Block–spiral magnetism: An exotic type of frustrated order

Significance

Magnetic frustration in spin or electronic models emerges from the failure of the system to fulfill simultaneously conflicting local requirements. The latter typically arise from the lattice geometry or are induced by special spin–spin interactions in the system at various distances. Here we show that the competing energy scales of the seemingly nonfrustrated orbital-selective Mott phase of the low-dimensional multiorbital Hubbard model can originate a “block–chiral magnetism,” i.e., a state with rigidly rotating spin–magnetic islands. Furthermore, we show how such an exotic spin state influences the electronic properties of the system, revealing parity-breaking quasiparticles.

Abstract

Competing interactions in quantum materials induce exotic states of matter such as frustrated magnets, an extensive field of research from both the theoretical and experimental perspectives. Here, we show that competing energy scales present in the low-dimensional orbital-selective Mott phase (OSMP) induce an exotic magnetic order, never reported before. Earlier neutron-scattering experiments on iron-based 123 ladder materials, where OSMP is relevant, already confirmed our previous theoretical prediction of block magnetism (magnetic order of the form $\uparrow \uparrow \downarrow \downarrow$). Now we argue that another phase can be stabilized in multiorbital Hubbard models, the block–spiral state. In this state, the magnetic islands form a spiral propagating through the chain but with the blocks maintaining their identity, namely rigidly rotating. The block–spiral state is stabilized without any apparent frustration, the common avenue to generate spiral arrangements in multiferroics. By examining the behavior of the electronic degrees of freedom, parity-breaking quasiparticles are revealed. Finally, a simple phenomenological model that accurately captures the macroscopic spin spiral arrangement is also introduced, and fingerprints for the neutron-scattering experimental detection are provided.

Frustrated magnetism is one of the main areas of research in contemporary condensed-matter physics. In the generic scenario, magnetic frustration emerges from the failure of the system to fulfill simultaneously conflicting local requirements. As a consequence, complex spin patterns can develop from geometrical frustration (as in triangular, Kagome, or pyrochlore lattices) or from special spin–spin interactions (long-range exchange, Dzyaloshinskii–Moriya coupling, Kitaev model, spin–orbit effects, and others). In real materials both scenarios often coexist. Competing mechanisms can lead to interesting phenomena, such as spiral order (1–3), spin ice (4), skyrmions (5), spin liquids (6, 7), and resonating valence bond states (8). Also, the electronic properties of such systems are of much interest: It was shown that the interplay of a spiral state on a metallic host can support Majorana fermions (9–14) and can also induce multiferroicity (15–21).

Another example of competing interactions is the orbital-selective Mott phase (OSMP) (22, 23). In multiorbital systems, this effect can lead to the selective localization of electrons on some orbitals. The latter coexist with itinerant bands of mobile electrons. This unique mixture of localized and itinerant components in multiorbital systems could be responsible for the (bad) metallic behavior of the parent compounds of iron-based superconductors. This is in stark contrast to cuprates, usually described by the single-band Hubbard model, where parent materials are insulators. Furthermore, the OSMP can also host exotic magnetic phases. It was shown that the competition between Hund and Hubbard interactions can stabilize unexpected block magnetism (24–28), namely antiferromagnetically (AFM)-coupled ferromagnetic (FM) islands. The size and shape of such islands depend on the electronic filling of the itinerant orbitals. This work focuses on two representative cases: 1) the $\pi /2$-block spin-pattern $\uparrow \uparrow \downarrow \downarrow$ and 2) the $\pi /3$-block spin-pattern $\uparrow \uparrow \uparrow \downarrow \downarrow \downarrow$. Note that these block patterns are not spin-density waves: The local expectation values of spin operators yield uniform magnetization throughout the system, unlike a spin wave that would have peaks and valleys. Our study, on the other hand, indicates a clear block structure with spins of the same magnitude at each site (26–28). Moreover, exact diagonalization results on small lattices (27) indicate that the block-OSMP ground state has a large overlap (at least $50\%$) with a state of the form $|\uparrow \uparrow \downarrow \downarrow \rangle -|\downarrow \downarrow \uparrow \uparrow \rangle$ (as exemplified for the $\pi /2$ block). As a consequence, our states can be viewed as Néel-like states of an enlarged magnetic unit cell.

The OSMP was shown (22) to be relevant for the low-dimensional family of iron-based ladders, the so-called 123 family ${\mathrm{A}\mathrm{F}\mathrm{e}}_2{\mathrm{X}}_3$, where A = Ba, K, Rb is an alkaline earth metal and X = S, Se is a chalcogen. From the magnetism perspective, two phases were experimentally reported: 1) For ${\mathrm{B}\mathrm{a}\mathrm{F}\mathrm{e}}_2{\mathrm{S}\mathrm{e}}_3$ inelastic neutron scattering (INS) identified (29) a 2$\times$2 block–magnetic phase in a $\uparrow \uparrow \downarrow \downarrow$ pattern along the legs. Neutron diffraction measurements and muon spin relaxation yield the same conclusion (30). 2) On the other hand, for ${\mathrm{B}\mathrm{a}\mathrm{F}\mathrm{e}}_2{\mathrm{S}}_3$ and ${\mathrm{R}\mathrm{b}\mathrm{F}\mathrm{e}}_2{\mathrm{S}\mathrm{e}}_3$ INS reported (31–33) 2$\times$1 blocks, FM along rungs and AFM along legs. Both of these phases are captured by the multiorbital Hubbard Hamiltonian (26) and also by its low-energy effective description (28), the generalized Kondo–Heisenberg model.

