Block–Spiral Magnetism: an Exotic Type of Frustrated Order J

Block–spiral magnetism: An exotic type of frustrated order J. Herbrycha,b,c,1 , J. Heverhagend,e, G. Alvarezf,g, M. Daghoferd,e, A. Moreoa,b , and E. Dagottoa,b

aDepartment of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996; bMaterials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831; cDepartment of Theoretical Physics, Faculty of Fundamental Problems of Technology, Wrocław University of Science and Technology, 50-370 Wrocław, Poland; dInstitute for Functional Matter and Quantum Technologies, University of Stuttgart, D-70550 Stuttgart, Germany; eCenter for Integrated Quantum Science and Technology, University of Stuttgart, D-70550 Stuttgart, Germany; fComputational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831; and gCenter for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, TN 37831

Edited by Andrew P. Mackenzie, Max Planck Institute for Chemical Physics of Solids, Dresden, Germany, and accepted by Editorial Board Member Angel Rubio May 15, 2020 (received for review January 20, 2020)

Competing interactions in quantum materials induce exotic states (AFM)-coupled ferromagnetic (FM) islands. The size and shape of matter such as frustrated magnets, an extensive field of research of such islands depend on the electronic filling of the itinerant from both the theoretical and experimental perspectives. Here, we orbitals. This work focuses on two representative cases: 1) the show that competing energy scales present in the low-dimensional π/2-block spin-pattern ↑↑↓↓ and 2) the π/3-block spin-pattern orbital-selective Mott phase (OSMP) induce an exotic magnetic ↑↑↑↓↓↓. Note that these block patterns are not spin-density order, never reported before. Earlier neutron-scattering experi- waves: The local expectation values of spin operators yield uni- ments on iron-based 123 ladder materials, where OSMP is relevant, form magnetization throughout the system, unlike a spin wave already confirmed our previous theoretical prediction of block mag- that would have peaks and valleys. Our study, on the other netism (magnetic order of the form ↑↑↓↓). Now we argue that hand, indicates a clear block structure with spins of the same another phase can be stabilized in multiorbital Hubbard models, magnitude at each site (26–28). Moreover, exact diagonaliza- the block–spiral state. In this state, the magnetic islands form a tion results on small lattices (27) indicate that the block-OSMP spiral propagating through the chain but with the blocks maintain- ground state has a large overlap (at least 50%) with a state of ing their identity, namely rigidly rotating. The block–spiral state is the form | ↑↑↓↓i − | ↓↓↑↑i (as exemplified for the π/2 block). As stabilized without any apparent frustration, the common avenue a consequence, our states can be viewed as Néel-like states of an to generate spiral arrangements in multiferroics. By examining enlarged magnetic unit cell. the behavior of the electronic degrees of freedom, parity-breaking The OSMP was shown (22) to be relevant for the low- quasiparticles are revealed. Finally, a simple phenomenological dimensional family of iron-based ladders, the so-called 123 fam- model that accurately captures the macroscopic spin spiral arrange- ily AFe2X3, where A = Ba, K, Rb is an alkaline earth metal and ment is also introduced, and fingerprints for the neutron-scattering X = S, Se is a chalcogen. From the magnetism perspective, two experimental detection are provided. phases were experimentally reported: 1) For BaFe2Se3 inelastic neutron scattering (INS) identified (29) a 2×2 block–magnetic multiorbital Hubbard model | frustrated magnetism | phase in a ↑↑↓↓ pattern along the legs. Neutron diffraction Fe-based superconductors

rustrated magnetism is one of the main areas of research Significance Fin contemporary condensed-matter physics. In the generic scenario, magnetic frustration emerges from the failure of the Magnetic frustration in spin or electronic models emerges system to fulfill simultaneously conflicting local requirements. from the failure of the system to fulfill simultaneously con- As a consequence, complex spin patterns can develop from geo- flicting local requirements. The latter typically arise from the metrical frustration (as in triangular, Kagome, or pyrochlore lattice geometry or are induced by special spin–spin inter- lattices) or from special spin–spin interactions (long-range actions in the system at various distances. Here we show exchange, Dzyaloshinskii–Moriya