Stability of the interorbital-hopping mechanism for ferromagnetism in multi-orbital Hubbard models

Abstract

The emergence of insulating ferromagnetic phase in iron oxychalcogenide chain system has been recently argued to be originated by interorbital hopping mechanism. However, the practical conditions for the stability of such mechanism still prevents the observation of ferromagnetic in many materials. Here, we study the stability range of such ferromagnetic phase under modifications in the crystal fields and electronic correlation strength, constructing a theoretical phase diagram. We find a rich emergence of phases, including a ferromagnetic Mott insulator, a ferromagnetic orbital-selective Mott phase, together with antiferromagnetic and ferromagnetic metallic states. We characterize the stability of the ferromagnetic regime in both the Mott insulator and the ferromagnetic orbital-selective Mott phase forms. We identify a large stability range in the phase diagram at both intermediate and strong electronic correlations, demonstrating the capability of the interorbital hopping mechanism in stabilizing ferromagnetic insulators. Our results may enable additional design strategies to expand the relatively small family of known ferromagnetic insulators.

Introduction

Transition metal (TM) systems involving multi-orbital correlated electrons continue attracting much attention due to their rich physical properties^1,2,3,4,5,6. In the standard Hubbard Hamiltonian of a multiorbital system, the interplay of the elements of the hopping matrix, the crystal fields Δ splitting orbitals, the Hubbard repulsion U, and the Hund coupling J_H linking all orbitals, often leads to several intriguing electronic phases arising from their competition, such as the molecular-orbital state in dimers⁵, the spin-singlet state^7,8, various forms of orbital ordering^9,10, and the recently much-addressed orbital-selective physics^{11,12,13,14,15,16}. An attractive example of orbital-selective states is the unusual orbital-selective Mott phase (OSMP) (Fig. 1a), involving a mixture of localized and itinerant behavior of the different orbitals when in the intermediate electronic correlation region^{17,18,19,20,21}.

Fig. 1: Orbital-selective Mott phase, the half-full mechanism, and model Hamiltonian.

a Sketch of the local density-of-states for the orbital-selective Mott phase (OSMP) and Mott insulator (MI) phases, in a multi-orbital system with electronic correlations. Here we use two electrons in two orbitals per site, as example. For the OSMP, one orbital forms a Mott state with a gap, while the other orbital remains itinerant and gapless, leading to a globally metallic phase. For the MI state, both orbitals form Mott states, resulting in an insulating phase. b Sketch of the FM superexchange mechanism discussed in ref. ³⁹ induced by the large entanglements between doubly occupied and half-filled orbitals. The half-filled orbitals are coupled by Hund’s coupling J_H. Orbitals are indicated by blue lines. Electrons with spin up or down are indicated by red or gray arrows, respectively. The two-way thin arrows indicate the overlap between inter-site orbitals. The virtual hopping process is highlighted by the red dashed ovals. c The effective Wannier functions (WF) of orbital γ₁ (d_yz) for Fe1 and γ₂ (${d}_{{x}^{2}-{y}^{2}}$) for Fe2 with bridges using the Se1 and Se2 p_x orbitals. The robust overlap between these WFs, which is related to the amplitude of hoppings t₁₂, are indicated by the dashed red ovals. Isosurface is set to be 3 here. Different colors represent different signs of the WF. The Wannier function plot is produced using WANNIER90 code⁴² and VESTA⁶⁵. d The three-orbital Hubbard model on a one-dimensional chain lattice geometry used in our study, including the kinetic component and electronic interaction. ${t}_{\gamma {\gamma }^{{\prime} }}$ represents hopping from orbital γ to orbital ${\gamma }^{{\prime} }$ between nearest neighbor sites i and i + 1. Crystal-field splittings are labeled by Δ_0/1/2. Orbitals are marked by γ_0/1/2. Hubbard repulsion U, ${U}^{{\prime} }$ between electrons at different orbitals, Hund’s coupling J_H, and the inter-orbital electron-pair hopping terms are also list, more details can be found in Model Hamiltonian part. Inset: The non-interacting band structure along the chain direction using the original parameters from ref. ³⁹.

It is precisely the intermediate coupling regime that harbors the most surprises because in this region hand-waving lines of reasoning are often not trustworthy since many couplings are similar in magnitude and reliable predictions are difficult, unless involving robust computational methodologies. This is the true regime of “complexity” in correlated electrons, where the expression complexity is used as denoting the emergence of unexpected properties from seemingly simple local interactions. This intermediate region will be the focus of this work.

Before moving into the focus of present contribution, we wish to briefly remind the reader of the practical importance of enlarging the family of ferromagnetic (FM) or ferrimagnetic insulators. Materials with these properties have important applications in the field of spintronics, where their low magnon damping and ability to exchange couple magnetism into neighboring materials are particularly useful^22,23,24. Compared to antiferromagnetic (AFM) insulators, FM insulators are relatively rare, particularly among oxides^25,26,27. Ferrimagnetic ferrite garnets like Y₃Fe5O₁₂ (yttrium iron garnet, or YIG) are commonly used in devices; however, some limitations of these materials have been noted and development of alternatives is an active and important area of research²⁸. Perhaps the earliest recognized example of a FM insulator was CrBr₃²⁹. The combination of FM order and cleavability has made this compound, its analogue CrI₃, and the related compounds CrSiTe₃ and CrGeTe₃ important materials in the field of functional van der Waals heterostructures^24,30,31. Although ferromagnetism is usually associated with TMs, several rare-earth-based FM insulators are also known, including the rock-salt structure compounds EuO, EuS, and mononitrides of several rare earth elements^32,33. Hexagonal ferrites and spinels such as CoFe₂O₄^34,35 are also of interest²⁸. FM semiconducting behavior has also been reported in spinel chalcogenides like CdCr₂S₄ and CrCr₂Se₄^27,36.

