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.
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Introduction
Transition metal (TM) systems involving multi-orbital correlated electrons continue attracting much attention due to their rich physical properties1,2,3,4,5,6. 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 JH 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 state7,8, various forms of orbital ordering9,10, and the recently much-addressed orbital-selective physics11,12,13,14,15,16. 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 region17,18,19,20,21.
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 useful22,23,24. Compared to antiferromagnetic (AFM) insulators, FM insulators are relatively rare, particularly among oxides25,26,27. 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 heterostructures24,30,31. 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 elements32,33. Hexagonal ferrites and spinels such as CoFe2O434,35 are also of interest28. FM semiconducting behavior has also been reported in spinel chalcogenides like CdCr2S4 and CrCr2Se427,36.
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 amplitudes37,38. 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 Ce2O2FeSe240,41, 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:
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
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 dyz 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 px orbital acts as a “bridge” between the 3d orbitals. To have an intuitive visual view of t12, the WFs related to this px 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 JH/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 = Hk + Hint. The tight-binding kinetic component is
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. Δγ 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 JH, and the on-site inter-orbital electron-pair hopping terms. Formally, it is given by:
where the standard relation \({U}^{{\prime} }=U-2{J}_{{{{{{{{\rm{H}}}}}}}}}\) is assumed and Piγ = ci↓γci↑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 nγ, and charge fluctuations δnγ, all at JH/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.
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 JH): 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)=〈Si ⋅ Sj〉 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 δnγ 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.
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 JH, 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 ργ(ω) vs frequency ω by using the dynamical DMRG, where the dynamical correlation vectors were obtained using the Krylov-space approach45,46. A broadening parameter η = 0.1 was chosen in our DMRG calculations, as in previous cases. The chemical potential is obtained from μ = (EN+1 − EN−1)/2, where EN is the ground state energy of the N-particle system. ργ(ω) is calculated from the portions of the spectra below and above μ, respectively, using:
with ψ0 as the ground state.
As displayed in Fig. 4, we calculated ργ(ω) for several Δ2’s, corresponding to different phases, at JH/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.
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 t2g orbitals localized and eg metallic47. In doped manganites, FM is induced by the double exchange mechanism48, where off-diagonal hopping between the half and empty eg 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.
By increasing the crystal field Δ2, the system has non-integer electronic density nγ in the three orbitals, see Fig. 5a, while the charge fluctuations δnγ 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 nγ 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 configuration51,52,53, 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 superconductivity13,57,58. 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 pressure60,61.
Methods
DMRG method
The model Hamiltonian discussed here was studied by using the DMRG method62,63, 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:
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:
where \({{{{{{{{\bf{S}}}}}}}}}_{i}=\mathop{\sum}\limits_{\begin{array}{c}\gamma \end{array}}{{{{{{{{\bf{S}}}}}}}}}_{i,\gamma }\).
The corresponding structure factor for spin is:
The site-average occupancy of orbitals is:
The orbital-resolved charge fluctuation is defined as:
The mean value of the squared spin for each orbital is defined as:
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
Code availability
The computer codes used in this study are available at https://g1257.github.io/dmrgPlusPlus/.
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Acknowledgements
The work of L.-F.L., Y.Z., M.A.M., A.F.M., A.M., and E.D. was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), Materials Sciences and Engineering Division. The work of G. A. was supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Science Center.
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L.L., Y.Z., and E.D. designed the project. L.L. carried out numerical calculations for the multiorbital Hubbard model. G.A. developed the DMRG++ computer program. L.L., Y.Z., M.A.M., A.F.M., A.M., and E.D. wrote the manuscript. All co-authors provided useful comments and discussion on the paper.
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Lin, LF., Zhang, Y., Alvarez, G. et al. Stability of the interorbital-hopping mechanism for ferromagnetism in multi-orbital Hubbard models. Commun Phys 6, 199 (2023). https://doi.org/10.1038/s42005-023-01314-w
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DOI: https://doi.org/10.1038/s42005-023-01314-w
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