Abstract
Ironbased superconductors display a variety of magnetic phases originating in the competition between electronic, orbital, and spin degrees of freedom. Previous theoretical investigations of the multiorbital Hubbard model in onedimension revealed the existence of an orbitalselective Mott phase (OSMP) with block spin order. Recent inelastic neutron scattering (INS) experiments on the BaFe_{2}Se_{3} ladder compound confirmed the relevance of the blockOSMP. Moreover, the powder INS spectrum revealed an unexpected structure, containing both lowenergy acoustic and highenergy optical modes. Here we present the theoretical prediction for the dynamical spin structure factor within a blockOSMP regime using the densitymatrix renormalizationgroup method. In agreement with experiments, we find two dominant features: lowenergy dispersive and highenergy dispersionless modes. We argue that the former represents the spinwavelike dynamics of the block ferromagnetic islands, while the latter is attributed to a novel type of local onsite spin excitations controlled by the Hund coupling.
Introduction
Inelastic neutron scattering (INS) measurements are crucial for the study of quantum magnetism in condensed matter physics. This powerful experimental technique provides detailed information of momentum and energy resolved spin excitations. The importance of INS studies is best illustrated in the case of high critical temperature superconductors. Shortly after the discovery of the copperoxide compounds it became evident that the standard BCS theory of the electron–phonon coupling could not explain the experimental findings. Simultaneously, INS results showed that superconductivity appears in close proximity to the antiferromagnetic (AFM) ordering of S = 1/2 Cu^{2+} moments providing robust evidence that the new pairing mechanism is based on spin fluctuations^{1,2,3,4}.
The discovery of ironbased superconductors (FeSC) added an extra complication to this “simple” picture. Although the phase diagrams of Cubased and Febased materials are qualitatively similar^{5}, there are important conceptual differences. The most significant are in the minimal models that describe the materials^{6,7}. While cuprates have a single Fermi surface (FS), the ironbased compounds have a complicated FS with electron and hole pockets originating in the five 3d orbitals of iron. As a consequence, the FeSC have to be described by means of multiorbital Hubbard models, involving not only a standard Hubbard U repulsion but also a Hund coupling. The competition between electronic, orbital, and spin degrees of freedom can lead to many exotic magnetic phases^{8,9,10,11,12,13}.
Past experience in cuprates showed that the analysis of lower dimensional systems, such as chains and ladders, can provide useful information to better contrast theory with experiments^{14}. One reason is that theoretical manybody calculations based on model Hamiltonians can be accurately performed in onedimension, particularly numerically. For this reason, it was exciting when a onedimensional family of compounds containing twoleg ladders was unveiled also in the ironsuperconductors context. Specifically, we refer to the lowdimensional FeSC in the 123 family, AFe_{2}X_{3}, where A are alkali metals A = K, Ba, Rb, Cs, and X are chalcogenides X = S, Se. These compounds are build of double chains (i.e., they are ladders) of edge sharing FeX_{4} tetrahedra^{15}. Recently, a superconducting state was identified under pressure for BaFe_{2}S_{3}^{16,17} and BaFe_{2}Se_{3}^{18,19}. The pressuredependent phase diagram of these materials resembles that of copperoxide ladders, e.g., the telephone number compound Sr_{14−x}Ca_{x}Cu_{24}O_{41}^{20}. Similar to their copperoxide counterparts, the iron123 family is insulating at ambient pressure. This behavior is unusual since, unlike the cuprates, the parent compounds of FeSC are typically bad metals. In addition, it was argued that orbitalselective Mott physics (OSMP)^{21} is consistent with results for BaFe_{2}Se_{3}^{22,23,24,25}. Within such a phase, itinerant and localized conduction electrons coexist.
It should be remarked that INS experiments on 123 materials have been performed up to now only on powder samples and, as a consequence, detailed data of the momentum dependence of the spin excitations over the whole Brillouin zone is not yet available. Nevertheless, the static (π, 0) stripe AFM order—with ferromagnetic rungs and antiferromagnetic legs—was identified for BaFe_{2}S_{3}^{26}, RbFe_{2}Se_{3}^{27}, CsFe_{2}Se_{3}^{28,29}, and also for KFe_{2}S_{3}^{22}. However, in the special case of BaFe_{2}Se_{3} remarkably an exotic block magnetism was found^{19,22,25,30,31} involving antiferromagnetically coupled ferromagnetic islands made of 2 × 2 iron clusters. This unusual magnetic state was also observed in the vicinity of superconductivity^{32,33,34} in twodimensional (2D) materials with \(\sqrt 5\, \times \sqrt 5\) ordered iron vacancies, such as Rb_{0.89}Fe_{1.58}Se_{2}^{35} and K_{0.8}Fe_{1.6}Se_{2}^{36,37,38}. In addition, for BaFe_{2}Se_{3}^{25}, BaFe_{2}S_{3}^{26}, and RbFe_{2}Se_{3}^{27} the INS revealed the existence of lowenergy acoustic and highenergy optical modes separated by an energy gap. It is important to remark that the generic features of the INS spectra of the aforementioned compounds are similar, but the physical origin of the acoustic modes can differ significantly—these modes reflect on the longdistance properties of the magnetic order in the system. Moreover, the origin and characteristics of the optical modes, that are induced by shortdistance properties, have not been clarified so far.