Consider now the primary specific goal of this publication. Most TM insulating materials are AFM while few display FM order. This can be understood from the simplicity and robustness of Anderson’s superexchange AFM theory which is based on second-order perturbation theory in the hopping amplitudes^37,38. However, recently a “half-full” mechanism involving large entanglements between doubly occupied and half-filled orbitals was proposed³⁹ to understand the puzzling origin of FM order along the chain direction in Ce₂O₂FeSe₂^40,41, as shown in Fig. 1b. Note that we employ the word “entanglement” between orbitals with different electronic population as indicating that in a quantum description these orbitals cannot be considered as independent of one another, but they are intrinsically coupled and this interplay of two orbitals with different electronic populations is at the heart of the interorbital hopping mechanism proposed.

Specifically, based on second-order perturbation theory in the hopping amplitudes, the total gain in energy of the FM configuration due to the robust hopping t₁₂ between half-occupied and fully occupied orbitals was found to be:

$${{\Delta }}{E}_{{{{{{{{\rm{FM}}}}}}}}}=-\frac{| {t}_{12}{| }^{2}}{U-3{J}_{{{{{{{{\rm{H}}}}}}}}}+\Delta }.$$

(1)

where Δ = Δ₁ − Δ₂ is the crystal-field splitting between the half-filled and double-occupied orbitals. On the other hand, in the AFM state the total energy gained from t₁₂ is

$${{\Delta }}{E}_{{{{{{{{\rm{AFM}}}}}}}}}=-\frac{| {t}_{12}{| }^{2}}{U-{J}_{{{{{{{{\rm{H}}}}}}}}}+{{\Delta }}}.$$

(2)

Hence, the interorbital electronic hopping t₁₂ by itself favors a FM interaction, basically driven by the Hund coupling. However, the prevailing intra-orbital hoppings t₁₁ or t₂₂ favor a superexchange AFM state with gains in energy such as ΔE ~ -$\frac{| {t}_{11}{| }^{2}}{U+{J}_{{{{{{{{\rm{H}}}}}}}}}}$. More details of analysis of second-order perturbation theory can be found in Supplementary Note II.

Then, the dominant magnetic order of the material under consideration is decided by the competition of these different channels. Our previous work showed that in some cases, the hopping t₁₂ can be large enough to stabilize a FM insulating phase, without the need to resort to the more frequently mentioned mechanism of double exchange, which produces robust FM but with metallic character, and the nearly 90^∘ bonds, typically associated with low critical temperatures. It is also worth remarking that while our previous theoretical work, as well as the present contribution, are in one dimension, the simple second-order perturbative foundations of the FM phase are robust and valid in higher dimensions as well.

To intuitively understand the effective interorbital hopping t₁₂ between the Fe1 d_yz and Fe2 ${d}_{{x}^{2}-{y}^{2}}$ orbitals, it is crucial to include Se’s p orbitals in the Wannier90 calculations⁴². By comparing all channels contributing to t₁₂, we found out that the dominant channel occurs when the Se’s p_x orbital acts as a “bridge” between the 3d orbitals. To have an intuitive visual view of t₁₂, the WFs related to this p_x channel are shown in Fig. 1c. In this sketch, the bending of the Fe-Se-Fe bond is important to achieve a nonzero t₁₂ matrix element.

While this mechanism has been observed for fixed hopping and crystal fields³⁹, some questions naturally arise on its stability and transferability to other materials. How does the FM state induced by this half-full mechanism evolve by varying the crystal-field values as well as the strength of the electronic correlations? At intermediate couplings, could the OSMP with FM order be instead stable in this half-full system with crystal field effects? What other magnetic or electronic phases can emerge by considering the competition of those parameters in an extended phase diagram? In other words, are the previous results³⁹ an anomaly or truly indicative of a general mechanism?

To address these broad goals, we investigated the crystal field and electronic correlations effects on the half-full FM ground state previously reported, by using the density matrix renormalization group (DMRG) technique on the multi-orbital Hubbard model. Fixing the Hund coupling to J_H/U = 0.25, realistic for materials of the iron family⁴, the magnetic and electronic phase diagram was theoretically constructed varying Δ and U/W. One of our main results is that the FM phase is stable in a large portion of the phase diagram. Namely, the previously reported FM phase due to a robust interhopping amplitude t₁₂ is here shown not to be fragile, as often spin liquid states tend to be, but representative of broad tendencies that until now were not considered by the community of experts. Specifically, we found both the recently discovered FM Mott insulator (MI) and the FM OSMP phases are stable at intermediate and strong electronic correlation, respectively. Our results potentially open a vast avenue of research and the possibility for the family of FM insulators to be considerably enlarged. We propose that via ab-initio techniques, a systematic exploration of materials with robust interorbital hopping could provide the first steps toward additional FM insulators in the near future.

In addition, several other magnetic electronic phases are also revealed in our study arising from the competition of hopping, crystal field splitting, and electronic correlations, involving an AFM2 metal (i.e. an AFM state with blocks of size 2), a FM metallic state, and staggered canonical AFM1 MI phases. This exemplifies the complexity that emerges from multiorbital models when they are analyzed with reliable computational techniques, particularly in the challenging intermediate coupling regime.