coupling, Kitaev model, spin– that the competing energy scales of the seemingly nonfrus- orbit effects, and others). In real materials both scenarios often trated orbital-selective Mott phase of the low-dimensional coexist. Competing mechanisms can lead to interesting phenom- multiorbital Hubbard model can originate a “block–chiral ena, such as spiral order (1–3), spin ice (4), skyrmions (5), spin magnetism,” i.e., a state with rigidly rotating spin–magnetic liquids (6, 7), and resonating valence bond states (8). Also, the islands. Furthermore, we show how such an exotic spin state electronic properties of such systems are of much interest: It influences the electronic properties of the system, revealing was shown that the interplay of a spiral state on a metallic parity-breaking quasiparticles. host can support Majorana fermions (9–14) and can also induce multiferroicity (15–21). Author contributions: J. Herbrych, M.D., A.M., and E.D. designed research; J. Herbrych and J. Heverhagen performed research; J. Herbrych, G.A., M.D., A.M., and E.D. con- Another example of competing interactions is the orbital- tributed new reagents/analytic tools; J. Herbrych and J. Heverhagen analyzed data; and selective Mott phase (OSMP) (22, 23). In multiorbital systems, J. Herbrych, J. Heverhagen, G.A., M.D., A.M., and E.D. wrote the paper.y this effect can lead to the selective localization of electrons The authors declare no competing interest.y on some orbitals. The latter coexist with itinerant bands of This article is a PNAS Direct Submission. A.P.M. is a guest editor invited by the Editorial mobile electrons. This unique mixture of localized and itiner- Board.y ant components in multiorbital systems could be responsible for Published under the PNAS license.y the (bad) metallic behavior of the parent compounds of iron- Data deposition: Computer codes used in this study are available at https://g1257.github. based superconductors. This is in stark contrast to cuprates, io/dmrgPlusPlus/. All data discussed in this paper are available to readers at https:// usually described by the single-band Hubbard model, where par- g1257.github.io/papers/104/.y ent materials are insulators. Furthermore, the OSMP can also 1 To whom correspondence may be addressed. Email: [email protected] host exotic magnetic phases. It was shown that the competition This article contains supporting information online at https://www.pnas.org/lookup/suppl/ between Hund and Hubbard interactions can stabilize unex- doi:10.1073/pnas.2001141117/-/DCSupplemental.y pected block magnetism (24–28), namely antiferromagnetically First published June 29, 2020.

16226–16233 | PNAS | July 14, 2020 | vol. 117 | no. 28 www.pnas.org/cgi/doi/10.1073/pnas.2001141117 Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 ebyhe al. et Herbrych Inter- (B) respectively. sites, three and action two of block–spirals and rotation spins (A) phase. 1. Fig. sym- parity break that quasiparticles can by properties described electronic accurately the be Moreover, We findings. the state. capture our this of that essence models missed phenomenological have simpler Sim- would two used: propose likely are techniques tools biased computational pler powerful when consequences only exotic whose appear not and system, level, the Hamiltonian in the Hub- at frustration obvious hidden a by is opposite There caused and interactions. bard systems, tendencies, multiorbital superexchange in tendencies, antiferromagnetic physics ferromagnetic Hund in by originates induced the order in spin present and scales This energy model OSMP. competing the subtly in of consequence frustration a apparent is any pat- without magnetic appears block–spiral tern the Furthermore, stabilization. their in scales 340 rare-earth length larger, the the block–spiral However, in The walls. states 1A. domain with Fig. similarities in have material panels report rotate, three rigidly we the blocks states the in state illustrated spin-to- block–spiral observe as the can in one rotation, where order. spin sys- noncollinear spirals develop the standard to in from and Different tendencies blocks magnetic simultaneous form of to state tem consequence a a Such state. as block–spiral exotic arises an report we namely, lized; effective low-energy model. Kondo–Heisenberg its generalized by the AFM also multiorbital (28), the description and and by (26) rungs captured are Hamiltonian along phases Hubbard these FM for of blocks, Both hand, legs. 2×1 along other (31–33) the reported On conclu- INS same 2) the (30). yield relaxation sion spin muon and measurements SP(aaantc region. 