Hund's Coupling Research Articles

Interplay of charge and spin fluctuations in a Hund's coupled impurity

In Hund's metals, the local ferromagnetic interaction between orbitals leads to an emergence of complex electronic states with large and slowly fluctuating magnetic moments. Introducing the Hund's coupled mixed valence quantum impurity, we gain analytic insight into recent numerical renormalization group studies. We show that valence fluctuations drastically impede the development of a large fluctuating moment over a wide range of temperatures and energy, characterized by quenched orbital degrees of freedom and a singular logarithmic behavior of the spin susceptibility $\chi_{\rm sp}''(\omega) \propto [\omega \ln(\omega/T_K^{\rm eff})^2]^{-1}$, closely resembling power-law scaling $\chi_{\rm sp}''(\omega) \sim \omega^{-\gamma}$. Such singular spin fluctuations are suspected to play an important role in future models of Hund's driven Cooper pairing.

Correlated metals and unconventional superconductivity in rhombohedral trilayer graphene: A renormalization group analysis

Motivated by recent experimental observations of correlated metallic phases and superconductivity in rhombohedral trilayer graphene (RTG), we perform an unbiased study of electronic ordering instabilities in hole-doped RTG. Specifically, we focus on electronic states energetically proximate to Van Hove singularities (VHSs), where a large density of states promotes different interaction-induced symmetry-breaking electronic orders. To resolve the Fermi surface near VHSs, we construct a fermionic hot-spot model and demonstrate that a perpendicular electric field can tune different nesting structures of the Fermi surface. Subsequently, we apply a renormalization group analysis to describe the low-energy phase diagrams of our model under both short-range repulsive interactions as well as realistic (long-range) Coulomb interactions. Our analysis shows instabilities towards either intervalley coherent metallic phases or superconducting phases. The dominant pairing channel depends crucially on the nature of Fermi surface nesting -- repulsive Coulomb interaction favors spin-singlet $d$-wave pairing for relatively small displacement field and spin-singlet $i$-wave pairing for larger displacement field. We argue that the phase diagram of RTG can be well-understood by modeling the realistic Coulomb interaction as the sum of repulsive density-density interaction and ferromagnetic spin-triplet intervalley coherence (IVC) Hund's coupling, while phonon-mediated electronic interactions have a negligible effect on this system, in sharp contrast to twisted graphene multilayers.

Spin-orbit coupling controlled ground states in the double perovskite iridates A2BIrO6 ( A= Ba, Sr; B= Lu, Sc)

Iridates with the $5{d}^{4}$ electronic configuration have attracted recent interest due to reports of magnetically ordered ground states despite longstanding expectations that their strong spin-orbit coupling would generate a $J=0$ electronic ground state for each ${\mathrm{Ir}}^{5+}$ ion. The major focus of prior research has been on the double perovskite iridates ${\mathrm{Ba}}_{2}{\mathrm{YIrO}}_{6}$ and ${\mathrm{Sr}}_{2}{\mathrm{YIrO}}_{6}$, where the nature of the ground states (i.e., ordered vs nonmagnetic) is still controversial. Here, we present neutron powder diffraction, high-energy-resolution fluorescence-detected x-ray absorption spectroscopy (HERFD-XAS), resonant inelastic x-ray scattering (RIXS), magnetic susceptibility, and muon spin relaxation data on the related double perovskite iridates ${\mathrm{Ba}}_{2}{\mathrm{LuIrO}}_{6}, {\mathrm{Sr}}_{2}{\mathrm{LuIrO}}_{6}, {\mathrm{Ba}}_{2}{\mathrm{ScIrO}}_{6}$, and ${\mathrm{Sr}}_{2}{\mathrm{ScIrO}}_{6}$ that enable us to gain a general understanding of the electronic and magnetic properties for this family of materials. Our HERFD-XAS and RIXS measurements establish $J=0$ electronic ground states for the ${\mathrm{Ir}}^{5+}$ ions in all cases, with similar values for Hund's coupling ${J}_{\mathrm{H}}$ and the spin-orbit coupling constant ${\ensuremath{\lambda}}_{\mathrm{SOC}}$. Our bulk susceptibility and muon spin relaxation data find no evidence for long-range magnetic order or spin freezing, but they do exhibit weak magnetic signals that are consistent with extrinsic local moments. Our results indicate that the large ${\ensuremath{\lambda}}_{\mathrm{SOC}}$ is the key driving force behind the electronic and magnetic ground states realized in the $5{d}^{4}$ double perovskite iridates, which agrees well with conventional wisdom.