Results

Model Hamiltonian

In this work, as an example of our broad ideas, we employ a canonical three-orbital Hubbard model defined on a one-dimensional (1D) chain lattice, including the kinetic energy and interaction terms written as H = H_k + H_int. The tight-binding kinetic component is

$${H}_{{{{{{{{\rm{k}}}}}}}}}=\mathop{\sum}\limits_{\begin{array}{c}i\sigma \\ \overrightarrow{{{{{{{{\boldsymbol{\alpha }}}}}}}}}\gamma {\gamma }^{{\prime} }\end{array}}{t}_{\gamma {\gamma }^{{\prime} }}^{\overrightarrow{{{{{{{{\boldsymbol{\alpha }}}}}}}}}}\left({c}_{i\sigma \gamma }^{{{{\dagger}}} }{c}_{i+\overrightarrow{\alpha }\sigma {\gamma }^{{\prime} }}^{}+H.c.\right)+\mathop{\sum}\limits_{i\gamma \sigma }{\Delta }_{\gamma }{n}_{i\gamma \sigma },$$

(3)

where the first term represents the hopping of an electron from orbital γ at site i to orbital ${\gamma }^{{\prime} }$ at the nearest neighbor site $i+\overrightarrow{{{{{{{{\boldsymbol{\alpha }}}}}}}}}$. ${c}_{i\sigma \gamma }^{{{{\dagger}}} }$(${c}_{i\sigma \gamma }^{}$) is the standard creation (annihilation) operator, γ and ${\gamma }^{{\prime} }$ represent the different orbitals, and σ is the z-axis spin projection. Δ_γ represent the crystal-field splitting of each orbital γ.

The electronic interaction portion of the Hamiltonian includes the standard intraorbital Hubbard repulsion U, the electronic repulsion ${U}^{{\prime} }$ between electrons at different orbitals, the Hund’s coupling J_H, and the on-site inter-orbital electron-pair hopping terms. Formally, it is given by:

$$\begin{array}{l}{H}_{{{{{{{{\rm{int}}}}}}}}}=U\mathop{\sum}\limits_{i\gamma }{n}_{i\uparrow \gamma }{n}_{i\downarrow \gamma }+\left({U}^{{\prime} }-\frac{{J}_{{{{{{{{\rm{H}}}}}}}}}}{2}\right)\mathop{\sum}\limits_{\begin{array}{c}i\\ \gamma < {\gamma }^{{\prime} }\end{array}}{n}_{i\gamma }{n}_{i{\gamma }^{{\prime} }}\\ -2{J}_{{{{{{{{\rm{H}}}}}}}}}\mathop{\sum}\limits_{\begin{array}{c}i\\ \gamma < {\gamma }^{{\prime} }\end{array}}{{{{{{{{\bf{S}}}}}}}}}_{i,\gamma }\cdot {{{{{{{{\bf{S}}}}}}}}}_{i,{\gamma }^{{\prime} }}+{J}_{{{{{{{{\rm{H}}}}}}}}}\mathop{\sum}\limits_{\begin{array}{c}i\\ \gamma < {\gamma }^{{\prime} }\end{array}}\left({P}_{i\gamma }^{{{{\dagger}}} }{P}_{i{\gamma }^{{\prime} }}+H.c.\right),\end{array}$$

(4)

where the standard relation ${U}^{{\prime} }=U-2{J}_{{{{{{{{\rm{H}}}}}}}}}$ is assumed and P_iγ = c_i↓γc_i↑γ.

Specifically, here we consider a three-orbital Hubbard model with four electrons per site (see the sketch in Fig. 1d), where the crystal-field splitting and hopping matrix are adopted from the real iron chain Ce₂O₂FeSe₂ system as a concrete example³⁹. To study the crystal-field splitting effects between half-filled and fully occupied orbitals we fixed the values of Δ₀ and Δ₁ as in³⁹ but vary Δ₂, as well as U/W. In the “DMRG” portion of the methods section, the reader can find the specific 3 × 3 hopping matrix and values of Δ₀ and Δ₁ employed. Note that all the DMRG calculations are performed at temperature T = 0.

Phase diagram with crystal field effects

First, using DMRG we constructed the general phase diagram varying the electronic correlations U/W and crystal field splitting Δ₂, as shown in Fig. 2. Although the original model parameters are derived from the five d orbitals of the real iron chain materials Ce₂O₂FeSe₂, the main conclusions are not limited to this specific material or to three-orbital models, but expand to any other system with large inter-orbital hopping between half-filling and fully occupied orbitals. Our primary aim is to test the interorbital hopping driven FM mechanism beyond Ce₂O₂FeSe₂. This phase diagram is obtained from measurements of the spin-spin correlation S(r), the site-average occupancy n_γ, and charge fluctuations δn_γ, all at J_H/U = 0.25 as already explained. The results are rich and include six different phases: (1) paramagnetic (PM) metallic (M) phase, (2) AFM2 M state, (3) FM OSMP, (4) AFM1 MI phase, (5) FM M state, and (6) FM MI phase. Note that in our case all the transitions between magnetic states are discontinuous in the order parameters, thus first order (with the caveat that we are in 1D thus the magnetic order is only quasi long-range order). The only exception is between a PM and ordered state: these tend to be smooth transitions.

Fig. 2: DMRG phase diagram.