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PHYSICS 2 where K = 4t11/U . It is worth noting that due to particle–hole A symmetry, the electronic filling relation between the two-orbital Hubbard (nH) and gKH (nK) models is nK = 3 − nH. 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 B 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 –nH/K) phase diagram: 1) At U < W the ground state is a paramagnetic metal. 2) For U & W the system enters the OSMP with coexisting metallic and Mott-insulating bands. 3) For sufficiently large values of interaction U W the system is in a FM state for all fillings. Fig. 2. Fourier decomposition of the spin order. (A) Analysis of classical 4) When U ∼ O(W ), namely when all energy scales compete, Heaviside-like block patterns (shaded area), i.e., ↑↑↓↓ and ↑↑↑↓↓↓. Line the system is primarily in the so-called block–magnetic state. dots represent the corresponding calculations using the generalized Kondo– Heisenberg model at U = W, nH = 1/2, and nH = 1/3 (π/2 and π/3 blocks, Depending on the filling of the itinerant band, the spins form respectively). (B) Shaded areas represent 1) perfect standard spiral with one various sizes of AFM-coupled FM spin islands. An important Fourier mode and 2) our perfect block–spiral spin pattern displaying two special case, found experimentally in BaFe2Se3, is the nK = 1/2 modes. Line dots are DMRG results for the block–spiral order at U/W = 1.95 (nH = 3/2 + 1) filling where spins form a ↑↑↓↓↑↑↓↓ pattern along and nH = 0.5. All results have pitch angle θ ' 1/3. The shoulder in the DMRG the legs, the π/2-block magnetic state. data is the fingerprint of the block–spiral. Note that in all panels we have The Fourier analysis of the perfect step-function pattern broadened the δ peaks of classical solutions for clarity. of the form ↑↑↓↓↑↑↓↓ yields only one Fourier mode at π/2, as shown in Fig. 2A. Our explicit DMRG calcula- sandwiched between block and FM states, lead to a block–spiral tions√ of the gKH spin structure factor S(q) = hSq · S−q i [Sq = P P spin state. (1/ L) exp(−iq`)S` with S` = Sγ,`] at U ∼ W confirms ` γ To better investigate the magnetic structure of the spiral that the dominant contribution to the magnetic ordering indeed within the OSMP we focus on the chirality correlation function originates in a π/2-block pattern. Equivalently, the analysis of a d d perfect ↑↑↑↓↓↓ pattern yields now two equal-height Fourier com- hκ` · κm i (2, 3, 52, 53), where ponents at π/3 and π. Calculations within gKH (Fig. 2A) display d a large, dominant peak at π/3 but also a smaller one at π. The κ` = S` × S`+d [3] small weight of the latter can be explained by the emergence of represents the vector product of two spin operators separated by optical modes of localized spin excitations present within mul- a distance d and consequently the angle between them. We stress tiorbital systems (27). These modes manifest in S(q) as a finite that in the following we consider the total spin at each site S` = offset at large values of the wavevector (Fig. 2B). Nevertheless, P S . In the generic case of AFM or FM order, and also in the the dominant shape of the block magnetism of the OSMP can γ γ,` OSMP block phase, the chirality correlation function vanishes, be qualitatively described by idealized Heaviside-like patterns. d d The small size of our blocks shows that they cannot be character- hκ` · κm i = 0, since the spins are collinear. On the other hand, d d ized as domain walls that are usually separated by much larger consecutive hκ` · κm i 6= 0 indicates a nontrivial spiral order. distances, but as a different magnetic order. In Fig. 3 C and F we present the interaction U dependence of the nearest-neighbor, d = 1, chirality correlation function 1 1 Block–Spiral State and Chirality Correlation. In between the block hκ` · κm i of the gKH model. It is evident from the presented and FM phases, we discovered a region where the spin static results that between the block and FM phases the chiral- structure factor S(q) has its maximum qmax at incommensu- ity acquires finite values. In addition, the spatial structure of 1 1 rate values of the wavevector q. In Fig. 3 A and D we explicitly hκ` · κm i displays a clear zig-zag–like pattern. To better investi- show the interaction U dependence of S(q) in this region (28) gate the spiral internal structure, in Fig. 4 we present the depen- for nK = 1/2 and nK = 1/3, respectively. It is