Influence of Multi‐Orbitals and Hund's Coupling Induced Pseudogap on Specific Heat Jump in Iron‐Pnictide High T c Superconductors

A three-orbital per site theoretical model study is attempted to analyze nature of specific heat in iron-based superconductors. Based on recent experimental data in these systems, it is found that Fe five 3d orbitals, Hund’s coupling and intra- and inter-electron orbital correlations play significant role in electronic properties of these systems. Thus, in present work, we employed model Hamiltonian, containing hopping between orbitals, Hund’s coupling and intra- and inter- orbital Coulomb interactions for iron pnictide. Employing Green’s function technique within BCS mean field approximation, expressions of superconducting energy gap parameter and quasiparticle energies are obtained. Considering these parameters, we calculated specific heat and analyzed its variation with temperature and other parameters of model Hamiltonian. Also, a pseudogap (PG) parameter, arising out of multi-orbitals’ Hund’s coupling, as supported by ARPES and STM experiments, is introduced. The expression of specific heat is numerically analyzed to pinpoint its variation below and above transition temperature (T c ). A self-consistent numerical analysis of the specific heat jump as a function of multi-orbital hopping, electron interactions and Hund’s coupling in terms of PG parameter is shown in detail. Finally, we compared outcomes with recent theoretical and experimental data available on specific heat variations in iron superconductors like LaOFeAs system. This article is protected by copyright. All rights reserved.

Anomalous Hall effect at half filling in twisted bilayer graphene

Magic-angle twisted bilayer graphene (tBLG) has been studied extensively owing to its wealth of symmetry-broken phases, correlated Chern insulators, orbital magnetism, and superconductivity. In particular, the anomalous Hall effect (AHE) has been observed at odd integer filling factors ($\nu=1$ and $3$) in a small number of tBLG devices, indicating the emergence of a zero-field orbital magnetic state with spontaneously broken time-reversal symmetry. However, the AHE is typically not anticipated at half filling ($\nu=2$) owing to competing intervalley coherent states, as well as spin-polarized and valley Hall states that are favored by an intervalley Hund's coupling. Here, we present measurements of two tBLG devices with twist angles slightly away from the magic angle (0.96$^{\circ}$ and 1.20$^{\circ}$), in which we report the surprising observation of the AHE at $\nu=+2$ and $-2$, respectively. These findings imply that a valley-polarized phase can become the ground state at half filling in tBLG rotated slightly away from the magic angle. Our results reveal the emergence of an unexpected ground state in the intermediately-coupled regime ($U/W \sim 1$, where $U$ is the strength of Coulomb repulsion and $W$ is the bandwidth), in between the strongly-correlated insulator and weakly-correlated metal, highlighting the need to develop a more complete understanding of tBLG away from the strongly-coupled limit.

Exotic physical properties in metallic perovskite LaRuO3 : Strong evidence for Hund metal

The Hund's coupling rule is of fundamental importance in atomic physics. It also plays a key role in the determination of physical properties in strongly correlated electron systems where the Fermi energy lies in multiple bands. We report a thorough investigation of a rarely studied $4d$ perovskite oxide ${\mathrm{LaRuO}}_{3}$ by a suite of measurements including transport, magnetization, specific heat, thermoelectric power, thermal conductivity, and neutron and x-ray diffraction. Our results are consistent with theoretical predictions, which makes ${\mathrm{LaRuO}}_{3}$ an exemplary Hund metal.

Doping Asymmetry and Layer-Selective Metal-Insulator Transition in Trilayer K_{3+x}C_{60}.

Thin films provide a versatile platform to tune electron correlations and explore new physics in strongly correlated materials. Epitaxially grown thin films of the alkali-doped fulleride K_{3+x}C_{60}, for example, exhibit intriguing phenomena, including Mott transitions and superconductivity, depending on dimensionality and doping. Surprisingly, in the trilayer case, a strong electron-hole doping asymmetry has been observed in the superconducting phase, which is absent in the three-dimensional bulk limit. Using density-functional theory plus dynamical mean-field theory, we show that this doping asymmetry results from a substantial charge reshuffling from the top layer to the middle layer. While the nominal filling per fullerene is close to n=3, the top layer rapidly switches to an n=2 insulating state upon hole doping, which implies a doping asymmetry of the superconducting gap. The interlayer charge transfer and layer-selective metal-insulator transition result from the interplay between crystal field splittings, strong Coulomb interactions, and an effectively negative Hund coupling. This peculiar charge reshuffling is absent in the monolayer system, which is an n=3 Mott insulator, as expected from the nominal filling.