Phase diagram of the three-orbital Hubbard model varying U/W and crystal-field splitting Δ₂, using DMRG and an L = 16 chain system with open boundary conditions. Note that the three-orbital character of the problem renders this system equivalent to a challenging highly entangled 3 × 16 ladder lattice, with hoppings at fairly long distances such as between top and bottom legs in this effective 3 × 16 lattice. We use the prototypical value J_H/U = 1/4. Different electronic and magnetic phases are indicated by solid regions and labels, including paramagnetic metal (PM M, in pink), ferromagnetic orbital-selective Mott phase (FM OSMP, in light blue), ferromagnetic Mott insulator (FM MI, in yellow), antiferromagnetic1 Mott insulator (AFM1 MI, in purple), ferromagnetic metal (FM M, in maroon) and antiferromagnetic2 metal (AFM2 M, in red). Due to strong competition between opposite tendencies, some mixed phases, marked by purple stars, are found near the boundaries, especially around Δ₂ ~ −0.25 eV and U/W ~ 8 (white region). The antiferromagnetic4 (AFM4) phase, marked by a purple circle, is also displayed in this region. Note that the boundaries should be considered only as crude approximations. We simply cannot have a very dense grid of points due to the high cost of DMRG. However, the existence of the six regions shown was clearly established by robust DMRG evidence near their centers, even if the boundaries are only crude estimations. All calculated data points are marked by gray stars. The two vertical dashed lines at U/W = 1.6 and 8.0 correspond to the results of Figs. 3 and 5.

At small electronic correlation (U/W ≲ 0.7), the system displays PM behavior with three itinerant orbitals, where the spin correlation S(r) involving two spin operators separated by a distance r decays rapidly with distance, as shown in Supplementary Note III. In this region, the hopping term plays the leading role, leading to the metallic behavior. At intermediate Hubbard coupling strength, the FM OSMP state—involving coexisting localized and itinerant electrons—is found to be stable in a large range of Δ₂. Furthermore, two magnetic and electronic phases are also obtained in this intermediate region by changing Δ₂. They are the AFM2 M and AFM1 MI phases, arising from the competition of Δ₂ and U/W. At large U/W, MI states with localized charges (n number of electrons either 1 or 2) dominate over a broad range of the crystal-field splitting Δ₂. Also an FM M phase is found in a small region due to the competition of Δ₂, U/W, and hopping amplitudes.

In this rich DMRG phase diagram, the FM phase is dominant in a large region of Δ₂ with the conduction type determined by the strength of the electronic correlations. At intermediate U/W, the system has simultaneously metallic and insulating orbitals, leading to the FM OSMP state. This OSMP is induced by the competition between hopping values of different orbitals and electronic correlations (Hubbard U, Hund coupling J_H): the electrons in the orbital with smaller hopping are localized at intermediate correlations U/W while the electrons in orbitals with larger hoppings remain itinerant i.e. metallic with non-integer filling. If we further increase the electronic correlations, the FM OSMP metallic state transitions to the FM MI phase where all three orbitals are fully Mott-localized at strong U/W. More details about Mott transition at Δ₂ ~ −0.8 eV is discussed in Supplementary Note IV. By decreasing the crystal-field splitting Δ₂ towards zero, the FM state changes to different AFM phases because in this regime the canonical AFM superexchange mechanism dominates. Note that there is a small OSMP region at large U/W and Δ₂ ~ −0.3 eV, where orbital γ = 1 is the localized one. This is slightly different from the large OSMP region at intermediate correlations, where orbital γ = 0 is the localized one. The reason is that, in the smaller OSMP region, Δ₂ is close to the crystal splitting value of orbital γ = 0, rendering stronger competition between orbital γ = 0 and γ = 2, while γ = 1 is easier to be localized.

FM-OSMP vs crystal field splitting

At intermediate Hubbard coupling strengths, the FM OSMP is found to be stable in a large region when decreasing Δ₂ into negative values (Fig. 2). Let us now focus on the crystal-field splitting effects at intermediate electronic correlation U/W = 1.6 in the phase diagram.

In the range −1.1 eV ≲ Δ₂ ≲ −0.2 eV and at U/W = 1.6, the spin-spin correlation S(r)=〈S_i ⋅ S_j〉 vs r indicates FM order along the chain geometry (see the results at Δ₂ = −0.8 and −0.4 eV in Fig. 3a). The distance is defined as $r=\left\vert i-j\right\vert$ with i and j site indexes. As shown in Fig. 3b, a sharp peak is also observed at q = 0 in the spin structure factor S(q), clearly indicating FM order. In addition, the occupation of orbital γ = 0 is locked at the integer 1 in this region of Δ₂, leading to the Mott-localized characteristic in this orbital, while the γ = 1 and γ = 2 orbitals have non-integer electronic densities, indicating metallic behavior, as displayed in Fig. 3c. To better understand the characteristics of the metallic vs. insulating behavior in different orbitals, we also studied the charge fluctuations δn_γ for different orbitals, as displayed in Fig. 3d. In the region (−1.1 eV ≲ Δ₂ ≲ −0.2 eV), the γ = 1 and γ = 2 orbitals have some charge fluctuations because of the itinerant nature of their electrons. However, the charge fluctuation of γ = 0 is basically zero due to its localized orbital characteristics. Furthermore, ${\langle {S}^{2}\rangle }_{0}$ saturates at 3/4, corresponding to a half-filled orbital with spin 1/2 at each site, while ${\langle {S}^{2}\rangle }_{1}$ and ${\langle {S}^{2}\rangle }_{2}$ are less than 3/4, as shown in Fig. 3e. Thus, the system displays the orbital selective Mott characteristics with one localized orbital (γ = 0) and two itinerant orbitals (γ = 1 and γ = 2) and with global FM order.

