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NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-28207-w ARTICLE vectors are denoted by a1 and a2, and the directed neighboring bonds are repre- sented by δ1 = a1, a2 and δ2 = a1 ± a2. In the interaction term HI, U is the effective Coulomb interaction, and nrσ is the density operator for electrons of spin σ at r. In the spin dependent hopping term, the sign εr takes the values ± 1 depending on which sublattice of the bipartite bilayer system r points to. The phase α arises from hopping matrix elements between dxz and dyz orbitals, which are allowed through the staggered octahedral rotations in the unit cell along side SOC27, 35. In the model, SOC enters via the phase of the c-axis hopping, which is smaller than the in-plane bandwidth, justifying the approximate use of singlet and triplet for labels of the different excitons. The model was studied at half-filling in the sense that it contains two bands (bonding and antibonding) in the model, which host two electrons as is appropriate for Sr3Ir2O723–25, 27. We solved the model using the RPA in the thermodynamic limit (detailed information is given in SI Section 2), which is valid for intermediately correlated materials even at finite temperatures28. The theoretically determined Néel temperature, in this case, is Tcal 1⁄4 424 K which is N slightly larger than the experimental value TN = 285 K. This is expected within the RPA we use here, as this ignores fluctuations than act to reduce the transition temperature. The dynamical spin structure factors in Fig. 3c, d are shown after convolution with the experimental resolution. A more complex model could include all t2g or all d orbitals, rather than just effective Jeff = 1/2 doublets. The success of our Jeff = 1/2 only model suggests that orbital degrees of freedom are entirely frozen out of the problem or manifest themselves in very subtle ways beyond current detection limits. Due to this, the Sr3Ir2O7 excitonic insulator state has no orbital component (other than in the trivial sense that the Jeff = 1/2 states in themselves are a coupled modulation of spin and orbital angular momentum). A possible SOC-induced orbital order is discussed in SI Section 6. Data availability The RIXS data generated in this study have been deposited in the Zenodo database under accession code 581298936. Code availability The code used in this study is available from the authors upon reasonable request. Received: 30 November 2021; Accepted: 10 January 2022; References 1. Mott, N. F. The transition to the metallic state. Philos. Mag. 6, 287–309 (1961). 2. Keldysh, L. & Kopaev, Y. V. Possible instability of semimetallic state toward Coulomb interaction. Sov. Phys. Solid State, USSR 6, 2219–2224 (1965). 3. Jérome, D., Rice, T. M. & Kohn, W. Excitonic insulator. Phys. Rev. 158, 462–475 (1967). 4. Halperin, B. I. & Rice, T. M. Possible anomalies at a semimetal-semiconductor transistion. Rev. Mod. Phys. 40, 755–766 (1968). 5. Bucher, B., Steiner, P. & Wachter, P. Excitonic insulator phase in TmSe0.45Te0.55. Phys. Rev. Lett. 67, 2717–2720 (1991). 6. Wakisaka, Y. et al. Excitonic insulator state in Ta2NiSe5 probed by photoemission spectroscopy. Phys. Rev. Lett. 103, 026402 (2009). 7. Eisenstein, J. Exciton condensation in bilayer quantum hall systems. Annu. Rev. Condens. Matter Phys. 5, 159–181 (2014). 8. Kogar, A. et al. Signatures of exciton condensation in a transition metal dichalcogenide. Science. 358, 1314–1317 (2017). 9. Lu, Y. et al. Zero-gap semiconductor to excitonic insulator transition in Ta2NiSe5. Nat. Commun. 8, 1–7 (2017). 10. Slater, J. C. Magnetic effects and the Hartree-Fock equation. Phys. Rev. 82, 538–541 (1951). 11. Moon, S. J. et al. Dimensionality-controlled insulator-metal transition and correlated metallic state in 5d transition metal oxides Srn+1IrnO3n+1: (n = 1, 2, and ∞). Phys. Rev. Lett. 101, 226402 (2008). 12. Kim, J. et al. Large spin-wave energy gap in the bilayer iridate Sr3Ir2O7: Evidence for enhanced dipolar interactions near the Mott metal-insulator transition. Phys. Rev. Lett. 109, 157402 (2012). 13. Lohöfer, M. et al. Dynamical structure factors and excitation modes of the bilayer Heisenberg model. Phys. Rev. B 92, 245137 (2015). 14. Moretti Sala, M. et al. Evidence of quantum dimer excitations in Sr3Ir2O7. Phys. Rev. B. 92, 024405 (2015). 15. Zhou, C., Yan, Z., Wu, H.-Q., Sun, K., Starykh, O. A. & Meng, Z. Y. Amplitude mode in quantum magnets via dimensional crossover. Phys. Rev. Lett. 126, 227201 (2021). 16. Su, Y. et al. Stable Higgs mode in anisotropic quantum magnets. Phys. Rev. B. 102, 125102 (2020). 17. Hogan, T. et al. Disordered dimer state in electron-doped Sr3Ir2O7. Phys. Rev. B. 94, 100401 (2016). 18. Gretarsson, H. et al. Two-magnon raman scattering and pseudospin-lattice interactions in Sr2IrO4 and Sr3Ir2O7. Phys. Rev. Lett. 116, 136401 (2016). 