In this work, we will address the spin dynamical properties of the exotic block magnetic state found in BaFe_{2}Se_{3}. The static (time independent) properties of this phase were previously qualitatively studied in ref.^{23} via a threeorbital Hubbard model in onedimension (1D) that unveiled an OSMP regime. Here, we will use the same Hamiltonian to investigate the momentum and energy resolved spin dynamics. To test the general features of our findings, we present also results obtained in a quasi1D ladder geometry. In agreement with experimental findings, we have observed two distinct modes of spin excitations: a lowenergy dispersive mode and highenergy dispersionless optical modes. The lowenergy acoustic mode reveals the frustrated nature of the block magnetism which can be described by a spin J_{1}–J_{2} Hamiltonian. On the other hand, we argue that the optical mode is controlled by local orbital physics and it cannot be properly captured by a Heisenberglike model. The main features of our analysis are simple and generic and should characterize any multiorbital model as long as its ground state is in a magnetic block phase.
Results
Model and observables
We will focus on a specific threeorbital Hubbard model on a onedimensional lattice, but our conclusions are generic for a broad group of models and materials in the OSMP magnetic blockphase regime. As mentioned before, the model chosen was previously studied with regards to its timeindependent properties, and it is known that it displays an OSMP regime in the ground state^{23}. The kinetic part of the Hamiltonian, H_{kin}, is defined as:
where \(c_{\ell ,\gamma ,\sigma }^\dagger\) creates an electron with spin σ = {↑, ↓} at orbital γ = {0, 1, 2} and site \(\ell = \{ 1, \ldots ,L\}\) of a 1D chain. \(n_{\ell ,\gamma ,\sigma } = c_{\ell ,\gamma ,\sigma }^\dagger c_{\ell ,\sigma ,\gamma }\) is the local \((\ell ,\gamma )\) electron density with spin σ. Note that another common labeling of these orbitals could be based on the canonical t_{2g} manifold, i.e., {yz, xz, xy}, respectively. \(t_{\gamma \gamma \prime }\) denotes a symmetric hopping amplitude matrix defined in the orbital space γ: t_{00} = t_{11} = −0.5, t_{22} = −0.15, t_{02} = t_{12} = 0.1, and t_{01} = 0, all in eV units (Fig. 1a displays a schematic representation of the Hamiltonian). The crystalfield splitting is set to Δ_{0} = −0_{1}, Δ_{1} = 0, and Δ_{2} = 0.8, also in eV units. The total kineticenergy bandwidth is W = 2.45 eV. These phenomenological values of parameters were chosen before^{23} to reproduce qualitatively the band structure properties of higher dimensional selenides at an electronic density \(\bar n = 4{\mathrm{/}}3\) per orbital, namely an electronlike pocket at k = 0 and holelike pockets at k = ±π (see Fig. 1b, and also ref.^{39} and references therein). It should be pointed out that the existence of an OSMP highlights the striking orbital sensitivity on electron correlations in multiorbital Hubbard models, and its presence is not limited to our use of 1D geometries nor to our choice of \(t_{\gamma \gamma \prime }\) hoppings. For example, the OSMP was proven to be relevant^{40} for 2D alkaline iron selenides as well, with and without \(\sqrt 5\, \times \sqrt 5\) ordered vacancies. We wish to emphasize that our predictions primarily depend on the existence of an OSMP magnetic blockphase state, rather than on the details of the Hamiltonian that leads to its stabilization. In this context, we believe that our results are universal for ironbased superconductors. To support this claim, we will present calculations for several models showing that all the many reported results lead essentially to the same qualitative conclusions.
The interaction portion of the Hamiltonian H_{int} is given by
where \(n_{\ell ,\gamma } = \mathop {\sum}\nolimits_\sigma {\kern 1pt} n_{\ell ,\gamma ,\sigma }\), the local spin \((\ell ,\gamma )\) is \(S_{\ell ,\gamma }\) = \((1{\mathrm{/}}2)\mathop {\sum}\nolimits_{a,b} {\kern 1pt} c_{\ell ,\gamma ,a}^\dagger \sigma ^{ab}c_{\ell ,\gamma ,b}\) (with σ^{ab} as a Pauli spin matrices), and \(P_{\ell ,\gamma }\) = \(c_{\ell , \uparrow ,\gamma }c_{\ell , \downarrow ,\gamma }\) is the pairhopping. We will consider an SU(2) symmetric system, i.e., U′ = U − 2J_{H}, where U stands for the onsite sameorbital repulsive Hubbard interaction. Finally, we set the Hund coupling to J_{H} = U/4, a value widely used before and considered to be realistic for Febased materials^{41,42}. We refer the interested reader to refs.^{23,43,44} for details of the J_{H}–U phase diagram of the above Hamiltonian. Here, if not stated differently, we will use U/W = 0.8 where previous studies found^{23} a blockOSMP, i.e., antiferromagnetically (AFM) coupled ferromagnetic (FM) blocks (magnetic unit cells), ↑↑↓↓↑↑↓↓, in the localized orbital γ = 2 (see Fig. 1c). Note that the block order is usually studied in the context of Heisenberglike spin Hamiltonians (such as dimerized^{25,35} or J_{1}–J_{2} models^{45}). Here, the block phase is a consequence of nontrivial electronic correlations within the OSMP phase. Since the latter is a feature of multiorbital systems that cannot be analyzed using purely spin systems, we believe that our setup is more suitable for the study of ironbased materials.
