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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 j ""##i − j ##""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 N´eel-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 j frustrated magnetism j 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 j PNAS j July 14, 2020 j vol. 117 j no. 28 www.pnas.org/cgi/doi/10.1073/pnas.2001141117 Downloaded by guest on September 26, 2021 measurements and muon spin relaxation yield the same conclu- metry, as expected within a spiral state. Also, we show that the sion (30). 2) On the other hand, for BaFe2S3 and RbFe2Se3 behavior of spins can be effectively modeled via a frustrated INS reported (31–33) 2×1 blocks, FM along rungs and AFM long-range spin-only Heisenberg model. Such a spin model can along legs. Both of these phases are captured by the multiorbital serve as a starting point for the spin-wave theory calculations Hubbard Hamiltonian (26) and also by its low-energy effective often used to compare theory vs. INS spectra. Our findings are description (28), the generalized Kondo–Heisenberg model. robust against system parameter changes and should character- In this work, we unveil another magnetic phase that we stabi- ize generic multiorbital systems in the OSMP regime, close to lized; namely, we report an exotic block–spiral state. Such a state ferromagnetism. arises as a consequence of simultaneous tendencies in the sys- tem to form magnetic blocks and to develop noncollinear order. Results Different from standard spirals where one can observe spin-to- Magnetism of OSMP. We discuss the properties of a multiorbital spin rotation, in the block–spiral state the blocks rigidly rotate, Hubbard model on a one-dimensional (1D) lattice. In the generic as illustrated in the three panels in Fig. 1A. The block–spiral SU(2)-symmetric form it can be written as states we report have similarities with states in the rare-earth material TbFeO3 (34) that display spin incommensurability and X y X 0 domain walls. However, the length scales in TbFeO3 are much HH = − tγγ cγ,`,σcγ0,`+1,σ + H:c: + ∆ n1,` larger, 340 A,˚ and magnetic fields or anisotropy are needed for γ,γ0,`,σ ` X X their stabilization. Furthermore, the block–spiral magnetic pat- + U nγ,`,"nγ,`,# + (U − 5JH=2) n0,`n1,` tern appears without any apparent frustration in the model and γ,` ` is a consequence of subtly competing energy scales present in the X X y OSMP. This spin order originates in ferromagnetic tendencies, − 2JH S0,` · S1,` + JH P0,`P1,` + H:c: : [1] induced by Hund physics in multiorbital systems, and opposite ` ` antiferromagnetic superexchange tendencies, caused by Hub- bard interactions. There is a hidden frustration in the system, not y Here, c (c ) creates (destroys) an electron with spin obvious at the Hamiltonian level, and whose exotic consequences γ,`,σ γ,`,σ appear only when powerful computational tools are used: Sim- projection σ = f", #g at orbital γ = f0, 1g of site ` = f1, ::: , Lg. ∆ n = P n pler biased techniques likely would have missed this state. We stands for crystal-field splitting, while γ,` σ γ,`,σ rep- propose two simpler phenomenological models that capture the resents the total density of electrons at (γ, `), with nγ,`,σ as essence of our findings.
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