Fig. 3: Observables at intermediate correlation.

a Spin-spin correlation S(r) = 〈S_i ⋅ S_j〉 (with $r=\left\vert i-j\right\vert$ in real space) and b the spin structure factor S(q), at different values of Δ₂, all at J_H/U = 0.25 and U/W = 1.6. c Orbital-resolved occupation number n_γ. d Charge fluctuations $\delta {n}_{\gamma }=\frac{1}{L}{\sum }_{i}(\langle {n}_{i,\gamma }^{2}\rangle -{\langle {n}_{i,\gamma }\rangle }^{2})$. e Averaged value of the total spin-squared ${\langle {S}^{2}\rangle }_{\gamma }$ vs. Δ₂, at J_H/U = 0.25 and U/W = 1.6. f The spin structure S(q) at 0, π/2, and π as a function of Δ₂. (c-f) Different electronic and magnetic phases are marked by solid regions and labels, including antiferromagnetic1 Mott insulator (AFM1 MI, in purple), ferromagnetic orbital-selective Mott phase (FM OSMP, in light blue), and antiferromagnetic2 metal (AFM2 M, in red). For all these results, the chain length is L = 16 and DMRG was used.

Decreasing further Δ₂ (≲ −1.1 eV), the system displays the canonical staggered AFM phase with the ↑-↓-↑-↓ configuration unveiled by S(r), while the spin structure factor S(q) shows also a sharp peak at q = π (see results for Δ₂ = −1.2 eV in Fig. 3b). Moreover, all three orbitals are integer occupied (n₀ = 1, n₁ = 1, and n₂ = 2) in this state. Furthermore, ${\langle {S}^{2}\rangle }_{0}$ and ${\langle {S}^{2}\rangle }_{1}$ saturates at 3/4, corresponding to the half-filled orbital, while ${\langle {S}^{2}\rangle }_{2}$ is zero, indicating a double-occupied orbital, as shown in Fig. 3e. Hence, the system is in an AFM1 MI state in this region of Δ₂. Because Δ = Δ₁ − Δ₂ is larger than the Hund coupling J_H, the extra electrons prefer to stay in the γ = 2 orbital even with a large interorbital hopping t₁₂. In this case, the system can be effectively regarded as a two-half-occupied orbital system, where the intraorbital hoppings lead to the Heisenberg AFM coupling, resulting in the expected AFM1 phase characteristic of one-orbital Hubbard U models, where the Anderson superexchange prevails.

As an extra clarification, the small AFM1 MI region at around Δ₂ ≲ −1.1 eV and U/W ~ 1, alternatively, may be a Slater insulator (purple with red dashed lines). Clearly, between the weak coupling limit that harbors the Slater insulator and the strong coupling limit where the Mott insulator is located, there is a complex crossover region with features common to both³. The most strict way to distinguish Mott vs Slater is to study the insulating vs metal characteristics, respectively, above the magnetic ordering temperatures. But performing DMRG at a finite temperature for a multiorbital model is beyond our computational resources. Moreover, in 1D a sharp magnetic critical temperature is not expected. In the multi-orbital model, this open question is worthy of more detailed investigations in the future.

Increasing Δ₂, the AFM tendency also increases, leading to competition with the FM tendency, resulting in more unusual phases. When Δ₂ > −0.2 eV, the spin-spin correlation S(r) at Δ₂ = 0.0 eV shows clearly the formation of antiferromagnetically coupled FM spin clusters with a ↑-↑-↓-↓ AFM2 pattern, as shown in Fig. 3a. Furthermore, the spin structure factor S(q) also displays a sharp peak at q = π/2, corresponding to the AFM2 phase, as shown in Fig. 3b. Moreover, the four electrons per site are in non-integer orbitals with large charge fluctuations, indicating metallic behavior (Figs. 3c and d). Moreover, the operator ${\langle {S}^{2}\rangle }_{\gamma }$ for different orbitals are all less than 3/4, in agreement with the anticipated metallic nature of this state, as shown in Fig. 3e. Finally, for the benefit of the readers, note that the intuitive origin of the AFM2 “block” patterns has been clarified in previous literature and will not be repeated here. Essentially, in extended phase diagrams varying the electronic density, the block states arise as a compromise between fully FM and staggered AFM1 states, i.e., between FM vs AFM tendencies. Interested readers are referred to refs. ^43,44. For the readers to better understand those phase transitions, we also show the spin structure S(q) at q = 0, π/2, and π, corresponding to the FM, AFM2, and AFM1 phases, respectively, as a function of Δ₂, as displayed in Fig. 3f.