In this work, we will investigate the zerotemperature frequency ωdependent spin structure factor (SSF) S(q, ω), defined as the Fourier transform of the realspace total (onsite, \(S_\ell = \mathop {\sum}\nolimits_\gamma {\kern 1pt} S_{\ell ,\gamma }\)) spin correlation functions (see Methods). Furthermore, we will study the contributions from the individual orbitals to the total SSF, i.e., \(S_{\gamma \gamma \prime }(q,\omega )\). γ = γ′ denotes the spin fluctuations within each of the orbitals, while γ ≠ γ′ are spin fluctuations between different orbitals. As a consequence \(S(q,\omega )\) = \(\mathop {\sum}\nolimits_\gamma {\kern 1pt} S_{\gamma \gamma }(q,\omega ) + \mathop {\sum}\nolimits_{\gamma \ne \gamma \prime } {\kern 1pt} S_{\gamma \gamma \prime }(q,\omega )\). From the experimental perspective, only the total SSF has a meaning^{46} because neutrons couple to electrons in all orbitals in neutron scattering experiments. However, the theoretical investigations of orbitalresolved SSF can provide further insight into the OSMP physics.
The Hamiltonians are diagonalized via the DMRG method, where the dynamical correlation functions are obtained with the help of dynamical DMRG techniques (see Methods and Supplementary Note 1 for details of the numerical simulations).
Dynamical spin structure factor
In Fig. 2, we present one of the main results of our effort: the frequencymomentum dependence of the dynamical SSF in the blockOSMP phase (i.e., at U/W = 0.8). Figure 2a depicts the total SSF, S(q, ω), while Fig. 2b shows only the contribution from the localized orbital, S_{22}(q, ω). Several conclusions can be obtained directly from the presented results: (i) A robust contribution to the total SSF arises from the localized orbital. Moreover, all the qualitative features of S(q, ω) are already present in S_{22}(q, ω). In fact, S(q, ω) and S_{22}(q, ω) become almost indistinguishable if normalized by the local magnetic moment squared (i.e., S^{2} = 3/4 for the S = 1/2 localized electron, and S^{2} = 2 for the total moment^{23}). (ii) The energy range for the spin dynamics is much smaller when compared with the energy bandwidth W = 2.45 eV of the Hamiltonian. (iii) Clearly the dynamical SSF has two distinct modes: a lowfrequency, ω ≲ ω_{c} = 0.08 eV, dispersive (acoustic) band and a highfrequency, ω ~ 0.11 eV, dispersionless (optical) band. Similar results were previously reported experimentally in INS investigations of BaFe_{2}Se_{3}^{25} (with 2 × 2 FM blocks), BaFe_{2}S_{3}^{26}, and RbFe_{2}Se_{3}^{27} (with 2 × 1 FM blocks). The different types of blocks in the INS investigations, and the similarity of results between neutrons and our calculations, suggest that our results apply to a broad variety of iron chalcogenides. Moreover, the INS measurements where performed on powder samples and, as a consequence, no detailed analysis of the spin excitations over all crystal momenta q (over the whole Brillouin zone) have been reported. In this respect, our results define clear theoretical predictions on what future singlecrystal experiments should display.
In Fig. 3a, we present the ω dependence of the total SSF at special values of the momenta q. It is evident that the acoustic mode is strongly momentum dependent in the range 0 < q/π ≲ 1/2, while it reduces drastically its intensity for q/π > 1/2. To understand these results, we can reanalyze the SSF spectrum using twosites as a rigid block, namely creating an effective magnetic unit cell of FM blocks with momentum \(\tilde q\). The acoustic mode as a function of \(\tilde q\) then is located between \(0 \, < \, \tilde q \, < \, \pi\), resembling a gapless continuum of spin excitations. Such an interpretation is consistent with “collective” spin waves based on FM blocks. On the other hand, the highenergy optical contribution is qindependent for q/π ≳ 1/2, with vanishing intensity in the q → 0 limit. As discussed later, this mode can be associated to local (onsite) spin excitations affecting the Coulomb potential portion of the Hamiltonian, independently of the dimensionality of the system. The qdependence of both modes is also clearly visible in the static SSF obtained from the energy integration of the dynamical SSF, i.e., \(S_\alpha (q)\) = \((1{\mathrm{/}}\pi ){\int} {\kern 1pt} {\mathrm{d}}\omega {\kern 1pt} S(q,\omega )\). In Fig. 3b, we present the acoustic (α = A) and optical (α = O) contribution to the total (α = T) static SSF, coming from the integration over the frequency ranges 0 < ω < ω_{c}, ω_{c} < ω < ∞, and 0 < ω < ∞, respectively. From the dynamical SSF spectra, it is evident that S_{O}(q) provides the sole contribution to the total static SSF for momentum 0.75 < q/π < 1. As a consequence, at least within a blockOSMP state it is remarkable that already in the static SSF one can observe the clear presence of an optical mode, a novel result which is intrinsic of block phases to our knowledge. In the same panel, we also present the total static SSF independently obtained from the expectation value of the ground state (GS), i.e., S_{stat}(q) = 〈GSS_{q}·S_{−q}\(\left {{\mathrm{GS}}} \right\rangle\), where S_{q} is the Fourier transform of the \(S_\ell\) operators for the same system size L. The agreement between S_{stat}(q) and S_{T}(q) serves as nontrivial accuracy test of the dynamical DMRG method, since the former can be obtained with much higher accuracy.