Theoretical analysis of single-ion anisotropy in d3 Mott insulators

An effective spin model for Mott insulators is determined by the symmetries involved among magnetic sites, electron fillings, and their interactions. Such a spin Hamiltonian offers insight to mechanisms of magnetic orders and magnetic anisotropy beyond the Heisenberg model. For a spin moment S bigger than 1/2, single-ion anisotropy is in principle allowed. However, for $d^3$ Mott insulators with large cubic crystal field splitting, the single-ion anisotropy is absent within the LS coupling, despite S = 3/2 local moment. On the other hand, preferred magnetic moment directions in $d^3$ materials have been reported, which calls for a further theoretical investigation. Here we derive the single-ion anisotropy interaction using the strong-coupling perturbation theory. The cubic crystal field splitting including $e_g$ orbitals, trigonal distortions, Hund's coupling, and spin-orbit coupling beyond the LS scheme are taken into account. For compressed distortion, the spin-orbit coupling at magnetic sites can favor either the easy-axis or the easy-plane while that of anions leads to easy-axis anisotropy. We apply the theory on $\rm{CrX}_3$ with X = Cl and I, and show the dependence of the single-ion anisotropy on the strength of the spin-orbit couplings of both magnetic and anion sites. Significance of the single-ion anisotropy in ideal two-dimensional magnets is also discussed.

Leading superconducting instabilities in three-dimensional models for Sr2RuO4

The superconductor Sr2RuO4 has been the subject of enormous interest over more than two decades, but until now the form of its order parameter has not been determined. Since groundbreaking NMR experiments revealed that the pairs are of dominant spin-singlet character, attention has focused on time-reversal symmetry breaking linear combinations of $s$-, $d$- and $g$-wave one-dimensional (1D) irreducible representations. However, a state of the form $d_{xz}+id_{yz}$ has also been proposed. We present a systematic study of the stability of various superconducting candidate states, assuming that pairing is driven by the fluctuation exchange mechanism, including a realistic three-dimensional Fermi surface, full treatment of both local and non-local spin-orbit couplings, and a wide range of interaction parameters $U,J,U',J'$. The leading superconducting instabilities are found to exhibit nodal even-parity $A_{1g} (s')$ or $B_{1g} (d_{x^2-y^2})$ symmetries, similar to the findings in two-dimensional models without longer-range Coulomb interaction which tends to favor $d_{xy}$ over $d_{x^2-y^2}$. Within the so-called Hund's coupling mean-field pairing scenario, the $E_g (d_{xz}/d_{yz})$ solution can be stabilized for large $J$ and specific forms of the spin-orbit coupling, but for all cases studied here the eigenvalues of other superconducting solutions are significantly larger when the full fluctuation exchange vertex is included in the pairing kernel. Additionally, we compute the spin susceptibility in relevant superconducting candidate phases and compare to recent neutron scattering and NMR Knight shift measurements.

Estimation of biquadratic and bicubic Heisenberg effective couplings from multiorbital Hubbard models

We studied a multi-orbital Hubbard model at half-filling for two and three orbitals per site on a two-site cluster via full exact diagonalization, in a wide range for the onsite repulsion U, from weak to strong coupling, and multiple ratios of the Hund coupling J H to U. The hopping matrix elements among the orbitals were also varied extensively. At intermediate and large U, we mapped the results into a Heisenberg model. For two orbitals per site, the mapping is into a S = 1 Heisenberg model where by symmetry both nearest-neighbor (S i ⋅ S j ) and are allowed, with respective couplings J 1 and J 2. For the case of three orbitals per site, the mapping is into a S = 3/2 Heisenberg model with (S i ⋅ S j ), , and terms, and respective couplings J 1, J 2, and J 3. The strength of these coupling constants in the Heisenberg models depend on the U, J H, and hopping amplitudes of the underlying Hubbard model. Our study provides a first crude estimate to establish bounds on how large the ratios J 2/J 1 and J 3/J 1 can be. We show that those ratios appear rather limited and, as a qualitative guidance, we conclude that J 2/J 1 is less than 0.4 and J 3/J 1 is less than 0.2, establishing bounds on effective models for strongly correlated Hubbard systems. Moreover, the intermediate Hubbard U regime was found to be the most promising to enhance J 2/J 1 and J 3/J 1.

