Spin-1/2 chains with alternating antiferromagnetic (AFM) and ferromagnetic (FM) couplings have attracted considerable interest due to the topological character of their spin excitations. Here, using density functional theory and density-matrix renormalization-group (DMRG) methods, we have systematically studied the dimerized chain system ${\mathrm{Na}}_2{\mathrm{Cu}}_2{\mathrm{TeO}}_6$ with a $d^9$ electronic configuration. Near the Fermi level, in the nonmagnetic phase the dominant states are mainly contributed by the Cu $3d_{x^2-y^2}$ orbitals highly hybridized with the O $2p$ orbitals, leading to an “effective” single-orbital low-energy model. By calculating the relevant hoping amplitudes, we explain the size and sign of the exchange interactions in ${\mathrm{Na}}_2{\mathrm{Cu}}_2{\mathrm{TeO}}_6$. In addition, a single-orbital Hubbard model is constructed for this dimerized chain system where the quantum fluctuations are taken into account. Both AFM and FM couplings (leading to an $\uparrow \text{−}\downarrow \text{−}\downarrow \text{−}\uparrow$ state) along the chain were found in our DMRG and Lanczos calculations, in agreement with density functional theory and neutron-scattering results. The hole pairing binding energy $\mathrm{\Delta }E$ is predicted to be negative at Hubbard $U\sim 11\mathrm{eV}$, suggesting incipient pairing tendencies.

The van der Waals oxide dichlorides $M\mathrm{O}{X}_2$ ($M=\mathrm{V}$, Ta, Nb, Ru, and Os; $X=\text{halogen}$ element), with different electronic densities, are attracting considerable attention. Ferroelectricity, spin-singlet formation, and orbital-selective Peierls phases were reported in this family with $d^1$ or $d^2$ electronic configurations, all believed to be caused by the strongly anisotropic electronic orbital degree of freedom. Here, using density functional theory and density matrix renormalization group methods, we investigate the electronic and magnetic properties of ${\mathrm{RuOCl}}_2$ and ${\mathrm{OsOCl}}_2$ with $d^4$ electronic configurations. Different from a previous study using ${\mathrm{VOI}}_2$ with $d^1$ configuration, these systems with $4d^4$ or $5d^4$ do not exhibit a ferroelectric instability along the $a$ axis. Due to the fully occupied $d_{xy}$ orbital in ${\mathrm{RuOCl}}_2$ and ${\mathrm{OsOCl}}_2$, the Peierls instability distortion disappears along the $b$ axis, leading to an undistorted $Immm$ phase (No. 71). Furthermore, we observe strongly anisotropic electronic and magnetic structures along the $a$ axis. For this reason, the materials of our focus can be regarded as “effective one-dimensional” systems even when they apparently have a dominant two-dimensional lattice geometry. The large crystal-field splitting energy (between $d_{xz/yz}$ and $d_{xy}$ orbitals) and large hopping between nearest-neighbor Ru and Os atoms suppresses the $J=0$ singlet state in $M{\mathrm{OCl}}_2$ ($M=\mathrm{Ru}$ or Os) with electronic density $n=4$, resulting in a spin-1 system. Moreover, we find staggered antiferromagnetic order with $\pi$ wave vector along the $M$-O chain direction ($a$ axis) while the magnetic coupling along the $b$ axis is weak. Based on Wannier functions from first-principles calculations, we calculated the relevant hopping amplitudes and crystal-field splitting energies of the $t_{2g}$ orbitals for the Os atoms to construct a multiorbital Hubbard model for the $M$-O chains. Staggered AFM with $\uparrow \text{−}\downarrow \text{−}\uparrow \text{−}\downarrow$ spin structure dominates in our density matrix renormalization group calculations, in agreement with density functional theory calculations. Our results for ${\mathrm{RuOCl}}_2$ and ${\mathrm{OsOCl}}_2$ provide guidance to experimentalists and theorists working on this interesting family of oxide dichlorides.

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 ${\mathrm{Ce}}_2{\mathrm{O}}_2\mathrm{Fe}{\mathrm{Se}}_2$, 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 ${\mathrm{Ce}}_2{\mathrm{O}}_2\mathrm{Fe}{\mathrm{Se}}_2$.