Orbital contribution
Before addressing the optical and acoustic modes in more detail, we will comment on the orbital γ contribution to S(q, ω). As already shown in Fig. 2, the main contribution to the total SSF originates in the localized orbital γ = 2. Our results (Fig. 3c) indicate that the spin fluctuations for the itinerant electrons (orbitals γ = 0 and γ = 1) follow the behavior of the localized orbital. As argued below, this is a consequence of the Hund coupling which aligns ferromagnetically spins at different orbitals. However, the nature of these orbitals is metallic and magnetic moments are not well formed. As a consequence, the spectral weight of the total itinerant contribution (two orbitals) is approximately the same as the localized (one orbital). On the other hand, the interorbital SSF \(S_{\gamma \ne \gamma \prime }\) have a large contribution only to the acoustic mode, especially near the q/π = 1/2 point.
Acoustic mode
Consider now the properties of the acoustic mode. Motivated by the results presented above, with the main contribution to the SSF arising from the localized orbital, we express the eigenstates in terms of the basis states of localized orbital ·〉_{γ=2} (see Methods section). Since the electrons are indeed localized with occupation n_{γ=2} = 1^{23} in the OSMP, in the lowenergy portion of the spectrum the basis states with empty and double occupied orbital γ = 2 should not be present. Within such a representation the GS of the blockOSMP phase can be identified as a superposition of \(\left { \uparrow \uparrow \downarrow \downarrow } \right\rangle _{\gamma = 2}\) and \(\left { \downarrow \downarrow \uparrow \uparrow } \right\rangle _{\gamma = 2}\) states, which constitutes ~50% of the true GS. One can improve further the qualitative description by investigating a simple toy model. Let us consider two FM coupled S = 1/2 spins as one S = 1 object, i.e., \(\left {\mathbf{1}} \right\rangle = \left { \uparrow \uparrow } \right\rangle _{\gamma = 2}\), \(\left {  {\mathbf{1}}} \right\rangle = \left { \downarrow \downarrow } \right\rangle _{\gamma = 2}\), and \(\left {\mathbf{0}} \right\rangle\) = \(1{\mathrm{/}}\sqrt 2 \left( {\left { \uparrow \downarrow } \right\rangle _{\gamma = 2} + \left { \uparrow \downarrow } \right\rangle _{\gamma = 2}} \right)\). In this setup, a 4site S = 1/2 system reduces to two antiferromagnetically coupled S = 1 spins. The ground state of the latter is simply
where \(c_a = c_b = 1{\mathrm{/}}\sqrt 3\) (see Fig. 4a for a schematic representation). Note that the above state, in agreement with numerics, is a singlet. The last two terms of Eq. (3) correspond to the “perfect” block order, i.e., \(\left { \uparrow \uparrow \downarrow \downarrow } \right\rangle _{\gamma = 2} + \left { \downarrow \downarrow \uparrow \uparrow } \right\rangle _{\gamma = 2}\), while the first term depicts the x–y component of the block order,
Our L = 4 Lanczos investigation of the full Hamiltonian (1) and (2) indicates that such a state has coefficients equal to \(\widetilde c_a^{\, 2} \simeq 1{\mathrm{/}}6\) and \(\widetilde c_b^{\,2} \simeq 1{\mathrm{/}}4\), which yields now a better overlap, ~70%, with the true GS. Finally, the first excited state—contributing to the acoustic mode—can be identified as a triplet of the form \(\left {\mathrm{A}} \right\rangle _{\gamma = 2}\) = \(\widetilde c_A\left( {\left { \uparrow \uparrow \downarrow \downarrow } \right\rangle _{\gamma = 2}  \left { \downarrow \downarrow \uparrow \uparrow } \right\rangle _{\gamma = 2}} \right)\), where \(\widetilde c_A^{\,2} \simeq 4{\mathrm{/}}9\) (Fig. 4b). This large overlap of \(\left {\mathrm{A}} \right\rangle _{\gamma = 2}\) with the full solution is also captured by the toy model since 1〉−1〉 − −1〉1〉 is one of the first excitations in our twosite S = 1 problem. Note that the above description of the \(\left {{\mathrm{GS}}} \right\rangle _{\gamma = 2}\) \(\left( {\left {\mathrm{A}} \right\rangle _{\gamma = 2}} \right)\) as a spin singlet (triplet) is not obvious from the signs of the localized orbital basis representation. While the above states capture the essence of the problem, the itinerant orbitals have to be included in the description to account for the true nature of the singlettriplet excitation.