The projected density of states (PDOS) ρ _γ at U/W = 1.6

To better understand the many different phases at intermediate electronic correlation, we calculated the orbital-resolved projected density of states ρ_γ(ω) vs frequency ω by using the dynamical DMRG, where the dynamical correlation vectors were obtained using the Krylov-space approach^45,46. A broadening parameter η = 0.1 was chosen in our DMRG calculations, as in previous cases. The chemical potential is obtained from μ = (E_N+1 − E_N−1)/2, where E_N is the ground state energy of the N-particle system. ρ_γ(ω) is calculated from the portions of the spectra below and above μ, respectively, using:

$${\rho }_{\gamma }(q,\omega < \mu )=\frac{1}{\pi }Im\left[\left\langle {\psi }_{0}\left\vert {c}_{i,\gamma }^{{{{\dagger}}} }\frac{1}{\omega +H-{E}_{g}+i\eta }{c}_{i,\gamma }\right\vert {\psi }_{0}\right\rangle \right],$$

(5)

$${\rho }_{\gamma }(q,\omega > \mu )=\frac{-1}{\pi }Im\left[\left\langle {\psi }_{0}\left\vert {c}_{i,\gamma }\frac{1}{\omega -H+{E}_{g}+i\eta }{c}_{i,\gamma }^{{{{\dagger}}} }\right\vert {\psi }_{0}\right\rangle \right],$$

(6)

$${\rho }_{\gamma }(\omega )={\rho }_{\gamma }(q,\omega < \mu )+{\rho }_{\gamma }(q,\omega > \mu ),$$

(7)

with ψ₀ as the ground state.

As displayed in Fig. 4, we calculated ρ_γ(ω) for several Δ₂’s, corresponding to different phases, at J_H/U = 0.25 and U/W = 1.6. At Δ₂ = −1.2 eV, the γ = 0 and γ = 1 orbitals display a localized behavior with a Mott gap, while γ = 2 shows fully occupied characteristics below the chemical potential μ, Fig. 4a. Hence, this is a MI state, in agreement with previous discussions based on the site-average occupancy and charge fluctuations for this region.

Fig. 4: Orbital-resolved density of states.

Density of states (DOS) ρ_γ(ω) of different orbitals for different values of Δ₂, at U/W = 1.6 and J_H/U = 0.25. a Antiferromagnetic1 Mott insulator (AFM1 MI) state at Δ₂ = −1.2 eV. b Ferromagnetic orbital-selective Mott phase (FM OSMP) phase at Δ₂ = −0.4 eV. c Antiferromagnetic2 metal (AFM2 M) state at Δ₂ = 0.0 eV. Δω = 0.1 eV and broadening parameter η = 0.1 eV were chosen in our DMRG calculations. The black dash line represents chemical potential μ, which is calculated by using μ = (E_N+1 − E_N−1)/2. For the inset in (b), smaller parameters Δω = 0.02 eV and η = 0.05 eV were used to calculate the DOS around the Fermi energy for comparison. The results indicate that the DOS is independent of the broadening parameter η. The purple dashed line represents the chemical potential μ, which is calculated by integrating the density of states for all ω’s to count the number of electrons.

At Δ₂ = −0.4 eV, the γ = 0 orbital displays localized behavior with a Mott gap. Meanwhile, the γ = 1 and γ = 2 orbitals have electronic states crossing the Fermi level, leading to metallic behavior, see Fig. 4b. Hence, the coexistence of localized and itinerant carriers supports the OSMP picture. By considering the spin-spin correlations S(r) and spin structure factor S(q), this state is an FM OSMP. In addition, the OSMP character is clear in our present work, supported by the results of charge occupation number and spin squared ${\langle {S}^{2}\rangle }_{\gamma }$: a half-filled γ = 0 orbital does not have any charge or magnetic fluctuations while the γ = 1 and γ = 2 orbitals have non-integer numbers with sizeable charge and magnetic fluctuations (Fig. 3c–e). In fact, the hole-doped manganites are also an FM OSMP system, with t_2g orbitals localized and e_g metallic⁴⁷. In doped manganites, FM is induced by the double exchange mechanism⁴⁸, where off-diagonal hopping between the half and empty e_g orbitals plays an important role, similar to the half-full hopping discussed here.

Because of the small weight of the DOS for the orbitals 1 and 2 when crossing the Fermi level (Fig. 4b), an insulating tendency can not be fully excluded. Then, we also calculated the DOS around the Fermi energy with smaller parameters Δω = 0.02 eV and η = 0.05 eV to confirm our results. In this case, no obvious difference with the previous results was found, as shown in the inset in Fig. 4b, indicating that the value of the broadening is not a problem. In addition, the chemical potential is shown in the inset of Fig. 4b. μ is obtained by integrating the density of states in ω from the far left to a value that matches the total number of electrons expected. Note that the calculated chemical potential μ has an error bar of about ±0.05 eV due to the grid used in the horizontal axis. At the Fermi level, the weight of γ = 0 orbital is virtually zero, while the γ = 1 and γ = 2 orbitals have small but non-zero weights, suggesting a clear difference between the three orbitals. In principle, if we have a dense enough ω, we could obtain more accurate μ by counting better the total occupied number of electrons. However, due to the very high cost of DMRG, we cannot have a very dense grid of ω to obtain ρ(ω) and μ with higher accuracy. Nevertheless, within these caveats, the results indicate that the DOS is independent of the broadening parameter η.

Furthermore, we also used the formula as defined in ref. ⁴⁹ to measure the gap as E(N + 1) + E(N − 1) − 2E(N) = −0.00098 eV which is basically zero, indicating that this is a metallic phase. We alert the readers that previous work by some members of this collaboration for another model observed orbital order and an insulating behavior with a small gap when the magnetic order is FM in the OSMP regime⁴⁹. Note that we can not exclude such a possibility that the system is in a non-orbital-selective insulating state with a very small gap, beyond our numerical accuracy. Here, we can only claim with certainty that in the present work with the grid of ω points used and with our measurement of orbital order correlations, and our measurement of the gap, we do not observe indications of orbital order and of a finite gap at the Fermi level. On the other hand, considering the small values of the DOS for orbitals 1 and 2, it is likely that nearby in parameter space an orbital-ordered insulating state could be stabilized. Searching for this phase is left for future efforts. Figure 4c clearly shows that all three orbitals are metallic with some electronic bands crossing μ, indicating itinerant electronic behavior, leading to an orbitally metallic state. This AFM2 metal was also predicted in the iron selenide chain based on the same model, where the crystal-field splitting Δ between half-filled and fully occupied orbitals is also small⁵⁰.