In this work, we unveil another magnetic phase that we stabilized; namely, we report an exotic block–spiral state. Such a state arises as a consequence of simultaneous tendencies in the system to form magnetic blocks and to develop noncollinear order. Different from standard spirals where one can observe spin-to-spin rotation, in the block–spiral state the blocks rigidly rotate, as illustrated in the three panels in Fig. 1A. The block–spiral states we report have similarities with states in the rare-earth material ${\mathrm{T}\mathrm{b}\mathrm{F}\mathrm{e}\mathrm{O}}_3$ (34) that display spin incommensurability and domain walls. However, the length scales in ${\mathrm{T}\mathrm{b}\mathrm{F}\mathrm{e}\mathrm{O}}_3$ are much larger, 340 Å, and magnetic fields or anisotropy are needed for their stabilization. Furthermore, the block–spiral magnetic pattern appears without any apparent frustration in the model and is a consequence of subtly competing energy scales present in the OSMP. This spin order originates in ferromagnetic tendencies, induced by Hund physics in multiorbital systems, and opposite antiferromagnetic superexchange tendencies, caused by Hubbard interactions. There is a hidden frustration in the system, not obvious at the Hamiltonian level, and whose exotic consequences appear only when powerful computational tools are used: Simpler biased techniques likely would have missed this state. We propose two simpler phenomenological models that capture the essence of our findings. Moreover, the electronic properties can be accurately described by quasiparticles that break parity symmetry, as expected within a spiral state. Also, we show that the behavior of spins can be effectively modeled via a frustrated long-range spin-only Heisenberg model. Such a spin model can serve as a starting point for the spin-wave theory calculations often used to compare theory vs. INS spectra. Our findings are robust against system parameter changes and should characterize generic multiorbital systems in the OSMP regime, close to ferromagnetism.

Fig. 1.

Schematic representation of spirals within the orbital-selective Mott phase. (A) Top to Bottom: standard spiral spin structure with site-to-site spins rotation and block–spirals of two and three sites, respectively. (B) Interaction $U$–filling $n_{\mathrm{H}/\mathrm{K}}$ phase diagram. Solid (striped) coloring represents the OSMP (paramagnetic) region.

Results

Magnetism of OSMP.

We discuss the properties of a multiorbital Hubbard model on a one-dimensional (1D) lattice. In the generic SU(2)-symmetric form it can be written as

$$\begin{array}{ll}\hfill H_{\mathrm{H}}& =-\sum _{\gamma ,\gamma \prime ,\ell ,\sigma }{t}_{\gamma \gamma \prime }\left(c_{\gamma ,\ell ,\sigma }^{†}{c}_{\gamma \prime ,\ell +1,\sigma }+\mathrm{H}.\mathrm{c}.\right)\hspace{0.17em}+\hspace{0.17em}\mathrm{\Delta }\sum _{\ell }{n}_{1,\ell }\hfill \\ \hfill & \hspace{1em}+U\sum _{\gamma ,\ell }{n}_{\gamma ,\ell ,\uparrow }{n}_{\gamma ,\ell ,\downarrow }+\left(U-5J_{\mathrm{H}}/2\right)\sum _{\ell }{n}_{0,\ell }{n}_{1,\ell }\hfill \\ \hfill & \hspace{1em}-2J_{\mathrm{H}}\sum _{\ell }{\mathbf{S}}_{0,\ell }\cdot {\mathbf{S}}_{1,\ell }+J_{\mathrm{H}}\sum _{\ell }\left(P_{0,\ell }^{†}{P}_{1,\ell }+\mathrm{H}.\mathrm{c}.\right).\hfill \end{array}$$ [1]

Here, $c_{\gamma ,\ell ,\sigma }^{†}$ ($c_{\gamma ,\ell ,\sigma }$) creates (destroys) an electron with spin projection $\sigma =\left\{\uparrow ,\downarrow \right\}$ at orbital $\gamma =\left\{\mathrm{0,1}\right\}$ of site $\ell =\left\{1,\dots ,L\right\}$. $\mathrm{\Delta }$ stands for crystal-field splitting, while $n_{\gamma ,\ell }={\sum }_{\sigma }{n}_{\gamma ,\ell ,\sigma }$ represents the total density of electrons at $\left(\gamma ,\ell \right)$, with $n_{\gamma ,\ell ,\sigma }$ as the density of electrons with $\sigma$-spin projection. $U$ is the standard, same-orbital repulsive Hubbard interaction, and $J_{\mathrm{H}}$ is the Hund exchange between spins ${\mathbf{S}}_{\gamma ,\ell }$ at different orbitals $\gamma$. Finally, the last term $P_{0,\ell }^{†}{P}_{1,\ell }$ stands for pair hopping between orbitals, where $P_{\gamma ,\ell }=c_{\gamma ,\ell ,\uparrow }{c}_{\gamma ,\ell ,\downarrow }$.

In the most generic case, the Fe-based materials should be modeled with five $3d$ orbitals (three $t_{2g}$, $d_{xy}$, $d_{yz}$, $d_{xz}$; and two $e_g$, $d_{x^2-y^2}$, $d_{z^2}$). However, it is widely believed (35, 36) that the $t_{2g}$ orbitals are the most relevant orbitals close to the Fermi surface. Furthermore, the $d_{yz}$ and $d_{zx}$ orbitals are often (or are close to being) degenerate and, as a consequence, one can design (28) two-orbital models. Such a choice represents a generic case of coexisting wide and narrow electronic bands, as often found in iron-based materials from the 123 family (23, 26, 37–39). The particular choice of the hopping matrix elements $t_{\gamma \gamma \prime }$ used here—specifically $t_{00}=-0.5\hspace{0.17em}\left[\mathrm{e}\mathrm{V}\right]$, $t_{11}=-0.15\hspace{0.17em}\left[\mathrm{e}\mathrm{V}\right]$, and $t_{01}=t_{10}=0$—and crystal-field splitting $\mathrm{\Delta }=0.8\hspace{0.17em}\left[\mathrm{e}\mathrm{V}\right]$ is motivated by several previous studies (26–28) on the magnetic properties of the OSMP. The above values yield a kinetic energy bandwidth $W=2.1\hspace{0.17em}\mathrm{e}\mathrm{V}$ which is used as an energy unit throughout this paper. Finally, to reduce the number of parameters of the Hamiltonian the value of Hund exchange is fixed to $J_{\mathrm{H}}=U/4$ throughout our investigation. The rationale behind this value comes from dynamical mean-field theory (using local-density approximation) calculations and is believed to be experimentally relevant (40–42). Also, it was shown (43) that in a wide range of Hund couplings, the OSMP properties are the same (26, 43).