19. Lu, X. et al. Doping evolution of magnetic order and magnetic excitations in ðSr1xLaxÞ3Ir2O7. Phys. Rev. Lett. 118, 027202 (2017). 20. Li, S. et al. Symmetry-resolved two-magnon excitations in a strong spin-orbit- coupled bilayer antiferromagnet. Phys. Rev. Lett. 125, 087202 (2020). 21. Mohapatra, S., van den Brink, J. & Singh, A. Magnetic excitations in a three- orbital model for the strongly spin-orbit coupled iridates: Effect of mixing between the j 1⁄4 12 and j 1⁄4 12 sectors. Phys. Rev. B. 95, 094435 (2017). 22. Okada, Y. et al. Imaging the evolution of metallic states in a correlated iridate. Nat. Mater. 12, 707–713 (2013). 23. Wang, Q. et al. Dimensionality-controlled mott transition and correlation effects in single-layer and bilayer perovskite iridates. Phys. Rev. B 87, 245109 (2013). 24. King, P. D. C. et al. Spectroscopic indications of polaronic behavior of the strong spin-orbit insulator Sr3Ir2O7. Phys. Rev. B. 87, 241106 (2013). 25. de la Torre, A. et al. Coherent quasiparticles with a small fermi surface in lightly doped Sr3Ir2O7. Phys. Rev. Lett. 113, 256402 (2014). 26. Suwa, H., Zhang, S.-S. & Batista, C. D. Exciton condensation in bilayer spin- orbit insulator. Phys. Rev. Res. 3, 013224 (2021). 27. Carter, J.-M., Shankar V., V. & Kee, H.-Y. Theory of metal-insulator transition in the family of perovskite iridium oxides. Phys. Rev. B 88, 035111 (2013). 28. Hirschmeier, D., Hafermann, H., Gull, E., Lichtenstein, A. I. & Antipov, A. E. Mechanisms of finite-temperature magnetism in the three-dimensional Hubbard model. Phys. Rev. B. 92, 144409 (2015). 29. Carter, J.-M. & Kee, H.-Y. Microscopic theory of magnetism in Sr3Ir2O7. Phys. Rev. B. 87, 014433 (2013). 30. Cao, G. et al. Anomalous magnetic and transport behavior in the magnetic insulator Sr3Ir2O7. Phys. Rev. B. 66, 214412 (2002). 31. Tokura, Y., Kawasaki, M. & Nagaosa, N. Emergent functions of quantum materials. Nat. Phys. 13, 1056–1068 (2017). 32. Kang, S. et al. Coherent many-body exciton in van der Waals antiferromagnet NiPS3. Nature. 583, 785–789 (2020). 33. Li, L. et al. Tuning the Jeff 1⁄4 12 insulating state via electron doping and pressure in the double-layered iridate Sr3Ir2O7. Phys. Rev. B. 87, 235127 (2013). 34. Kim, B. J. et al. Novel Jeff = 1/2 Mott state induced by relativistic spin-orbit coupling in Sr2IrO4. Phys. Rev. Lett. 101, 076402 (2008). 35. Cao, G. & Schlottmann, P. The challenge of spin–orbit-tuned ground states in iridates: a key issues review. Rep. Prog. Phys. 81, 042502 (2018). 36. Mazzone, D. G. et al. Data repository for: Antiferromagnetic excitonic insulator state in Sr3Ir2O7 https://doi.org/10.5281/zenodo.5812988 (2022). Acknowledgements We thank D.F. McMorrow, B. Normand, Ch. Rüegg, and J.P. Hill for fruitful discussions. Work performed at Brookhaven National Laboratory was supported by the US Depart- ment of Energy, Division of Materials Science, under Contract No. DE-SC0012704. This research used resources of the Advanced Photon Source, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under Contract No. DE-AC02-06CH11357. This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725. D.G.M. acknowledges support from the Swiss National Science Foundation, Fellowship No. P2EZP2_175092. H.S. acknowledges sup- port from JSPS KAKENHI Grant No. JP19K14650. X.L. acknowledges the support from the National Natural Science Foundation of China under grant No. 11934017. K.J. and Y.G.S. acknowledge the support from the National Natural Science Foundation of China (Grants No. U2032204), and the K.C. Wong Education Foundation (GJTD-2018-01). Y.G.S. acknowledges the Chinese National Key Research and Development Program (No. 2017YFA0302901) and the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (Grant No. XDB33000000). J.L. acknowledges support from the National Science Foundation under Grant No. DMR-1848269. J.Y. acknowledges funding from the State of Tennessee and Tennessee Higher Education Commission (THEC) through their support of the Center for Materials Processing. H.M. was sponsored by the Laboratory Directed Research and Development Program of Oak Ridge National Laboratory, managed by UT-Battelle, LLC, for the U.S. Department of Energy. Author contributions The project was conceived and initiated by M.P.M.D, C.D.B., J.L. and X.L. D.G.M., Y.S., G.F., J.Y., H.M., M.H.U. and D.M.C. prepared and performed the experiments. K.J. and Y.Shi synthesized the samples. D.G.M. and Y.S. analyzed the experimental data, H.S. and S.S.Z. provided the theoretical calculations. The results were interpreted by D.G.M., Y.S., H.S., J.S., J.L. C.D.B, and M.P.M.D. The paper was written by D.G.M., Y.S., H.S., X.L., C.D.B. and M.P.M.D. with the input from all co-authors. Competing interests The authors declare no competing interests. NATURE COMMUNICATIONS | (2022)13:913 | https://doi.org/10.1038/s41467-022-28207-w | www.nature.com/naturecommunications 7PDF Image | Antiferromagnetic excitonic insulator state in Sr3Ir2O7
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