Strong U/W region

By increasing further the electronic correlation strength, the OSMP itinerant orbitals would become fully Mott-localized by the Hubbard strength U, resulting in a MI phase. To better understand the crystal-field effects on the FM MI state, we will focus on the discussion of the main results for different values of Δ₂ at strong electronic correlation U/W = 8 in the phase diagram.

The FM MI state is found to dominate in a large region of Δ₂, as shown in Fig. 5. When Δ₂ ≲ −0.5 eV, the γ = 0 and γ = 1 are half-filled (n₀ = 1 and n₁ = 1) and γ = 2 is a fully occupied orbital (n₂ = 2) without any charge fluctuation, as shown in Fig. 5a, b. Meanwhile, ${\langle {S}^{2}\rangle }_{0}$ and ${\langle {S}^{2}\rangle }_{1}$ are fixed at 3/4, while ${\langle {S}^{2}\rangle }_{2}$ is zero, indicating a strong Mott-localized behavior, as shown in Fig. 5c. Furthermore, the critical Hubbard U for the metal-insulator transition of the FM phase decreases as the crystal-field splitting Δ = Δ₁ − Δ₂ (between doubly occupied and half-filled orbitals) increases. In this case, the FM MI state becomes more stable with larger crystal-field splitting Δ.

Fig. 5: Observables at strong correlation.

a Orbital-resolved occupation number n_γ, b charge fluctuations $\delta {n}_{\gamma }=\frac{1}{L}{\sum }_{i}(\langle {n}_{i,\gamma }^{2}\rangle -{\langle {n}_{i,\gamma }\rangle }^{2})$, and c averaged value of the total spin-squared 〈S²〉, as a function of Δ₂, at J_H/U = 0.25 and U/W = 8. d The spin structure S(q) at q = 0, π/2, and π as a function of Δ₂. a–d Different electronic and magnetic phases are indicated by solid regions and labels, including ferromagnetic Mott insulator (FM MI, in yellow), ferromagnetic metal (FM M, in maroon), antiferromagnetic2 metal (AFM2 M, in pink), and antiferromagnetic1 Mott insulator (AFM1 MI, in purple). Here, the length of the chain is L = 16 and DMRG was used.

By increasing the crystal field Δ₂, the system has non-integer electronic density n_γ in the three orbitals, see Fig. 5a, while the charge fluctuations δn_γ are large, leading to a metallic phase. In particular, when Δ₂ is in the region, closer to Δ₀ = −0.277 eV and Δ₁ = −0.203 eV, the competition between orbitals is the strongest. When Δ₂ ≳ 0.0 eV, the three orbitals begin to localize with integer electronic density n_γ without charge fluctuations, leading to a MI state, as shown in Fig. 5b, c. In addition, the γ = 0 orbital becomes doubly occupied (n₀ = 2), while the γ = 1 and γ = 2 orbitals are half-filled (n₁ = 1 and n₂ = 1). Furthermore, we also present the spin structure S(q) at q = 0, π/2, and π, corresponding to the FM, AFM2, and AFM1 phases, respectively, for different values of Δ₂ to help the readers better understand the phase transitions, as shown in Fig. 5d. More details about spin-spin correlation and spin structure factor is shown in Supplementary Note VI.

Discussion

In multiorbital systems with more than half-filled orbitals, it was recently shown that the interorbital hopping between half-filled and fully occupied orbitals can lead to an FM insulating phase³⁹. To understand how this interorbital hopping driven FM mechanism is affected by the crystal-field splitting Δ = Δ₁ − Δ₂ between half-filled and fully occupied orbitals, as well as by the strength of the Hubbard repulsion U/W, here we comprehensively studied the Δ₂ vs U/W phase diagram of a three-orbital lattice model defined on a chain, by using DMRG many-body techniques. By modifying the value of Δ₂ over a broad range, the FM phase was found to be quite stable in the phase diagram, showing that the results of our previous study are not fragile, but indicative of robust tendencies, which also apply in higher dimensions. The FM state was found to display both FM OSMP or FM MI behavior, at intermediate or strong electronic correlations, respectively. In addition, several additional magnetic electronic phases were also revealed in our phase diagram arising from the competition of hoppings, crystal fields, and electronic correlations, involving AFM2 M, FM M, and AFM1 MI states.

Our results indicate that the FM order induced by this half-filled mechanism should be robust. This FM mechanism may explain the FM order in some other materials besides the one we studied before, such as in iron chains with d⁶ configuration^51,52,53, one-dimensional S = 1 Ni-based chains⁵⁴, and Fe₃GeTe₂⁵⁵, where their unique lattice geometry provides the possibility of strong overlap between half-filled and fully occupied orbitals, thus creating a robust value for the interorbital coupling. On the other hand, it must be kept in mind that the intra-orbital hoppings (t₁₁ and t₂₂) lead to AFM tendencies, thus a competition of tendencies will produce the final outcome. What other phases can be obtained by increasing the values of those inter-orbital hoppings in the large intra-orbital hopping case, as well as for different electronic correlations remains to be investigated (especially the evolution of the FM OSMP state).