Bad metal and negative compressibility transitions in a two-band Hubbard model

We analyze the paramagnetic state of a two-band Hubbard model with finite Hund's coupling close to integer filling at $n=2$ in two spacial dimensions. Previously, a Mott metal-insulator transition was established at $n=2$ with a coexistence region of a metallic and a bad metal state in the vicinity of that integer filling. The coexistence region ends at a critical point beyond which a charge instability persists. Here we investigate the transition into negative electronic compressibility states for an extended filling range close to $n=2$ within a slave boson setup. We analyze the separate contributions from the (fermionic) quasiparticles and the (bosonic) multiparticle incoherent background and find that the total compressibility depends on a subtle interplay between the quasiparticle excitations and collective fields. Implementing a Blume-Emery-Griffiths model approach for the slave bosons, which mimics the bosonic fields by Ising-like pseudospins, we suggest a feedback mechanism between these fields and the fermionic degrees of freedom. We argue that the negative compressibility can be sustained for heterostructures of such strongly correlated planes and results in a large capacitance of these structures. The strong density dependence of these capacitances allows to tune them through small electronic density variations. Moreover, by resistive switching from a Mott insulating state to a metallic state through short electric pulses, transitions between fairly different capacitances are within reach.

Noncubic local distortions and spin-orbit excitons in K2IrCl6

The cubic antifluorite ${\mathrm{K}}_{2}{\mathrm{IrCl}}_{6}$ has recently garnered renewed attention due to its relevance to Kitaev magnetism. Combining Raman spectroscopy with numerical calculations, we investigate its electronic structure and ensuing low-lying excitations as well as lattice instabilities. For temperatures below ${T}^{*}\ensuremath{\approx}180$ K, we observe several lattice anomalies: (i) a gradual appearance of the symmetry-forbidden phonons; (ii) a central-mode-like excitation; and (iii) a soft-mode-like behavior of the ${\mathrm{\ensuremath{\Gamma}}}^{5+}$ mode involving vibrations of the ${\mathrm{K}}^{+}$ ion relative to the ${\mathrm{Cl}}^{\ensuremath{-}}$ ion. All these features indicate the occurrence of local noncubic distortions. At high energies, we observe spin-orbit (SO) excitons made of five peaks at $\ensuremath{\omega}=0.62--0.79$ eV as well as a weak electronic excitation at $\ensuremath{\omega}=0.12--0.86$ eV. Our numerical calculations reproduce their spectral energy and shape with the electronic parameters: SO coupling $\ensuremath{\lambda}=465$ meV; Hund's coupling ${J}_{H}=300$ meV; tetragonal distortion strength ${\mathrm{\ensuremath{\Delta}}}_{t}=30$ meV and on-site Coulomb interaction $U=2.2$ eV. The multiple SO excitons are interpreted in terms of bounded SO excitons of the $|{j}_{\mathrm{eff}}=\frac{3}{2},\ifmmode\pm\else\textpm\fi{}\frac{1}{2}\ensuremath{\rangle}$ states arising from the coupling between the SO excitons and electron-hole excitations. Our results showcase that ${\mathrm{K}}_{2}{\mathrm{IrCl}}_{6}$ is on the brink of a structural phase transition.

Fermi Surface Expansion above Critical Temperature in a Hund Ferromagnet.

Using a cluster extension of the dynamical mean-field theory, we show that strongly correlated metals subject to Hund's physics exhibit significant electronic structure modulations above magnetic transition temperatures. In particular, in a ferromagnet having a large local moment due to Hund's coupling (Hund's ferromagnet), the Fermi surface expands even above the Curie temperature (T_{C}) as if a spin polarization occurred. Behind this phenomenon, effective "Hund's physics" works in momentum space, originating from ferromagnetic fluctuations in the strong-coupling regime. The resulting significantly momentum-dependent (spatially nonlocal) electron correlations induce an electronic structure reconstruction involving a Fermi surface volume change and a redistribution of the momentum-space occupation. Our finding will give a deeper insight into the physics of Hund's ferromagnets above T_{C}.