Although simplified, descriptions such as those above of the lowenergy spectrum can yield nontrivial consequences. A similar ground state to our \(\left {{\mathrm{GS}}} \right\rangle _{\gamma = 2}\) with π/2 pitch angle was previously observed in the frustrated ferromagnetic S = 1/2 J_{1}–J_{2} Heisenberg model with ferromagnetic J_{1} and antiferromagnetic J_{2}^{45,47,48,49,50}. In Fig. 5a, we present a comparison of the multiorbital system Eqs. (1) and (2) SSF vs J_{1}–J_{2} results obtained for J_{2}/J_{1} = 1. Within the latter, the dynamical SSF yields a continuum of excitations with maximum intensity at q/π = 1/2 and vanishing intensity in the q/π → 1 limit. In fact, the dynamical SSF of the J_{1}–J_{2} model is very similar to the acoustic mode found in our multiorbital system, i.e., compare panels (b) and (c) of Fig. 5. To strengthen this argument, in Fig. 5b and c, we present the dynamical SSF factor plotted against the quantum dispersion relation of the J_{1}–J_{2} model ω(q) = \(\epsilon _q\) − \(\epsilon _{{\mathrm{GS}}}\), where \(\epsilon _q\) is the energy of the lowest eigenstate at a given q. To match the energy scales, we set J_{1} = J_{2} = 0.6J_{eff}, where \(J_{{\mathrm{eff}}} = 4{\kern 1pt} t_{22}^2{\mathrm{/}}U\) is the natural superexchange scale within the localized orbital, as a crude approximation. As clearly shown in Fig. 5b, ω(q) quantitatively captures the main features of the acoustic portion of the spectrum.
We remark that the present comparison with the J_{1}–J_{2} model is at a phenomenological level, since this effective description of the lowest mode of the spin dynamics was not rigorously derived from our multiorbital Hamiltonian Eqs. (1) and (2). The acoustic mode reflects the frustrated nature of the magnetism within the blockOSMP phase. Also, the J_{1}–J_{2} model may be relevant in a wide range of interaction U within the OSMP phase, beyond the block ordering region 0.4 ≲ U/W ≲ 1.5. For example, previous results showed that in the range 1.5 ≲ U/W ≲ 20, the system is in a ferromagneticOSMP^{23}, where the spins within the localized orbital γ = 2 have ferromagnetic ordering. Clearly, a J_{1}–J_{2} model with small or vanishing J_{2} will also exhibit a similar ordering. Finally, note that although the alternative S = 1 toy model is useful in the description of elementary states of the blockOSMP system, its validity is limited for the dynamical spin response: it is well known that the dynamical SSF of the S = 1 AFM Heisenberg model exhibits “sharp” magnon lines, in contrast to the S(q, ω) of the S = 1/2 model that contains a continuum of excitation (at least at lowω), in agreement with our results for the threeorbital Hamiltonian.
Optical mode
Let us now turn to the highenergy optical mode of the dynamical SSF spectrum. The states contributing to this mode are also triplet excitations. In the L = 4 Lanczos analysis, we found that this highenergy mode arises from a state of the form \(\left {\mathrm{O}} \right\rangle _{\gamma = 2}\) ≃ \(1{\mathrm{/}}2\left( {\left { \downarrow \uparrow \uparrow \downarrow } \right\rangle _{\gamma = 2} + \left { \uparrow \downarrow \downarrow \uparrow } \right\rangle _{\gamma = 2}} \right)\). It is evident that \(\left {\mathrm{O}} \right\rangle\) breaks the FM magnetic unit cells present in the GS. Note, again, that the discussed states do not have doubly occupied or empty sites, reflecting the Mott nature of orbital γ = 2. It should be also pointed out that using a small L = 4 system with OBC, we have found another state that contributes to the optical mode, i.e., \(\left {\widetilde {\mathrm{O}}} \right\rangle _{\gamma = 2}\) = \(1{\mathrm{/}}2\left( {\left { \downarrow \uparrow \downarrow \uparrow } \right\rangle _{\gamma = 2} + \left { \uparrow \downarrow \uparrow \downarrow } \right\rangle _{\gamma = 2}} \right)\). However, such a state is not present in the system with periodic boundary conditions.