It was previously shown (28) that the magnetic properties of the OSMP can be qualitatively, and even quantitatively, described by the effective Hamiltonian obtained by Schrieffer–Wolff transforming the subspace with strictly one electron per site at the localized orbital $\gamma =1$, leading to the generalized Kondo–Heisenberg (gKH) model

$$\begin{array}{ll}\hfill H_{\mathrm{K}}& =-t_{\mathrm{00}}\sum _{\ell ,\sigma }\left(c_{0,\ell ,\sigma }^{†}{c}_{0,\ell +1,\sigma }+\mathrm{H}.\mathrm{c}.\right)+U\sum _{\ell }{n}_{0,\ell ,\uparrow }{n}_{0,\ell ,\downarrow }\hfill \\ \hfill & \hspace{1em}+K\sum _{\ell }{\mathbf{S}}_{1,\ell }\cdot {\mathbf{S}}_{1,\ell +1}-2J_{\mathrm{H}}\sum _{\ell }{\mathbf{S}}_{0,\ell }\cdot {\mathbf{S}}_{1,\ell },\hfill \end{array}$$ [2]

where $K=4t_{11}^2/U$. It is worth noting that due to particle–hole symmetry, the electronic filling relation between the two-orbital Hubbard ($n_{\mathrm{H}}$) and gKH ($n_{\mathrm{K}}$) models is $n_{\mathrm{K}}=3-n_{\mathrm{H}}$.

In this work, we primarily reach our conclusions on the basis of the gKH Hamiltonian, Eq. 2. However, in SI Appendix we reproduce the main findings with the full two-orbital Hubbard model Eq. 1. All Hamiltonians are diagonalized via the single-site density matrix renormalization group (DMRG) method (44–47) (with up to 1,200 states kept), where the dynamical correlation functions are obtained using the dynamical-DMRG technique (48–50), i.e., calculating spectral functions directly in frequency space with the correction-vector method (51) and Krylov decomposition (50). Open boundary conditions are assumed.

Let us briefly describe the several magnetic phases of the OSMP, where the phase reported here is in yellow in Fig. 1B. For details we refer the interested reader to ref. 28. In Fig. 1B we present a sketch of the interaction-filling ($U$–$n_{\mathrm{H}/\mathrm{K}}$) phase diagram: 1) At $U<W$ the ground state is a paramagnetic metal. 2) For $U\gtrsim W$ the system enters the OSMP with coexisting metallic and Mott-insulating bands. 3) For sufficiently large values of interaction $U\gg W$ the system is in a FM state for all fillings. 4) When $U\sim \mathcal{O}\left(W\right)$, namely when all energy scales compete, the system is primarily in the so-called block–magnetic state. Depending on the filling of the itinerant band, the spins form various sizes of AFM-coupled FM spin islands. An important special case, found experimentally in ${\mathrm{B}\mathrm{a}\mathrm{F}\mathrm{e}}_2{\mathrm{S}\mathrm{e}}_3$, is the $n_{\mathrm{K}}=1/2$ ($n_{\mathrm{H}}=3/2+1$) filling where spins form a $\uparrow \uparrow \downarrow \downarrow \uparrow \uparrow \downarrow \downarrow$ pattern along the legs, the $\pi /2$-block magnetic state.

The Fourier analysis of the perfect step-function pattern of the form $\uparrow \uparrow \downarrow \downarrow \uparrow \uparrow \downarrow \downarrow$ yields only one Fourier mode at $\pi /2$, as shown in Fig. 2A. Our explicit DMRG calculations of the gKH spin structure factor $S\left(q\right)=\langle {\mathbf{S}}_q\cdot {\mathbf{S}}_{-q}\rangle$ [${\mathbf{S}}_q=\left(1/\sqrt{L}\right){\sum }_{\ell }\exp \left(-iq\ell \right){\mathbf{S}}_{\ell }$ with ${\mathbf{S}}_{\ell }={\sum }_{\gamma }{\mathbf{S}}_{\gamma ,\ell }$] at $U\sim W$ confirms that the dominant contribution to the magnetic ordering indeed originates in a $\pi /2$-block pattern. Equivalently, the analysis of a perfect $\uparrow \uparrow \uparrow \downarrow \downarrow \downarrow$ pattern yields now two equal-height Fourier components at $\pi /3$ and $\pi$. Calculations within gKH (Fig. 2A) display a large, dominant peak at $\pi /3$ but also a smaller one at $\pi$. The small weight of the latter can be explained by the emergence of optical modes of localized spin excitations present within multiorbital systems (27). These modes manifest in $S\left(q\right)$ as a finite offset at large values of the wavevector (Fig. 2B). Nevertheless, the dominant shape of the block magnetism of the OSMP can be qualitatively described by idealized Heaviside-like patterns. The small size of our blocks shows that they cannot be characterized as domain walls that are usually separated by much larger distances, but as a different magnetic order.

Fig. 2.