Coulomb correlations and magnetic properties of L10 FeCo: A DFT+DMFT study

We consider electronic correlation effects and their impact on magnetic properties of tetragonally distorted chemically ordered FeCo alloys (L1$_0$ structure) being a promising candidate for rare-earth-free permanent magnets. We employ a state-of-the-art method combining density functional and dynamical mean-field theory. According to our results, the predicted Curie temperature reduces with increase of lattice parameters ratio $c/a$ and reaches nearly 850 K at ${c/a=1.22}$. For all considered $c/a$ from 1 to $\sqrt{2}$, we find well-localized magnetic moments on Fe sites, which are formed due to strong correlations originating from Hund's coupling. At the same time, magnetism of Co sites is more itinerant with a much less lifetime of local magnetic moments. However, these short-lived local moments are also formed due to Hund's exchange. Electronic states at Fe sites are characterized by a non-quasiparticle form of self-energies, while the ones for Co sites are found to have a Fermi-liquid-like shape with quasiparticle mass enhancement factor ${m^*/m\sim 1.4}$, corresponding to moderately correlated metal. The strong electron correlations on Fe sites leading to Hund's metal behaviour can be explained by peculiarities of the density of states, which has pronounced peaks near the Fermi level, while weaker many-body effects on Co sites can be caused by stronger deviation from half-filling of their $3d$ states. The obtained momentum dependence of magnetic susceptibility suggests that the ferromagnetic ordering is the most favourable one except for the near vicinity of the fcc structure and the magnetic exchange is expected to be of RKKY type.

The density functional theory study of substitution effect in antiferromagnetic USn0.5Sb1.5

The electronic structure of antiferromagnetic (AF) USn0.5Sb1.5 crystallizing in the tetragonal Cu2Sb-type structure is investigated by means of ab-initio calculations using the full-potential linearized augmented plane wave method (FP-LAPW) method. Using Local-spin density approximation (LSDA) with the inclusion of spin-orbit coupling (LSDA ​+ ​SO) and Hubbard corrections (LSDA ​+ ​SO ​+ ​U-J) in the calculations, the most stable magnetic ground with an ordered magnetic moment of ∼ 1.7 μB is predicted for the system. Meaningful changes in the lattice parameters, magnetic moments, and magnetic ground state instabilities of USn0.5Sb1.5 are discussed in terms of effects brought by the Coulomb interaction, and the Hund coupling. The calculated electronic properties involving Densities of States (DOS), Electronic Band Structures (EBS), Electron Localization Functions (ELF), and Fermi Surfaces (FS) are presented and compared to those of the parent USb2 compound. The electronic properties, particularly FS and ELF, are found to be strongly modified as a result of the Sn-substitution. The reformation of anisotropy in the FS and ELF upon changing magnetic states (paramagnetic → antiferromagnetic) is examined.

Evolution of interorbital superconductor to intraorbital spin-density wave in layered ruthenates

The ruthenate family of layered perovskites has been a topic of intense interest, with much work dedicated to the superconducting state of ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$. Another long-standing puzzle is the lack of superconductivity in its sister compound, ${\mathrm{Sr}}_{3}{\mathrm{Ru}}_{2}{\mathrm{O}}_{7}$, which constrains the possible mechanisms of ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$. Here we address a microscopic mechanism that unifies the orders in these materials. Beginning from a model of ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ featuring interorbital spin-triplet pairing via Hund's and spin-orbit couplings, we find that bilayer coupling alone enhances, while staggered rotations destroy interorbital superconductivity. A magnetic field then shifts van Hove singularities, allowing intraorbital spin-density wave order to form in ${\mathrm{Sr}}_{3}{\mathrm{Ru}}_{2}{\mathrm{O}}_{7}$. Our theory predicts that ${\mathrm{Sr}}_{3}{\mathrm{Ru}}_{2}{\mathrm{O}}_{7}$ without staggered rotations exhibits interorbital superconductivity with a possibly higher transition temperature.

Sign-free determinant quantum Monte Carlo study of excitonic density orders in a two-orbital Hubbard-Kanamori model

While excitonic instabilities in multiorbital systems recently have come under scrutiny in a variety of transition-metal compounds, understanding emergence of these instabilities from strong electronic interactions has remained a challenge. Here, we present a sign-problem-free determinant quantum Monte Carlo study of excitonic density orders in a half-filled two-orbital Hubbard-Kanamori model with broken orbital degeneracy, which accounts for the role of Hund's coupling in transition-metal compounds. For strong inverted (negative) Hund's exchange, we find numerical evidence for the emergence of excitonic density order, with competition between anti-ferro-orbital order and $\mathbf{Q} = (\pi,\pi)$ excitonic density order as a function of orbital splitting and Hund's coupling. While inverted Hund's coupling stabilizes a spin-singlet excitonic density phase for weak orbital splitting, positive Hund's coupling favors a spin-triplet excitonic density phase.