To understand properly the optical mode it is not enough to focus solely on the localized orbital. A detailed analysis of the remaining “metallic” orbitals γ = 0, 1 indicates that: (i) the \(\left {{\mathrm{GS}}} \right\rangle\) and the \(\left {\mathrm{A}} \right\rangle\) states obey the Hund’s rule: spins in different orbitals of the same site are ferromagnetically aligned (see Fig. 4a for a schematic representation). (ii) However, the \(\left {\mathrm{O}} \right\rangle\) states, Fig. 4c, do not fulfill this rule because part of the spins are antiferromagnetically aligned. As a consequence, the main difference in energy between the \(\left {{\mathrm{GS}}} \right\rangle\) and \(\left {\mathrm{O}} \right\rangle\) arises from the local (onsite) Hund exchange portion of the electronic interaction. We confirm this by calculating separately the expectation values of all terms contributing to the Hamiltonian (see Methods section). The main difference between the energy of the \(\left {{\mathrm{GS}}} \right\rangle\) and \(\left {\mathrm{A}} \right\rangle\) arises from the kinetic portion. On the other hand, the difference in \(\left {\mathrm{O}} \right\rangle\) originates, as expected, from the Hund coupling part of the interaction energy. The local onsite nature of the optical mode is also visible in the orbitalresolved SSF. In Fig. 3c, we present the spin correlations between different orbitals at different sites, i.e., S_{mix}. As clearly visible, the S_{mix}(q → π) → 0, indicating a drastic reduction of spectral weight at large momentum. These findings indicate that the optical mode is not present in the interorbital intersite spin correlations. As a consequence, the only remaining possibility of the origin of the optical mode are the intrasite fluctuations between orbitals. Our investigation of orbitalresolved SSF (Fig. 3c) shows that each orbital contributes to the optical mode with a similar weight. Finally, the lack of momentum dependence of the optical mode (at least for q/π > 1/2) suggests that such excitations are local (onsite) fluctuations of spin between different orbitals at the same site.
In addition, we have shown that the frequency ω_{O} = \(\epsilon _{\mathrm{O}}\) − \(\epsilon _{{\mathrm{GS}}}\) of the corresponding \(\left {\mathrm{O}} \right\rangle\) excitation is directly proportional to the value of the Hund exchange J_{H}, contrary to the \(\left {\mathrm{A}} \right\rangle\) excitation with energy \(\omega _{\mathrm{A}} = \epsilon _{\mathrm{A}}  \epsilon _{{\mathrm{GS}}}\). In Fig. 6a, we present the dynamical SSF at q/π = 1/2 for various values of U within the blockOSMP calculated via the Lanczos method on L = 4 sites, at a fixed J_{H}/U = 1/4. Our results in Fig. 6b indicate that this behavior is valid throughout the entire blockOSMP phase, 0.4 ≲ U/W ≲ 1.5.
Ladder geometry
Finally, let us comment on the lattice geometry dependence of our results. In Fig. 7, we present the SSF for the twoleg ladder twoorbital Hamiltonian introduced in ref.^{39} for the BaFe_{2}Se_{3} compound. The lattice is sketched in Fig. 7a and hopping values are given in the Methods section. It was previously shown^{39} that at density \(\bar n = 1.75{\mathrm{/}}2\), J_{H} = U/4, and U/W_{L} = 2 with W_{L} = 3.82 eV the system is in an enlarged block phase, similar to the 2 × 2 block state of BaFe_{2}Se_{3}^{25}. Before addressing specific results, it is important to remark that the DMRG numerical studies of multiorbital ladders require expensive computations. This is because the intersite interorbital hoppings behave effectively as longdistance hoppings in the equivalent onedimensional representation, leading to larger entanglement for the ground state (see Supplementary Note 1 for details). The calculation of dynamical quantities is certainly a challenge and even the static expectation values have to be carefully analyzed with regards to the number of states kept (here M = 1000 states are used). As a consequence, the results presented for the twoorbital twoleg ladder below may not be as accurate as those for the chains.
On a ladder, there are two separate contributions to the SSF arising from the bonding (q_{y} = 0) and antibonding (q_{y} = π) sectors. For the twoorbital twoleg ladder results, presented in Fig. 7b, we find a q_{y} = 0 dispersive mode at lowω, with a continuum of spin excitations similar to the acoustic mode of the chain geometry. At ω ≃ 0.075 eV, we find an energy narrow q_{y} = π mode. According to our analysis of the 1D system, a similar spectrum can be found in the J_{1}–J_{2} model on the ladder with FM rung coupling J_{⊥} = J_{1} (Fig. 7c, d). Both the J_{1}–J_{2} spin model and multiorbital model on the ladder studied here exhibit the 2 × 2 block state, i.e., AFMcoupled blocks of four FM aligned spins on two neighboring rungs (see Fig. 7a). Such a state has a peak in the static SSF at q_{x} = π/2 in the bonding contribution (q_{y} = 0), see Fig. 7c. Note that the maximum of the acoustic mode appear at ω ≠ 0, which suggests a nonzero spin gap, common in ladders. Finally, at higher frequencies (ω ≃ 0.13 eV) in the q_{y} = 0 sector we find a flat mode of excitations, similar to the optical mode present in the chain analysis. It is again evident that the latter is not captured by the J_{1}–J_{2} model.
Discussion
Let us compare the INS data for BaFe_{2}Se_{3} reported in ref.^{25} against our results. Note that this compound is insulating^{22}, while our system Eqs. (1) and (2) for the parameters considered in this work, U/W = 0.8 and J_{H}/U = 1/4, is a (bad) metal in the blockOSMP phase, becoming insulator only for U/W ≳ 1.5 in the ferromagneticOSMP phase^{51}. Our Hamiltonian reproduces the OSMP state and the magnetic block phase of BaFe_{2}Se_{3}, and although the charge dynamics of our model does not capture the experimentally observed insulating nature of the real material, it is still appealing to study the spin physics. The lack of other multiorbital models that can reproduce both the spin and charge sector of lowdimensional iron selenides makes it appealing to carry out detailed theoretical calculations of the spin dynamics within this model and compare with the experiments.