Fourier decomposition of the spin order. (A) Analysis of classical Heaviside-like block patterns (shaded area), i.e., $\uparrow \uparrow \downarrow \downarrow$ and $\uparrow \uparrow \uparrow \downarrow \downarrow \downarrow$. Line dots represent the corresponding calculations using the generalized Kondo–Heisenberg model at $U=W$, $n_{\mathrm{H}}=1/2$, and $n_{\mathrm{H}}=1/3$ ($\pi /2$ and $\pi /3$ blocks, respectively). (B) Shaded areas represent 1) perfect standard spiral with one Fourier mode and 2) our perfect block–spiral spin pattern displaying two modes. Line dots are DMRG results for the block–spiral order at $U/W=1.95$ and $n_{\mathrm{H}}=0.5$. All results have pitch angle $\theta \simeq 1/3$. The shoulder in the DMRG data is the fingerprint of the block–spiral. Note that in all panels we have broadened the $\delta$ peaks of classical solutions for clarity.

Block–Spiral State and Chirality Correlation.

In between the block and FM phases, we discovered a region where the spin static structure factor $S\left(q\right)$ has its maximum $q_{max}$ at incommensurate values of the wavevector $q$. In Fig. 3 A and D we explicitly show the interaction $U$ dependence of $S\left(q\right)$ in this region (28) for $n_{\mathrm{K}}=1/2$ and $n_{\mathrm{K}}=1/3$, respectively. It is evident from the presented results that the maximum of the spin structure factor continuously changes with interaction $U$ interpolating between block magnetism at $U\sim W$ and the FM state at $U\gg W$ (e.g., at fixed $n_{\mathrm{K}}=1/2$, from $q_{max}=\pi /2$ to $q_{max}=0$). The real-space spin–spin correlation functions $\langle {\mathbf{S}}_{L/2}\cdot {\mathbf{S}}_{\ell }\rangle$ (Fig. 3 B and E) reveal an oscillatory structure throughout the chain, with period $\theta =q_{max}$, decaying in amplitude at large spatial separations, within the incommensurate region in the phase diagram. Such behavior may naively suggest a spin-density wave. For the latter, the Fourier transform of the spin correlations should yield only one Fourier mode, due to $\langle {\mathbf{S}}_{L/2}\cdot {\mathbf{S}}_{\ell }\rangle \propto \cos \left(q_{max}\ell \right)$. As we will show, this interpretation is incorrect and the competing interactions present in OSMP systems, as well as the location of this phase sandwiched between block and FM states, lead to a block–spiral spin state.

Fig. 3.

Interaction $U/W$ dependence of correlation functions. (A and D) Static spin structure factor $\langle {\mathbf{S}}_q\cdot {\mathbf{S}}_{-q}\rangle$. (B and E) Real-space spin–spin correlation function $\langle {\mathbf{S}}_{L/2}\cdot {\mathbf{S}}_{\ell }\rangle$. (C and F) Nearest-neighbor chirality correlation function $\langle {\mathit{\kappa }}_{L/2}^1\cdot {\mathit{\kappa }}_{\ell }^1\rangle$. A–C represent results for filling $n_{\mathrm{K}}=1/2$, while D–F depict results for $n_{\mathrm{K}}=1/3$. All results were calculated using the DMRG and the generalized Kondo–Heisenberg model on $L=48$ sites.

To better investigate the magnetic structure of the spiral within the OSMP we focus on the chirality correlation function $\langle {\mathit{\kappa }}_{\ell }^d\cdot {\mathit{\kappa }}_m^d\rangle$ (2, 3, 52, 53), where

$${\mathit{\kappa }}_{\ell }^d={\mathbf{S}}_{\ell }\times {\mathbf{S}}_{\ell +d}$$ [3]

represents the vector product of two spin operators separated by a distance $d$ and consequently the angle between them. We stress that in the following we consider the total spin at each site ${\mathbf{S}}_{\ell }={\sum }_{\gamma }{\mathbf{S}}_{\gamma ,\ell }$. In the generic case of AFM or FM order, and also in the OSMP block phase, the chirality correlation function vanishes, $\langle {\mathit{\kappa }}_{\ell }^d\cdot {\mathit{\kappa }}_m^d\rangle =0$, since the spins are collinear. On the other hand, consecutive $\langle {\mathit{\kappa }}_{\ell }^d\cdot {\mathit{\kappa }}_m^d\rangle \hspace{0.17em}\ne \hspace{0.17em}0$ indicates a nontrivial spiral order.

In Fig. 3 C and F we present the interaction $U$ dependence of the nearest-neighbor, $d=1$, chirality correlation function $\langle {\mathit{\kappa }}_{\ell }^1\cdot {\mathit{\kappa }}_m^1\rangle$ of the gKH model. It is evident from the presented results that between the block and FM phases the chirality acquires finite values. In addition, the spatial structure of $\langle {\mathit{\kappa }}_{\ell }^1\cdot {\mathit{\kappa }}_m^1\rangle$ displays a clear zig-zag–like pattern. To better investigate the spiral internal structure, in Fig. 4 we present the dependence of the chirality correlation with the distance $d$ between spins. In SI Appendix we provide the full interaction $U$ dependence of the next–nearest-neighbor, $d=2$, chirality correlation function. Here, as illustration we focus on the representative cases of $U/W=2.0$ for $n_{\mathrm{K}}=1/2$ and $U/W=1.2$ for $n_{\mathrm{K}}=1/3$.

Fig. 4.

Chiral correlation function $\langle {\mathit{\kappa }}_{\ell }^d\cdot {\mathit{\kappa }}_m^d\rangle$. (A) Dependence of chirality ${\mathit{\kappa }}_{\ell }^d={\mathbf{S}}_{\ell }\times {\mathbf{S}}_{\ell +d}$ on distance $d$ between spins. Results were calculated using the generalized Kondo–Heisenberg model, $L=48$, $n_{\mathrm{K}}=1/2$, and $U/W=2.0$. (B) Same as A but for $n_{\mathrm{K}}=1/3$ and $U/W=1.2$. In both A and B arrows represent schematics of the block order for given filling.