The material $\alpha \text{−}{\mathrm{RuCl}}_3$, with a two-dimensional Ru honeycomb sublattice, has attracted considerable attention because it may be a realization of the Kitaev quantum spin liquid. Recently, a new honeycomb material, $\alpha \text{−}{\mathrm{RuI}}_3$, was prepared under moderately high pressure, and it is stable under ambient conditions. However, different from $\alpha \text{−}{\mathrm{RuCl}}_3, \alpha \text{−}{\mathrm{RuI}}_3$ was reported to be a paramagnetic metal without long-range magnetic order down to 0.35 K. Here, the structural and electronic properties of the quasi-two-dimensional $\alpha \text{−}{\mathrm{RuI}}_3$ are theoretically studied. First, based on first-principles density functional theory calculations, the ABC stacking honeycomb-layer $R\overline{3}$ (No. 148) structure is found to be the most likely stacking order for $\alpha \text{−}{\mathrm{RuI}}_3$ along the $c$ axis. Furthermore, both $R\overline{3}$ and $P\overline{3}1c$ are dynamically stable because no imaginary frequency modes were obtained in the phononic dispersion spectrum without Hubbard $U$. Moreover, the different physical behavior of $\alpha \text{−}{\mathrm{RuI}}_3$ compared to $\alpha \text{−}{\mathrm{RuCl}}_3$ can be understood naturally. The strong hybridization between Ru $4d$ and I $5p$ orbitals decreases the “effective” atomic Hubbard repulsion, leading the electrons of ${\mathrm{RuI}}_3$ to be less localized than in ${\mathrm{RuCl}}_3$. As a consequence, the effective electronic correlation is reduced from Cl to I, leading to the metallic nature of $\alpha \text{−}{\mathrm{RuI}}_3$. Based on the $\mathrm{DFT}+U$ ($U_{\mathrm{eff}}=2$ eV) plus spin-orbital coupling, we obtained a spin-orbit Mott insulating behavior for $\alpha \text{−}{\mathrm{RuCl}}_3$ and, with the same procedure, a metallic behavior for $\alpha \text{−}{\mathrm{RuI}}_3$, in good agreement with experimental results. Furthermore, when introducing large (unrealistic) $U_{\mathrm{eff}}=6$ eV, the spin-orbit Mott gap opens in $\alpha \text{−}{\mathrm{RuI}}_3$ as well, supporting the physical picture we are proposing. Our results provide guidance to experimentalists and theorists working on two-dimensional transition metal tri-iodide layered materials.

Using ab initio density functional theory, we study the electronic and magnetic properties of the van der Waals chain material OsCl₄. In the nonmagnetic state, a strongly anisotropic band structure was observed, in agreement with its anticipated one-dimensional crystal geometry. Based on Wannier functions, we found that the four electrons of the 5d Os atom form a low-spin S = 1 state, with a large crystal field between the $d_{xz/yz}$ and d_xy orbitals, corresponding to a strong Jahn–Teller distortion ($Q_3<\hspace{0.17em}0$ ). As a consequence, the magnetic properties are mainly contributed by the $d_{xz/yz}$ states. Furthermore, when a Mott gap develops after the introduction of the Hubbard U and Hund coupling J, we found that the staggered spin order is the most likely magnetic state, namely, spins arranged as (↑-↓-↑-↓) with π wavevector along the chain. In addition, the energy differences between various spin states are small, suggesting a weak magnetic exchange coupling along the chain. Our results provide guidance to experimentalists and theorists working on quasi-one-dimensional osmium halides chain materials.

The competition between spin-orbit coupling (SOC) $\lambda$ and electron-electron interaction $U$ leads to a plethora of novel states of matter, extensively studied in the context of $t_{2g}^4$ and $t_{2g}^5$ materials, such as ruthenates and iridates. Excitonic magnets—the antiferromagnetic state of bounded electron-hole pairs-are prominent examples of phenomena driven by those competing energy scales. Interestingly, recent theoretical studies predicted that excitonic magnets can be found in the ground state of SOC $t_{2g}^4$ Hubbard models. Here we present a detailed computational study of the magnetic excitations in that excitonic magnet, employing one-dimensional chains (via density matrix renormalization group) and small two-dimensional clusters (via Lanczos). Specifically, first we show that the low-energy spectrum is dominated by a dispersive (acoustic) magnonic mode, with extra features arising from the $\lambda =0$ state in the phase diagram. Second, and more importantly, we found a novel magnetic excitation forming a high-energy optical mode with the highest intensity at wave-vector $q\to 0$. In the excitonic condensation regime at large $U$, we also have found a novel high-energy $\pi$ mode composed solely of orbital excitations. These features do not appear all together in any of the neighboring states in the phase diagram and thus constitute unique fingerprints of the $t_{2g}^4$ excitonic magnet, of importance in the analysis of neutron and resonant inelastic x-ray scattering experiments.