Within the spinwave theory the lowω portion of the INS spectra was interpreted^{25} as a dispersive mode which reflects the frustrated nature of the π/2order. In addition, the highenergy optical modes were interpreted as local excitation of spins within the 2 × 2 plaquette. A similar rationale was used to explain the INS result of the doped compound Rb_{0.89}Fe_{1.58}Se_{2}^{35}. The spinwave theory of BaFe_{2}Se_{3} reproduces^{25} all of the modes and also properly captures the frequency bandwidth of the spin excitations. However, only ~2/3 of the total spectral weight expected for localized 3d electrons is obtained. Also note that within the considered spin models of refs.^{25,35} unphysically large dimerization spinexchange couplings are required^{52,53} to stabilize the π/2 spin pattern.
From the perspective of our results, the interpretation of the INS spin spectra of lowdimensional ladder iron chalcogenides is different from spinwave theory. The latter assumes that all excitations occur between localized spins, while in our system we have a mixture of localized and itinerant electrons. Moreover, as shown above, the SSF of multiorbital systems not only contains dispersive acoustic modes but also local excitations controlled by the Hund exchange, at least within the blockOSMP. The interorbital nature of such modes cannot be properly captured by localized Heisenberg models. Our results, on both chain and ladder geometries, indicate that spin models can only properly capture dispersive modes resulting from the peculiar spin order of a given phase as in the π/2 state of BaFe_{2}Se_{3}. However, we argue that only one of the low lying optical modes of this compound arises from a weakly dispersive (probably beyond experimental resolution of powder sample) q_{y} = π excitation. Within our interpretation of the SSF spectra, the second optical mode is of a different nature, involving interorbital spin fluctuations on each site. Such a picture is consistent with our multiorbital ladder results.
Concerning the spectral weight, for the chosen parameters U/W = 0.8 and J_{H}/U = 1/4 in Eqs.(1) and (2), we observe the magnetic moment \(\left\langle {S^2} \right\rangle\) ~ 2 (maximal possible for \(\bar n = 4{\mathrm{/}}3\)). This is consistent with previous Hartree–Fock calculations^{39} of the blockOSMP phase within a fiveorbital ladder system, which reported \(\left\langle {S^2} \right\rangle\) ~ 6 for \(\bar n = 6{\mathrm{/}}5\) (again the maximal value). As a consequence, our results do not reproduce the missing spectral weight observed in experiments^{25}. However, the magnetic moments evolve within the blockOSMP^{23} (see also Supplementary Note 2 for additional results) and only saturate to its maximal value at U/W ≳ 0.6, namely in the middle of the blockphase. Since the exact value of U and J_{H} are not know for BaFe_{2}Se_{3}, it is possible in theoretical investigations to stabilize the blockOSMP phase with a reduced \(\left\langle {S^2} \right\rangle\) < 2 (Fig. 6b). Moreover, note that recently it was argued^{54} that insufficient energy (time) resolution in INS experiments produces moments that can be smaller than the actual instantaneous moments. In this context, faster Xraybased techniques such as photoemission spectroscopy (PES), Xray absorption spectroscopy (XAS), and Xray emission spectroscopy (XES) are needed to resolve this issue.
In conclusion, we have investigated the dynamical spin structure factor of a onedimensional threeorbital Hubbard model in the block orbitalselective Mott phase, as well as a ladder twoorbital Hubbard model also in a similar block state. This has been a computationally demanding effort even with the powerful DMRG, and to our knowledge this is the first time that results of this quality are produced. We have shown that our Hamiltonian captures nontrivial features of a broad family of lowdimensional iron chalcogenides, in particular for the ladder BaFe_{2}Se_{3} compound for which π/2block order was reported. We have found two different types of modes in the spin spectra: (i) lowfrequency dispersive (acoustic) spin excitations and (ii) optical dispersionless excitations at higher energy. The acoustic band reflects the nature of magnetic order of the system, namely for the blockOSMP the frustrated π/2ordering can be captured by the quantum J_{1}–J_{2} frustrated Heisenberg model, as also shown here. The optical band arises from onsite interorbital spin fluctuations controlled by the Hund exchange coupling. Finally, our 1D dynamical SSF is in qualitative agreement with the powder INS spectrum of BaFe_{2}Se_{3} (see Supplementary Note 3). Although the latter has only a quasi1D geometry, with small but nonzero couplings perpendicular to the ladder, the ωdependent spectra should be dominated by the predominantly 1D nature of the system. As a consequence, the location in momentum and energy space is properly resolved by our model Hamiltonian Eqs. (1) and (2) for both of the modes.
Our results are general and should apply to a variety of block states in multiorbital quasi1D systems. They should all contain an acoustic band (with pitch wavevector compatible with the size of the magnetic block), a strong asymmetry in the distribution of weight of this acoustic band in different portions of the Brillouin zone, and optical modes with at least one of them related to atomic transitions regulated by the Hund coupling.