As already mentioned, the nearest-neighbor ($d=1$) chiral correlation for both considered fillings contains additional patterns modulating the usual decay. Specifically, for $n_{\mathrm{K}}=1/2$ ($n_{\mathrm{K}}=1/3$) the correlation function oscillates every two (three) sites. Interestingly, these patterns change their nature when the next–nearest-neighbor ($d=2$) chirality is considered: 1) The values of these chiralities increase $\langle {\mathit{\kappa }}_{\ell }^2\cdot {\mathit{\kappa }}_m^2\rangle >\langle {\mathit{\kappa }}_{\ell }^1\cdot {\mathit{\kappa }}_m^1\rangle$, and 2) for the case of $n_{\mathrm{K}}=1/2$ the $\mathit{\kappa }$ correlation is now a smooth function of distance, while $n_{\mathrm{K}}=1/3$ still exhibits some three-site oscillations. Investigating the next–next-nearest-neighbor case, $d=3$, gives additional information. While for the $n_{\mathrm{K}}=1/2$ filling the $d=3$ correlations are smaller than $d=1$ and $d=2$, for $n_{\mathrm{K}}=1/3$ they are larger and (as for $d=2$ at $n_{\mathrm{K}}=1/2$) they are now a smooth function of distance. This seemingly erratic behavior of $\langle {\mathit{\kappa }}_{\ell }^d\cdot {\mathit{\kappa }}_m^d\rangle$ correlations varying $d$ cannot be simply explained by a mere uniform change of the pitch angle $\theta$. The latter changes only with the interaction $\theta =\theta \left(U\right)$. On the other hand, as evident from Fig. 3 C and F, the internal structure of $\langle {\mathit{\kappa }}_{\ell }^1\cdot {\mathit{\kappa }}_m^1\rangle$ depends only on the electronic filling $n_{\mathrm{K}}$ (see also SI Appendix for $d=2$ results).

To better explain this behavior let us focus on the dimer correlation defined (54) as

$$D_{\pi /2}=\frac{2}{L}\sum _{\ell =L/4}^{3L/4}{\left(-1\right)}^{\ell -1}\langle {\mathbf{S}}_{\ell }\cdot {\mathbf{S}}_{\ell +1}\rangle .$$ [4]

The above operator compares the number of FM and AFM bonds in the bulk of the system (SI Appendix contains the full real-space dependence of $\langle {\mathbf{S}}_{\ell }\cdot {\mathbf{S}}_{\ell +1}\rangle$). For true FM or AFM ordered states, $|\uparrow \uparrow \uparrow \uparrow \dots \rangle$ or $|\uparrow \downarrow \uparrow \downarrow \dots \rangle$, respectively, each nearest-neighbor bond has the same sign: positive for FM $\uparrow \uparrow$ and negative for AFM $\uparrow \downarrow$. Consequently, $D_{\pi /2}=0$. On the other hand, in the $\pi /2$-block state, $|\uparrow \uparrow \downarrow \downarrow \dots \rangle$, the FM and AFM bonds alter in staggered fashion, rendering $D_{\pi /2}\hspace{0.17em}\ne \hspace{0.17em}0$. In Fig. 5 we present the interaction $U$ dependence of $D_{\pi /2}$ for $n_{\mathrm{K}}=1/2$. Furthermore, in Fig. 5 we present also the wavevector where the static structure factor is maximized, $q_{max}$, and the value of the ${\stackrel{\sim }{\kappa }}_{L/3}^d={\langle {\mathit{\kappa }}_{\ell }^d\cdot {\mathit{\kappa }}_m^d\rangle }_{|\ell -m|=L/3}$ correlator for $d=1$ and $d=2$. Starting in the paramagnet at small $U$ both $D_{\pi /2}$ and ${\stackrel{\sim }{\kappa }}_{L/3}^d$ vanish, with $q_{max}=\pi$ just depicting the usual short-range staggered correlations of weak-$U$ physics. In the opposite limit of strong interaction, $U\gg W$, $D_{\pi /2}={\stackrel{\sim }{\kappa }}_{L/3}^d=0$ as well, a consequence of a simple FM state with $q_{max}=0$. In the most interesting case of competing interaction $U\sim W$, the dimer correlation $D_{\pi /2}$ acquires a finite value maximized at $U\simeq W$. The latter reflects a perfect $\pi /2$-block magnetic state. Interestingly, one can observe a continuous transition of $D_{\pi /2}$ between the block and FM phases in the region where a finite chirality ${\stackrel{\sim }{\kappa }}_{L/3}^d\hspace{0.17em}\ne \hspace{0.17em}0$ was found and where $q_{max}$ takes incommensurate values.

Fig. 5.

Phase diagram varying the interaction $U/W$. Presented are 1) maximum of static spin structure factor $q_{max}$ (squares), 2) nearest-neighbor ($d=1$) and next–nearest-neighbor ($d=2$) chirality ${\stackrel{\sim }{\kappa }}_{L/3}^d$ (open and solid circles, respectively), and 3) dimer correlation $D_{\pi /2}$ (diamonds). Results were calculated using the generalized Kondo–Heisenberg model, $L=48$ sites, and $n_{\mathrm{K}}=1/2$.