evident from the dence of the chirality correlation with the distance d between presented results that the maximum of the spin structure factor spins. In SI Appendix we provide the full interaction U depen- continuously changes with interaction U interpolating between dence of the next–nearest-neighbor, d = 2, chirality correlation block magnetism at U ∼ W and the FM state at U W (e.g., function. Here, as illustration we focus on the representative at fixed nK = 1/2, from qmax = π/2 to qmax = 0). The real-space cases of U /W = 2.0 for nK = 1/2 and U /W = 1.2 for nK = 1/3. spin–spin correlation functions hSL/2 · Sì (Fig. 3 B and E) reveal As already mentioned, the nearest-neighbor (d = 1) chiral cor- an oscillatory structure throughout the chain, with period θ = relation for both considered fillings contains additional patterns qmax, decaying in amplitude at large spatial separations, within modulating the usual decay. Specifically, for nK = 1/2 (nK = the incommensurate region in the phase diagram. Such behav- 1/3) the correlation function oscillates every two (three) sites. ior may naively suggest a spin-density wave. For the latter, the Interestingly, these patterns change their nature when the next– Fourier transform of the spin correlations should yield only one nearest-neighbor (d = 2) chirality is considered: 1) The values 2 2 1 1 Fourier mode, due to hSL/2 · Sì ∝ cos(qmax`). As we will show, of these chiralities increase hκ` · κm i > hκ` · κm i, and 2) for this interpretation is incorrect and the competing interactions the case of nK = 1/2 the κ correlation is now a smooth func- present in OSMP systems, as well as the location of this phase tion of distance, while nK = 1/3 still exhibits some three-site

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PHYSICS d d cial cases. In SI Appendix we present results for hκ` · κm i obtained with the full two-orbital Hubbard model Eq. 1 and also the incommensurability of S(q) 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 ' W it was previously shown (39)—for the three- orbital Hubbard model—that the density of states (DOS) at the Fermi level F is reduced, indicating a pseudogap-like behavior. Our calculations of the single-particle spectral function A(q, ω) and DOS(ω) (SI Appendix) using the gKH model are presented in Fig. 6A and confirm this picture. They also show the strength Fig. 5. Phase diagram varying the interaction U/W. Presented are 1) max- of our effective Hamiltonian: The behavior of the gKH model imum of static spin structure factor (squares), 2) nearest-neighbor qmax γ = 0 (d = 1) and next–nearest-neighbor (d = 2) chirality κd (open and solid cir- perfectly matches the itinerant orbital of the full two- eL/3 orbital Hubbard result presented in Fig. 6C. It is worth noting cles, respectively), and 3) dimer correlation Dπ/2 (diamonds). Results were calculated using the generalized Kondo–Heisenberg model, L = 48 sites, and that to properly match the electron and hole parts of A(k, ω) nK = 1/2. between the models we exploit the particle–hole symmetry of gKH and present results for nK = 3/2 (instead of nK = 1/2 for which the spectrum would be simply mirrored, i.e., ω → −ω and k → π − k). Also, we reiterate here that although the system is hκd · κd i ` m correlations. Remarkably, during the rotation the overall metallic in nature, the band structure is vastly different overall FM islands-nature of the state is preserved, yielding from the simple cosine-like result of U → 0. Distinctive features a finite Dπ/2 6 =0 all of the way to the FM state at U W . in A(q, ω) at the Fermi vector kF, and a large renormalization Such an unexpected scenario is also encoded in the inequali- of the overall band structure at higher energies, indicate a com- κ2 > κ1 κ2 > κ3 ties eL/3 eL/3 and eL/3 eL/3 observed in Fig. 4A. This is plex interplay between various degrees of freedom and energy qualitatively different from a standard spiral state where the scales. spin rotates from site to site (Fig. 1A, Top sketch). In our case, Upon increasing the interaction U and entering the block– instead, the spiral is made of individual blocks, and it is the spiral region, A(q, ω) changes drastically. In Fig. 6B we show rep- entire block that rotates from block to block (Fig. 1A, Middle resentative results for a θ/π ' 0.3 block–spiral state at U /W = 2 S(q) sketch). Furthermore, the detailed analysis of reveals a and nK = 1/2. Two conclusions are directly evident from the π − q B small secondary Fourier mode at max (Fig. 2 ). As already presented results: 1) The pseudogap at F is closed, but some mentioned, the long-wavelength components of S(q) are hid- additional gaps at higher energies opened. 