.