On the basis of the above results, a coherent picture emerges explaining the nature of the magnetic state between the block and FM limits. Consider first filling $n_{\mathrm{K}}=1/2$. At $U\simeq W$ the ground state is a block–magnetic phase, where two-site FM islands (blocks) are AFM coupled. Increasing the interaction $U$, the spins start to rotate with respect to each other, inducing finite $\langle {\mathit{\kappa }}_{\ell }^d\cdot {\mathit{\kappa }}_m^d\rangle$ correlations. Remarkably, during the rotation the overall FM islands-nature of the state is preserved, yielding a finite $D_{\pi /2}\ne 0$ all of the way to the FM state at $U\gg W$. Such an unexpected scenario is also encoded in the inequalities ${\stackrel{\sim }{\kappa }}_{L/3}^2>{\stackrel{\sim }{\kappa }}_{L/3}^1$ and ${\stackrel{\sim }{\kappa }}_{L/3}^2>{\stackrel{\sim }{\kappa }}_{L/3}^3$ observed in Fig. 4A. This is qualitatively different from a standard spiral state where the spin rotates from site to site (Fig. 1A, Top sketch). In our case, instead, the spiral is made of individual blocks, and it is the entire block that rotates from block to block (Fig. 1A, Middle sketch). Furthermore, the detailed analysis of $S\left(q\right)$ reveals a small secondary Fourier mode at $\pi -q_{max}$ (Fig. 2B). As already mentioned, the long-wavelength components of $S\left(q\right)$ are hidden behind the OSMP optical mode contribution (27) and a detailed analysis of experimental dynamical spin spectra will be needed to fully reveal the presence of our predicted block–spiral states. Because these additional modes are not consistent with a mere standard spiral but instead appear in the Fourier analysis of the perfectly sharp block $\uparrow \uparrow \downarrow \downarrow$ state modulated by the spiral $\cos$-like component and are also consistent with our analysis of $\stackrel{\sim }{\kappa }$, they represent the fingerprints of our block–spiral states.

Such a block–spiral state can also be observed at filling $n_{\mathrm{K}}=1/3$. Here, $\pi /3$ blocks of three sites $\uparrow \uparrow \uparrow \downarrow \downarrow \downarrow$ develop a finite dimer correlation of the form $D_{\pi /3}\propto {\sum }_{\ell }f\left(\ell \right)\langle {\mathbf{S}}_{\ell }\cdot {\mathbf{S}}_{\ell +1}\rangle$, where $f\left(\ell \right)$ accounts for the specific form of the bond sign pattern, i.e., $\left\{\mathrm{1,1},-\mathrm{1,1,1},-1,\dots \right\}$ (SI Appendix). A finite $D_{\pi /3}$ together with ${\stackrel{\sim }{\kappa }}_{L/3}^3>{\stackrel{\sim }{\kappa }}_{L/3}^2>{\stackrel{\sim }{\kappa }}_{L/3}^1$ is compatible with a spiral state of rotating three-site blocks (Fig. 1A, Bottom sketch).

Finally, let us comment on the finite-size dependence of our findings. Analysis of system sizes up to $L=96$ sites (SI Appendix) indicates that the discussed block–spiral states display short-ranged order but with a robust correlation length of $\xi \sim 15$ sites, where $\langle {\mathit{\kappa }}_{\ell }^d\cdot {\mathit{\kappa }}_{\ell +x}^d\rangle \propto \exp \left(-x/\xi \right)$. However, it was argued (2, 55) for the case of the FM long-range Heisenberg model that realistic small SU(2)-breaking anisotropies, often present in real materials, can induce true (quasi–)long-range order in a spiral state. Also, such an anisotropy will choose the plane of rotation of the spiral, i.e., in plane or out of plane with regard to the chain direction. As we argue in the next section, such (frustrated) long-range Heisenberg Hamiltonians can (at least qualitatively) capture the main physics of the block–spiral unveiled here.

We emphasize that the same conclusions are reached in the multiorbital Hubbard model, although because the effort is much more computationally demanding, it was limited to special cases. In SI Appendix we present results for $\langle {\mathit{\kappa }}_{\ell }^d\cdot {\mathit{\kappa }}_m^d\rangle$ obtained with the full two-orbital Hubbard model Eq. 1 and also the incommensurability of $S\left(q\right)$ for a three-orbital Hubbard model.

Quasiparticle Excitations.

A distinctive feature of the OSMP is the coexistence of localized electrons (spins in an insulating band) and itinerant electrons (a metallic band). In the block–magnetic phase at $U\simeq W$ it was previously shown (39)—for the three-orbital Hubbard model—that the density of states (DOS) at the Fermi level ${ϵ}_{\mathrm{F}}$ is reduced, indicating a pseudogap-like behavior. Our calculations of the single-particle spectral function $A\left(q,\omega \right)$ and $\mathrm{D}\mathrm{O}\mathrm{S}\left(\omega \right)$ (SI Appendix) using the gKH model are presented in Fig. 6A and confirm this picture. They also show the strength of our effective Hamiltonian: The behavior of the gKH model perfectly matches the $\gamma =0$ itinerant orbital of the full two-orbital Hubbard result presented in Fig. 6C. It is worth noting that to properly match the electron and hole parts of $A\left(k,\omega \right)$ between the models we exploit the particle–hole symmetry of gKH and present results for $n_{\mathrm{K}}=3/2$ (instead of $n_{\mathrm{K}}=1/2$ for which the spectrum would be simply mirrored, i.e., $\omega \to -\omega$ and $k\to \pi -k$). Also, we reiterate here that although the system is overall metallic in nature, the band structure is vastly different from the simple cosine-like result of $U\to 0$. Distinctive features in $A\left(q,\omega \right)$ at the Fermi vector $k_{\mathrm{F}}$, and a large renormalization of the overall band structure at higher energies, indicate a complex interplay between various degrees of freedom and energy scales.

Fig. 6.

Spectral function $A\left(q,\omega \right)$. (A and B) Block phase ($n_{\mathrm{K}}=3/2,U/W=1.0$) (A) and block–spiral phase ($n_{\mathrm{K}}=3/2,U/W=2.0$) (B) as calculated for the generalized Kondo–Heisenberg model using $L=48$ sites. (C and D) The same results obtained with the two-orbital Hubbard model ($L=48$). (C) Block phase ($n_{\mathrm{H}}=2.50,U/W=1.0$). (D) block–spiral phase ($n_{\mathrm{H}}=2.50,U/W=2.0$). (A–D) Horizontal (vertical) lines depict the Fermi level ${ϵ}_{\mathrm{F}}$ (Fermi wavevector $k_{\mathrm{F}}$), while (i)PES stands for (inverse-) photoemission spectroscopy. (A–D, Right) Density of states $\mathrm{D}\mathrm{O}\mathrm{S}\left(\omega \right)={\sum }_{q}A\left(q,\omega \right)$.