2) A(q, ω) in the vicin- den behind the OSMP optical mode contribution (27) and a ity of the Fermi level, ω ∼ F, develops two bands, intersecting at detailed analysis of experimental dynamical spin spectra will the q = 0 and q = π points, with maximum at q ' θ/2. The bands be needed to fully reveal the presence of our predicted block– represent two quasiparticles: left and right movers reflecting the spiral states. Because these additional modes are not consistent two possible rotations of the spirals. It is obvious from the above with a mere standard spiral but instead appear in the Fourier results that the quasiparticles break the parity symmetry; i.e., analysis of the perfectly sharp block ↑↑↓↓ state modulated by going from q → −q momentum changes the quasiparticle charac- the spiral cos-like component and are also consistent with our ter, as expected for a spiral state. Somewhat surprisingly, A(q, ω) analysis of κe, they represent the fingerprints of our block–spiral does not show any gap as would typically be associated with the states. finite dimerization Dα that we observe. However, it should be Such a block–spiral state can also be observed at filling noted that for quantum localized S = 1/2 spins the quarter-filling nK = 1/3. Here, π/3 blocks of three sites ↑↑↑↓↓↓ develop a P (nK = 1/2) implies a filling of two-fifths of the lower Kondo band finite dimer correlation of the form Dπ/3 ∝ ` f (`)hS` · S`+1i, (due to the energy difference between local Kondo singlets and where f (`) accounts for the specific form of the bond sign triplets). The dimerization gap expected at π/2 would thus open pattern, i.e., {1, 1, −1, 1, 1, −1, ... } (SI Appendix). A finite 3 2 1 away from the Fermi level and would consequently not confer Dπ/3 together with κeL/3 > κeL/3 > κeL/3 is compatible with a substantial energy gain to the electrons. Similarly, no dimeriza- spiral state of rotating three-site blocks (Fig. 1A, Bottom tion gap is found in the nK ' 0.33 case, i.e., the π/3-block–spiral sketch). (not presented). We thus conclude that quantum fluctuations of Finally, let us comment on the finite-size dependence of our the localized and itinerant spins are here strong enough to sup- findings. Analysis of system sizes up to L = 96 sites (SI Appendix) press the dimerization gap. The above conclusions can be also indicates that the discussed block–spiral states display short- reached from results obtained with the full two-orbital Hubbard ranged order but with a robust correlation length of ξ ∼ 15 model (Fig. 6D). d d sites, where hκ` · κ`+x i ∝ exp(−x/ξ). However, it was argued (2, 55) for the case of the FM long-range Heisenberg model Discussion and Effective Model that realistic small SU(2)-breaking anisotropies, often present It was previously shown (27) that the frustrated FM-AFM in real materials, can induce true (quasi–)long-range order in J1-J2 Heisenberg model (with |J1| ∼ J2) qualitatively captures a spiral state. Also, such an anisotropy will choose the plane the physics of the nonspiral block–magnetic state. Here, in of rotation of the spiral, i.e., in plane or out of plane with Fig. 7A we show that the spin structure factor S(q) of the regard to the chain direction. As we argue in the next section, block–spiral state can be accurately described by an extension of such (frustrated) long-range Heisenberg Hamiltonians can (at that model: the frustrated long-range Heisenberg Hamiltonian. least qualitatively) capture the main physics of the block–spiral Although this phenomenological model is not derived here from unveiled here. the basic Hamiltonians, it accurately reproduces the interac- We emphasize that the same conclusions are reached in the tion U dependence throughout the incommensurate region (e.g., multiorbital Hubbard model, although because the effort is compare Figs. 3A and 7B). The intuitive understanding of the ori- much more computationally demanding, it was limited to spe- gin of the effective spin model is as follows: At U ∼ W the system