Electronic structure and magnetism of the Hund's insulator CrI3

${\mathrm{CrI}}_{3}$ is a two-dimensional ferromagnetic (FM) van der Waals material with a charge gap of 1.1--1.2 eV. In this paper, the electronic structure and magnetism of ${\mathrm{CrI}}_{3}$ are investigated by using density functional theory and dynamical mean-field theory. Our calculations successfully reproduce a charge gap of $\ensuremath{\sim}1.1\phantom{\rule{0.16em}{0ex}}\mathrm{eV}$ in the paramagnetic (PM) state when a Hund's coupling ${J}_{\mathrm{H}}=0.7\phantom{\rule{0.16em}{0ex}}\mathrm{eV}$ is included with an onsite Hubbard $U=5\phantom{\rule{0.16em}{0ex}}\mathrm{eV}$. In contrast, with a large $U$ value of 8 eV and negligible Hund's coupling ${J}_{\mathrm{H}}, {\mathrm{CrI}}_{3}$ is predicted to be a moderately correlated metal in the PM state. We conclude that ${\mathrm{CrI}}_{3}$ is a Mott-Hund's insulator due to the half-filled configuration of the Cr $3d {t}_{2g}$ orbitals. The Cr $3d {e}_{g}$ orbitals are occupied by $\ensuremath{\sim}1$ electron, which leads to strong valence fluctuations so that the Cr $3d$ orbitals cannot be described by a single state. Moreover, at finite temperature, the calculated ordered static magnetic moment in the FM state is significantly larger in the $R\overline{3}$ phase than in the $C2/m$ phase. This observation indicates that the structural phase transition from the $C2/m$ phase to the $R\overline{3}$ phase with decreasing temperature is driven by FM spin fluctuations.

Comparing the influence of Floquet dynamics in various Kitaev-Heisenberg materials

In this paper we examine the possibility of Floquet engineering in the three candidate Kitaev materials ${\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}$, $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}$, and $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$. We derive an effective Floquet Hamiltonian and give an approximation for heating processes arising from doublon holon propagation. This suggests that compounds with stronger Hund's rule coupling are less prone to heating. We then investigate the impact of light frequency and amplitude on magnetic interaction terms up to third nearest neighbor and find that third neighbor Heisenberg coupling ${J}_{3}$ is very susceptible to tuning by circularly polarized light. Finally, we discuss uses of linear polarized light in selectively tuning single bond directions.

Prediction of orbital-selective Mott phases and block magnetic states in the quasi-one-dimensional iron chain Ce2O2FeSe2 under hole and electron doping

The recent detailed study of quasi-one-dimensional iron-based ladders, with the $3d$ iron electronic density $n = 6$, has unveiled surprises, such as orbital-selective phases. However, similar studies for $n=6$ iron chains are still rare. Here, a three-orbital electronic Hubbard model was constructed to study the magnetic and electronic properties of the quasi-one-dimensional $n=6$ iron chain Ce$_2$O$_2$FeSe$_2$, with focus on the effect of doping. Specifically, introducing the Hubbard $U$ and Hund $J_{H}$ couplings and studying the model via the density matrix renormalization group, we report the ground-state phase diagram varying the electronic density away from $n=6$. For the realistic Hund coupling $J_{H}/U = 1/4$, several electronic phases were obtained, including a metal, orbital-selective Mott, and Mott insulating phases. Doping away from the parent phase, the competition of many tendencies leads to a variety of magnetic states, such as ferromagnetism, as well as several antiferromagnetic and magnetic "block" phases. In the hole-doping region, two different interesting orbital-selective Mott phases were found: OSMP1 (with one localized orbital and two itinerant orbitals) and OSMP2 (with two localized orbitals and one itinerant orbital). Moreover, charge disproportionation phenomena were found in special doping regions. We argue that our predictions can be tested by simple modifications in the original chemical formula of Ce$_2$O$_2$FeSe$_2$.