Upon increasing the interaction $U$ and entering the block–spiral region, $A\left(q,\omega \right)$ changes drastically. In Fig. 6B we show representative results for a $\theta /\pi \simeq 0.3$ block–spiral state at $U/W=2$ and $n_{\mathrm{K}}=1/2$. Two conclusions are directly evident from the presented results: 1) The pseudogap at ${ϵ}_{\mathrm{F}}$ is closed, but some additional gaps at higher energies opened. 2) $A\left(q,\omega \right)$ in the vicinity of the Fermi level, $\omega \sim {ϵ}_{\mathrm{F}}$, develops two bands, intersecting at the $q=0$ and $q=\pi$ points, with maximum at $q\simeq \theta /2$. The bands represent two quasiparticles: left and right movers reflecting the two possible rotations of the spirals. It is obvious from the above results that the quasiparticles break the parity symmetry; i.e., going from $q\to -q$ momentum changes the quasiparticle character, as expected for a spiral state. Somewhat surprisingly, $A\left(q,\omega \right)$ does not show any gap as would typically be associated with the finite dimerization $D_{\alpha }$ that we observe. However, it should be noted that for quantum localized $S=1/2$ spins the quarter-filling ($n_{\mathrm{K}}=1/2$) implies a filling of two-fifths of the lower Kondo band (due to the energy difference between local Kondo singlets and triplets). The dimerization gap expected at $\pi /2$ would thus open away from the Fermi level and would consequently not confer substantial energy gain to the electrons. Similarly, no dimerization gap is found in the $n_{\mathrm{K}}\simeq 0.33$ case, i.e., the $\pi /3$-block–spiral (not presented). We thus conclude that quantum fluctuations of the localized and itinerant spins are here strong enough to suppress the dimerization gap. The above conclusions can be also reached from results obtained with the full two-orbital Hubbard model (Fig. 6D).

Discussion and Effective Model

It was previously shown (27) that the frustrated FM-AFM $J_1$-$J_2$ Heisenberg model (with $\left|J_1\right|\sim J_2$) qualitatively captures the physics of the nonspiral block–magnetic state. Here, in Fig. 7A we show that the spin structure factor $S\left(q\right)$ of the block–spiral state can be accurately described by an extension of that model: the frustrated long-range Heisenberg Hamiltonian. Although this phenomenological model is not derived here from the basic Hamiltonians, it accurately reproduces the interaction $U$ dependence throughout the incommensurate region (e.g., compare Figs. 3A and 7B). The intuitive understanding of the origin of the effective spin model is as follows: At $U\sim W$ the system is in a block–magnetic state with quasi–long-range (QLR) spin correlation of $\pi /N_{\mathrm{b}}$ nature, where $N_{\mathrm{b}}$ is the size of the block. We found excellent agreement between the gKH and Heisenberg models with $J_1=-1$, $J_2=1/4$, $J_3=1/10$ (see also Fig. 7A, Inset for the $J_2$-$J_3$ dependence of $q_{max}$). The latter yields $\alpha \simeq 2$ in $\left|J_r\right|\propto 1/r^{\alpha }$. As a consequence, the block–spiral magnetism is described by the class of Haldane–Shastry models (56, 57). In fact, we speculate that due to the oscillating character of the electron-mediated exchange couplings, the spin system which encompasses all phenomena is given by

$$\begin{array}{ll}\hfill H& =\sum _{\ell ,r}\hspace{0.17em}{J}_r\hspace{0.17em}{\mathbf{S}}_{\ell }\cdot {\mathbf{S}}_{\ell +r},\hfill \\ \hfill \text{with}\hspace{1em}{J}_1<0,& \hspace{1em}{J}_2>0,\hspace{1em}{J}_{r>2}=\left|J_1\right|\frac{{\left(-1\right)}^{r-1}}{r^{\alpha }}.\hfill \end{array}$$ [5]

It was shown (58–60) that the above model has (for the zero-magnetization sector, $S_{\mathrm{t}\mathrm{o}\mathrm{t}}^z=0$) a QLR ground state with $\pi /2$ correlations and also can support spiral states. Such frustrated long-range Hamiltonians are not suitable for direct DMRG calculations due to the area law of entanglement. Thus, alternatively in SI Appendix we show small system size Lanczos diagonalization results for Eq. 5. However, as presented in Fig. 7, the first three terms $J_1$-$J_2$-$J_3$ (which can still be computed accurately with DMRG) already give satisfactory results even for the chirality correlator (compare Figs. 3C and 7C).

Fig. 7.

Effective Heisenberg model. (A) Comparison of $S\left(q\right)$ between the gKH model in the spiral state ($L=48,n_{\mathrm{K}}=1/2,U/W=2.0$) and the long-range Heisenberg (lrH) model ($L=48,J_2/\left|J_1\right|=0.25,J_3/\left|J_1\right|=0.1$). Note results are normalized by the magnetic moment $S^2=\mathbf{S}\left(\mathbf{S}+1\right)$; i.e., $S^2=1.375$ and $S^2=3/4$ for gKH and lrH, respectively. (B and C) Static spin structure factor $S\left(q\right)$ (B) and chirality correlation function $\langle {\mathit{\kappa }}_{\ell }^d\cdot {\mathit{\kappa }}_m^d\rangle$ (C) of the $J_1$-$J_2$-$J_3$ Heisenberg model (with ferromagnetic $J_1=-1$), calculated for $L=120$ sites, $J_3/\left|J_1\right|=0.1$, and various $J_2/\left|J_1\right|=0.00,0.05,\dots ,1.00$ (top to bottom). (A, Inset) Position of the maximum of the static structure factor $q_{max}$ of the lrH model as a function of $J_2/\left|J_1\right|$, for $J_3/\left|J_1\right|=0.00,0.05,0.10$, and 0.15.