16230 | www.pnas.org/cgi/doi/10.1073/pnas.2001141117 Herbrych et al. Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 ftelclzdcmoet,oecudnieybleethat believe naively al. et Herbrych could one because Since metal component), 2) bad a localized system. likely the the (albeit metallic of of overall nature is system long-range our and effec- the Hamiltonian the controls multiorbital in which full parameters original few magnetization As the the a 62). model: to (61, only spin order lead are tive QLR there Hamiltonian the consequence, of FM a periodicity long-range the the in modifications of sector beyond netization nonspiral goes above discussed described the model) spin effective (long-range the 3C for Figs. even (compare results correlator satisfactory chirality give already DMRG) with rately terms three Eq. first for results ization nopse l hnmn sgvnby given of is character phenomena which oscillating all system the encompasses spin to the couplings, due exchange (56, that electron-mediated models speculate the Haldane–Shastry we of fact, class In the 57). by described is netism lc.W on xeln gemn ewe h K and gKH α the between 7A, (QLR) Fig. agreement with quasi–long-range excellent models Heisenberg with found We state block. of block–magnetic correlation a spin in is (A–D, spectroscopy. photoemission (inverse-) for stands (i)PES (n using model Kondo–Heisenberg generalized uain u oteae a fetnlmn.Tu,alternatively Thus, entanglement. of cal- in law DMRG area frustrated the direct to Such for due suitable culations states. zero- not spiral are the support Hamiltonians (for long-range can has also model and above correlations the sector, that (58–60) magnetization shown was It 6. Fig. H ' ial,w aetoadtoa omns )Tescenario The 1) comments: additional two make we Finally, IAppendix SI = 2 2.50, in pcrlfunction Spectral Inset with | U J /W r ∝ | o the for = J 1/r lc–prlpae(n phase block–spiral (D) 1.0). 1 eso ml ytmsz aco diagonal- Lanczos size system small show we J < 1 α H -J N π/ J 0, sacneune h lc–prlmag- block–spiral the consequence, a As . 2 A(q, 2 -J = S -J oee,a rsne nFg ,the 7, Fig. in presented as However, 5. tot z J b 3 A B X bokcs.Cagsi h mag- the in Changes case. 2-block π/ J 3 `,r 2 ω eedneof dependence 1 aue where nature, 0 = > wihcnsilb optdaccu- computed be still can (which .(A ). = J 0, −1, r L rudsaewith state ground QLR a ) and S ` J J · r lc hs (n phase Block B) S 2 S >2 ` 1 = tot z +r L n 7C). and = = , /4, eae otefiln in filling the to related H 8sts (C sites. 48 q |J N max = 1 b 2.50, J | (−1) 3 .Telte yields latter The ). stesz fthe of size the is 1 = r {J U α /W K r and /10 2 −1 = Right , α} 3/2, = h aerslsotie ihtetoobtlHbadmdl(L model Hubbard two-orbital the with obtained results same The D) . sealso (see oiotl(etcl ie eitteFrilevel Fermi the depict lines (vertical) Horizontal (A–D) 2.0). = est fsae DOS(ω states of Density ) U f /W (U 2 π/ [5] = ) n lc–prlpae(n phase block–spiral and (A) 1.0) D C ny nw elztoso adn–hsr-iemdlwith the model not Haldane–Shastry-like if a few, α the of com- of realizations iron-based one multiorbital known be in only, the family—could realized 123 knowledge the work—as from our this pounds in to discussed Also, with system faster. slowly decays decays exchange which spin the latter distance, Heisenberg the frustrated long-range Within a model. unveiled: we (phenomenologi- model effective spin exotic cal) the in apparent interac- become the effects and between exchange multiorbital Hund competition robust in tion i.e., a by present frustration, dominated mechanism systems hidden double-exchange–like is of symmetry the effect appears SU(2) here an the reported nearest-neighbor state as and block–spiral only geometry, the have Instead, chain we preserved. itself. a system model on our classical) in interactions (often contrary, the orig- the of and On wall frustration these domain strong of a all in in resembles inates struc- However, structure (67). magnetic block-like perovskite order the induce spiral-like hexagonal cases the can of the top field on for magnetic ture case external magnets. the the rare-earth where also and is compounds This transition-metal many symmetry- (63–66) reported in explicit were waves spin-density of helical exam- long-period For ple, or couplings. Dzyaloshinskii–Moriya or interactions) like (geometrical terms Spiral breaking competing frustration phenomenon. empha- of by this We consequence in induced a other. correlation usually each of FM are to role states which crucial respect in the with magnetism size rotate block–spiral rigidly the islands identified have We Conclusion of interaction. effect RKKY the nontrivial of nature the the modifies state—qualitatively because magnetic (quasi)ordered case the interaction not the is this However, exchange spin J (RKKY) Ruderman–Kittel–Kasuya–Yosida the r 2 = are ymbl lcrn ol xli oeo u results. our of some explain could electrons mobile by carried U . htgvrssprxhnetnece.Teenontrivial These tendencies. superexchange governs that ∼ ) = 1/r P q 2 U A(q, ncnrs oteuulRK interaction RKKY usual the to contrast in , cetn ealcsaeceitn iha with coexisting state metallic a —creating PNAS ω ). | K = uy1,2020 14, July 3/2, U /W = F | scluae o the for calculated as (B) 2.0) Friwavevector (Fermi o.117 vol. = 8.(C 48). | o 28 no. lc phase Block ) CsCuCl k F | ,while ), 16231 3 ,