Methods
DMRG method
The Hamiltonians discussed here were studied using primarily the densitymatrix renormalizationgroup (DMRG) method^{55,56} within the singlecenter site approach^{57}, where the dynamical correlation functions are evaluated via the dynamical DMRG^{58,59,60}, i.e., calculating spectral functions directly in frequency space with the correctionvector method^{61} with Krylov decomposition^{60}. The computer package DMRG++ developed at ORNL was used. For a chain geometry, in both stages of the DMRG algorithm, we keep up to M = 800 states. This allow us to simulate accurately system sizes up to L = 24 sites for dynamical quantities (truncation <10^{−8} for all frequencies ω) and L = 32 for static quantities (truncation <10^{−10} for the GS). For the ladder geometry results, we use a standard twosite central block approach with M = 1000 states (truncation <10^{−3}, showing that the twoleg ladder twoorbital results are qualitatively correct, because of its close resemble to the rest, but their quantitative accuracy can be further improved in future efforts). In the Supplementary Note 1, we present the scaling of our results with system size L, number of states kept M, and broadening η of Eq. (5).
Dynamical SSF
The zero temperature, T = 0, total spin structure factor (SSF) S(q, ω) is defined as:
with ω^{−} = ω − iη, and \(\left {{\mathrm{GS}}} \right\rangle\) is the ground state with energy \(\epsilon _{{\mathrm{GS}}}\). In the above equation, \(\tilde S_\ell = \mathop {\sum}\nolimits_\gamma {\kern 1pt} S_{\ell ,\gamma }\) is the total spin onsite \(\ell\) for the total SSF S(q, ω), or \(\tilde S_\ell = S_{\ell ,\gamma }\) for the orbitalresolved SSF \(S_{\gamma \gamma \prime }(q,\omega )\).
Furthermore, in the above equation, we adopted the wavevector definition appropriate for open boundary conditions (OBC), i.e., q = kπ/(L + 1) with k = 1, …, L. As a consequence, in this work we used approximate (exact in the thermodynamic limit L → ∞) values of the wavevectors, e.g., q = π ≡ πL/(L + 1).
Localized basis representation
The eigenstates \(\left \phi \right\rangle\) of the threeorbital system can be written as
where \(\left n \right\rangle\) represents the orthonormal basis (particle configuration) of all orbitals and \(\left {n_\gamma } \right\rangle\) (with γ = 0, 1, 2) represents the particle configuration on given orbital γ. Note that \(\mathop {\sum}\nolimits_n {\kern 1pt} c_n^2\) = \(\mathop {\sum}\nolimits_{n_1,n_2,n_3} {\kern 1pt} c^2\left( {n_1,n_2,n_3} \right)\) = \(1\) and \(\left\langle {n_\gamma n_{\gamma \prime }^\prime } \right\rangle = \delta _{nn\prime }\delta _{\gamma \gamma \prime }\). One can rewrite the above equation as
where j ≡ n_{2} represents—within OSMP—the localized orbital and
are vectors. The set of \(\left\{ {\left {\tilde c_j} \right\rangle } \right\}\) vectors represent an orthogonal vectorspace with \(\mathop {\sum}\nolimits_j \left\langle {\tilde c_j\tilde c_j} \right\rangle = 1\). Finally, the weight of the \(\left j \right\rangle _{\gamma = {\mathrm{ }}2}\) configuration in the \(\left \phi \right\rangle\) eigenstate is given by the norm of the \(\left {\tilde c_j} \right\rangle\) vector, i.e., \(\left\langle {\tilde c_j\tilde c_j} \right\rangle = \left\ {\tilde c_j} \right\ \equiv \tilde c_j^2\).
Energy contribution
In Table 1, we present the expectation values of the several terms present in the Hamiltonian Eqs. (1) and (2) for the ground state and also states which contribute to the acoustic and optical modes.
Twoorbital twoleg ladder Hamiltonian
The symmetric hoppings for the twoorbital twoleg ladder system are defined^{39} in orbital space as follows (see sketch in Fig. 7a):
all expressed in units of eV. The interaction portion of the Hamiltonian is the same as in the 1D system Eq. (2).
Code availability
Computer codes used in this study are available at https://g1257.github.io/dmrgPlusPlus/.
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
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Acknowledgements
J.H, A.M., and E.D. were supported by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), Materials Sciences and Engineering Division. N.K. was supported by the National Science Foundation Grant No. DMR1404375. The work of G.A. was conducted at the Center for Nanophase Materials Science, sponsored by the Scientific User Facilities Division, BES, DOE, under contract with UTBattelle.
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J.H. and E.D. planned the project. J.H. performed all DMRG calculations for the multiorbital Hubbard model, N.K. performed all Lanczos and J_{1}–J_{2} model calculations, while A.N. and G.A. developed the DMRG++ computer program. J.H., A.M., and E.D. wrote the manuscript. All coauthors provided comments on the paper.
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Herbrych, J., Kaushal, N., Nocera, A. et al. Spin dynamics of the block orbitalselective Mott phase. Nat Commun 9, 3736 (2018). https://doi.org/10.1038/s41467018061816
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DOI: https://doi.org/10.1038/s41467018061816
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