Finally, we make two additional comments: 1) The scenario (long-range effective spin model) described above goes beyond the discussed nonspiral $\pi /2$-block case. Changes in the magnetization sector of the long-range FM Hamiltonian lead to modifications in the periodicity of the QLR order (61, 62). As a consequence, there are only a few parameters in the effective spin model: the magnetization $S_{\mathrm{t}\mathrm{o}\mathrm{t}}^z$ related to the filling in the original full multiorbital Hamiltonian and $\left\{J_2,\alpha \right\}=f\left(U\right)$ which controls the long-range nature of the system. 2) Since our system is overall metallic (albeit likely a bad metal because of the localized component), one could naively believe that the Ruderman–Kittel–Kasuya–Yosida (RKKY) spin exchange $J_r$ carried by mobile electrons could explain some of our results. However, this is not the case because the nontrivial effect of the interaction $U$—creating a metallic state coexisting with a (quasi)ordered magnetic state—qualitatively modifies the nature of the RKKY interaction.

Conclusion

We have identified the block–spiral magnetism in which FM islands rigidly rotate with respect to each other. We emphasize the crucial role of correlation in this phenomenon. Spiral states are usually a consequence of frustration (geometrical or induced by competing interactions) or of explicit symmetry-breaking terms like Dzyaloshinskii–Moriya couplings. For example, long-period helical spin-density waves were reported (63–66) in many transition-metal compounds and rare-earth magnets. This is also the case for the hexagonal perovskite ${\mathrm{C}\mathrm{s}\mathrm{C}\mathrm{u}\mathrm{C}\mathrm{l}}_3$, where the external magnetic field can induce block-like structure on top of the spiral-like order (67). However, in all of these cases the magnetic structure resembles a domain wall and originates in strong frustration of the (often classical) model itself. On the contrary, in our system we have only nearest-neighbor interactions on a chain geometry, and the SU(2) symmetry is preserved. Instead, the block–spiral state reported here appears as an effect of hidden frustration, i.e., competition between the double-exchange–like mechanism present in multiorbital systems dominated by a robust Hund exchange and the interaction $U$ that governs superexchange tendencies. These nontrivial effects become apparent in the exotic effective (phenomenological) spin model we unveiled: a long-range frustrated Heisenberg model. Within the latter the spin exchange decays slowly with distance, $\sim 1/r^2$, in contrast to the usual RKKY interaction which decays faster. Also, to our knowledge the multiorbital system discussed in this work—as realized in iron-based compounds from the 123 family—could be one of the few, if not the only, known realizations of a Haldane–Shastry-like model with $\alpha =2$.

Furthermore, due to properties unique to the OSMP, primarily the coexistence of metallic and insulating bands, the block–spiral state displays exotic behavior in the electronic degrees of freedom. For example, new quasiparticles appear due to the parity breaking of the spiral state (left and right movers). Similar physics can be found in systems with spin–orbit coupling (68, 69). Here, again, this is an effect of competing energy scales. Another interesting possibility is the existence of multiferroic behavior in our system. It is known (17–21) that in materials such as quasi-1D compounds ${\mathrm{L}\mathrm{i}\mathrm{C}\mathrm{u}}_2{\mathrm{O}}_2$, ${\mathrm{L}\mathrm{i}\mathrm{C}\mathrm{u}\mathrm{V}\mathrm{O}}_2$, or ${\mathrm{P}\mathrm{b}\mathrm{C}\mathrm{u}\mathrm{S}\mathrm{O}}_4$ (OH)₂, the spin spirals drive the system to ferroelectricity. Moreover, the phenomenon described here, robust spiral magnetism without a charge gap, is at the heart of one of the proposed systems where topological Majorana phases can be induced (9–14).

Finally, let us comment on our results from the perspective of the real iron-based materials. As already mentioned, a nontrivial magnetic order, such as spirals, in the vicinity of high critical temperature superconductivity can lead to topological effects. This is the case of the two-dimensional (2D) material ${\mathrm{F}\mathrm{e}\mathrm{T}\mathrm{e}}_{1-x}{\mathrm{S}\mathrm{e}}_x$ where zero-energy vortex bound states (Majorana fermions) have been reported (70–74). Furthermore, similar to our findings, it was argued (75) that the frustrated magnetism of ${\mathrm{F}\mathrm{e}\mathrm{T}\mathrm{e}}_{1-x}{\mathrm{S}\mathrm{e}}_x$ can be captured by a long-range $J_1$-$J_2$-$J_3$ spin Hamiltonian. From this perspective, it seems appropriate to assume that the phenomenon described in our work extends beyond 1D systems. Unfortunately, the lack of sufficiently reliable computational methods to treat 2D quantum models limits our understanding of multiorbital effects in 2D. An intermediate promising route is the low-dimensional ladders from the family of 123 compounds where accurate DMRG calculations are possible. Early density functional theory and Hartree–Fock results suggest that the effects of correlations are important (37, 76) and that noncollinear magnetic order can develop in the ground state (37). The recent proposal of exploring ladder tellurides with a predicted higher value of $U/W$ provides another avenue to consider (76). As a consequence, we encourage crystal growers with expertise in iron-based materials to explore in detail the low-dimensional family of 123 compounds, including doped samples, because our results suggest that exotic physics may come to light.

Data Availability.

Computer codes used in this study are available at https://g1257.github.io/dmrgPlusPlus/. All data discussed in this paper are available to readers at https://g1257.github.io/papers/104/.