PHYSICS physics can be found in systems with spin–orbit coupling (68, A 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 LiCu2O2, LiCuVO2, or PbCuSO4 (OH)2, the spin spirals drive the system to ferroelectricity. More- over, 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 B 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 FeTe1−x Sex 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 FeTe1−x Sex can be captured by a long-range J1-J2-J3 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 reli- able 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 pos- C sible. 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/. Fig. 7. Effective Heisenberg model. (A) Comparison of S(q) between the gKH model in the spiral state (L = 48, n = 1/2, U/W = 2.0) and the long- K ACKNOWLEDGMENTS. We thank M. L. Baez, C. Batista, and M. Mierzejewski range Heisenberg (lrH) model (L = 48, J2/|J1| = 0.25, J3/|J1| = 0.1). Note for fruitful discussions. J. Herbrych, A. Moreo, and E. Dagotto were sup- 2 2 results are normalized by the magnetic moment S = S(S + 1); i.e., S = ported by the US Department of Energy (DOE), Office of Science, Basic 1.375 and S2 = 3/4 for gKH and lrH, respectively. (B and C) Static spin Energy Sciences (BES), Materials Sciences and Engineering Division. In addi- d d structure factor S(q)(B) and chirality correlation function hκ` · κmi (C) of tion, J. Herbrych acknowledges grant support by the Polish National Agency the J1-J2-J3 Heisenberg model (with ferromagnetic J1 = −1), calculated for of Academic Exchange under Contract PPN/PPO/2018/1/00035. The devel- opment of the DMRG++ code by G.A. was supported by the Scientific L = 120 sites, J3/|J1| = 0.1, and various J2/|J1| = 0.00, 0.05, ... , 1.00 (top to Discovery through Advanced Computing (SciDAC) program funded by the bottom). (A, Inset) Position of the maximum of the static structure factor US Department of Energy (DOE), Office of Science, Advanced Scientific of the lrH model as a function of /| |, for /| | = 0.00, 0.05, 0.10, qmax J2 J1 J3 J1 Computer Research and Basic Energy Sciences, Division of Materials Sci- and 0.15. ence and Engineering, which was conducted at the Center for Nanophase Materials Science, sponsored by the Scientific User Facilities Division, Basic Energy Sciences, DOE, under contract with University of Tennessee–Battelle. Furthermore, due to properties unique to the OSMP, primar- J. Heverhagen and M.D. were supported by the Deutsche Forschungsge- ily the coexistence of metallic and insulating bands, the block– meinschaft via the Emmy-Noether program (DA 1235/1-1) and FOR1807 spiral state displays exotic behavior in the electronic degrees of (DA 1235/5-1) and by the state of Baden-Wurttemberg¨ through Baden- Wurttemberg¨ High Performance Computing (bwHPC). Calculations have freedom. For example, new quasiparticles appear due to the par- been partly carried out using resources provided by Wroclaw Centre for ity breaking of the spiral state (left and right movers). Similar Networking and Supercomputing.

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16232 | www.pnas.org/cgi/doi/10.1073/pnas.2001141117 Herbrych et al. Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 4 .R ht,Dniymti omlto o unu eomlzto groups. Schollw renormalization U. quantum for 45. formulation matrix Density White, R. S. 44. iron- the for model orbital Three Dagotto, E. Moreo, A. Nicholson, A. Daghofer, M. 35. Garc J. D. 25. of out transition Garc Mott J. Orbital-selective D. Dai, X. 24. Capone, M. Hassan, R. S. Medici, de’ L. 23. one- in dynamics electromagnetic and order Chiral Onoda, S. Sato, M. Furukawa, S. 18. 3 .Z Zhang Y.-Z. 43. Luo Q. 37. Luo Q. 36. Artyukhin S. 34. Wang M. 33. Wang M. 32. Mourigal M. 29. Herbrych J. 28. Herbrych J. 27. Rinc J. 26. Caron M. J. 22. quantum frustrated the Scaramucci of phases A. Multiferroic 21. Zheludev, A. Feng, Y. orders. Povarov, spin Y. spiral K. with Multiferroics 20. Seki, S. Tokura, Y. 19. in interaction Dzyaloshinskii-Moriya the of Role Dagotto, E. 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PHYSICS