Our study not only focuses on insulators but also on OSMP metals. Because in our case the hopping t₁₁ (between γ = 1 orbitals) is much smaller than others, the γ = 1 orbitals could easily be localized while the other two orbitals still remain metallic in the intermediate correlated region, leading to the FM OSMP. Recently, a FM OSMP was reported in the Fe₃GeTe₂ system by neutron experiments⁵⁶. A more detailed study for the Fe₃GeTe₂ system would be interesting to perform. Furthermore, in iron-based superconductors, the OSMP is believed by some groups to play a key role to understand superconductivity^13,57,58. Hence, the next step is naturally to try to find additional real materials with the FM OSMP state to study whether superconductivity can be found in that state, complementing our search for FM insulators based on the interorbital hopping mechanism here discussed.

All these future directions of research require systematic work based on density functional theory (to calculate hopping and crystal fields and to identify the relevant orbitals) plus model calculations for correlations. Our study indicates the half-full mechanism could lead to robust FM order under effects of crystal-field splitting. Furthermore, AFM2 M and AFM1 MI phases were also found in our DMRG study at intermediate or strong Hubbard strengths, where those phases were also found or predicted in some other iron chain systems with the same electronic density. Our work is a natural starting point point for a variety of future studies to realize, both in theory and experiment, the important role of inter-hopping between half-filled and full occupied orbitals. This area of research has been barely touched until now.

Recently, the stability of the OSMP state with respect to the addition of interorbital hoppings was discussed by using the single-site dynamical mean-field theory but without including magnetism at zero and finite temperature⁵⁹. The next step would be interesting to study this matter related to finite temperature by including magnetism in more detail in future DMRG investigations.

In order to find new strong FM insulating states experimentalists should focus on materials with large orbital entanglement and strong crystal-field splitting between half-filled and fully occupied orbitals. The FM OSMP also can be obtained in a range of appropriate crystal field splittings, as in our calculations. Furthermore, crystal field splitting sensitively depends on chemical bonds and crystal structures, which could be in practice tuned by strain or pressure^60,61.

Methods

DMRG method

The model Hamiltonian discussed here was studied by using the DMRG method^62,63, where the DMRG++ computer package was employed⁶⁴. In our DMRG calculations, we used an L = 16 sites cluster chain geometry with open-boundary conditions. In addition, the electronic filling n = 4 in the three orbitals was considered. Furthermore, at least 1200 states were kept during our DMRG calculations and up to 21 sweeps were performed during the finite-size algorithm evolution. Truncation error remains below 10⁻⁶ for all of our results. An example input file and additional details can be found in Supplementary Note I. Note that all the DMRG calculations are performed at zero temperature T = 0. Size effect is discussed in Supplementary Note V.

The kinetic part of the Hamiltonian

The hopping matrix for the three-orbital chain system is defined³⁹ in orbital space as follows:

$$\begin{array}{l}{t}_{\gamma {\gamma }^{{\prime} }}=\left[\begin{array}{ccc}0.187&-0.054&0.020\\ 0.054&0.351&-0.349\\ 0.020&0.349&-0.433\end{array}\right].\\ \end{array}$$

(8)

The crystal field splitting of the two orbitals are fixed as Δ₀ = −0.277 and Δ₁ = −0.203 eV, while Δ₂ was allowed to vary in our DMRG calculations. The total kinetic energy bandwidth W is 2.085 eV. All parameters mentioned above, involving the hopping matrix and crystal fields, were extracted from our previous work³⁹.

Observables

To obtain the phase diagram of the three-orbital 1D Hubbard model varying U/W and Δ₂, several observables were measured using the DMRG many-body technique.

The spin-spin correlation is defined as:

$${S}_{i,j}=\langle {{{{{{{{\bf{S}}}}}}}}}_{i}\cdot {{{{{{{{\bf{S}}}}}}}}}_{j}\rangle .$$

(9)

where ${{{{{{{{\bf{S}}}}}}}}}_{i}=\mathop{\sum}\limits_{\begin{array}{c}\gamma \end{array}}{{{{{{{{\bf{S}}}}}}}}}_{i,\gamma }$.

The corresponding structure factor for spin is:

$$S(q)=\frac{1}{L}\mathop{\sum}\limits_{\begin{array}{c}j,k\end{array}}{e}^{-iq(j-k)}\langle {{{{{{{{\bf{S}}}}}}}}}_{k}\cdot {{{{{{{{\bf{S}}}}}}}}}_{j}\rangle ,$$

(10)

The site-average occupancy of orbitals is:

$${n}_{\gamma }=\frac{1}{L}\langle {n}_{i\gamma \sigma }\rangle .$$

(11)

The orbital-resolved charge fluctuation is defined as:

$$\delta {n}_{\gamma }=\frac{1}{L}\mathop{\sum}\limits_{i}(\langle {n}_{\gamma ,i}^{2}\rangle -{\langle {n}_{\gamma ,i}\rangle }^{2}).$$

(12)

The mean value of the squared spin for each orbital is defined as:

$${\langle {{{{{{{{\bf{S}}}}}}}}}^{2}\rangle }_{\gamma }=\frac{1}{L}\mathop{\sum}\limits_{\begin{array}{c}i\end{array}}\langle {{{{{{{{\bf{S}}}}}}}}}_{i,\gamma }\cdot {{{{{{{{\bf{S}}}}}}}}}_{i,\gamma }\rangle .$$

(13)