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ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-28207-w The involvement of the longitudinal mode in magnetic long- range order is also evident from the temperature dependent quasi-elastic intensity. While most spectra feature the expected gradual enhancement in the quasi-elastic channel upon increas- ing temperature (Fig. 4e for (0, 0) and S2 for other reciprocal- lattice positions), the (0.5, 0.5) spectrum at qc = 0.5 displays a pronounced rise of intensity around TN (Fig. 4f). Note that nei- ther qc = 0 nor qc = 0.5 correspond to the magnetic Bragg peak location, because the bilayer separation is incommensurate with respect to the c-axis lattice constant. Since in our setup qc = 0 is closer to a magnetic Bragg peak than qc = 0.5, we can exclude critical scattering from the long-range antiferromagnetic order as a significant contributor to this intensity as it would predict the opposite intensity behavior to what we observe (a more extensive demonstration of this is in SI Section 5). Thus the observed quasi- elastic anomaly at TN is indicative of substantial longitudinal mode condensation. The excitonic insulator character of the ground state is further supported by a large increase in resistivity below T (see Fig. 4f)30, as the condensation of the excitonic N mode leads to a reduction in the electronic carriers participating in electrical transport. This property is distinct from what is expected for a strongly-coupled Mott insulator (i.e., the large U limit of Fig. 1) where all charge-related processes are frozen out. The resistivity increase below TN could, in principle, also arise from Slater-type interactions, which can open a charge gap upon Methods Samples. Sr3Ir2O7 single crystals were synthesized using the flux method33. Starting materials of IrO2, SrCO3, and SrCl2 ⋅ 6H2O were mixed with a molar ratio of 1:2:20, and heated at 1200 ∘C for 10 h in a platinum crucible. The melt was then cooled to 800°C at a rate of 3 ∘C/h, before quenching to room temperature. We index reciprocal space using a pseudo-tetragonal unit cell with a = b = 3.896 Å and c = 20.88 Å at room temperature. Resonant inelastic X-ray scattering (RIXS) setup. RIXS spectra were measured at the 27-ID-B station of the Advanced Photon Source at Argonne National Laboratory. The incident x-ray beam was tuned to the Ir L3-edge at 11.215 keV and monochromated using a Si (884) channel-cut monochromator. The exact x-ray energy was refined via resonant energy of a standard IrO2 and the Sr3Ir2O7 sample and was set 3 eV below the resonant edge. Scattered photons were analyzed using a spherically bent diced silicon (844) analyzer with a curvature radius of 2 m. The energy and Q-resolution were 32.0(2) meV and 0.105 Å−1 full-width at half- maximum (FWHM), respectively. A small background contribution arising from air scattering was removed by subtracting a constant value from the measured intensity. The value was determined by fitting the intensity on the energy-gain side of the spectra. The L values in Fig. 2c–e were chosen such that they correspond to specific reciprocal-lattice positions with respect to the Ir-Ir interlayer spacing (see also Fig. 2a), i.e., G + qc = Ld/c, where G is an integer, qc the reduced c-axis reciprocal lattice position in terms of the Ir-Ir spacing, d = 4.07 Å the shortest Ir-Ir interlayer spacing and c = 20.88 Å the out-of-plane lattice constant. qc equals 0, 0.25 and 0.5 for L = 25.65, 26.95 and 28.25, respectively. The magnetic dispersions in Fig. 3a, b were measured along (H1, K1, 25.65) and (H2, K2, 28.25) with H1 and K1 ranging between 0.5 and 1 and H2 and K2 between 0 and 0.5. The particular Brillouin zones were chosen to ensure a scattering geometry close to 90∘, minimizing Thompson scattering. For (0, 0, 25.65), (1, 1, 25.65), (0, 0, 26.92) and (0, 0, 28.25), 2θ = 85.5, 90.2, 90.9 and 96.8∘, respectively. The sample was aligned in the horizontal (H, H, L) scattering plane, such that both dispersions could be probed through a sample rotation of Δχ ≤ 4.1∘ relative to the surface normal. Analysis of the RIXS data. The spectra were analyzed by decomposing them into four components: (1) A quasi-elastic contribution (possibly containing contribu- tions from phonons) which was modeled using a pseudo-Voigt energy resolution function, along with an additional low energy feature, which was modeled using the resolution functions at ± 32 meV, whose relative weights were constrained to follow the Bose factor. (2) The transverse magnetic mode was accounted for by a pseudo- Voigt function multiplied with an error function to capture the high-energy tail arising from the interactions with continuum. The interactions are enhanced when the modes and the continuum are less separated in energy, which leads to a reduced quasiparticle lifetime. In this case, we used a damped harmonic oscillator (with Bose factor) that was convoluted with the resolution function, which was further multiplied by an error function. (3) The longitudinal mode was described by either a pseudo-Voigt function or a damped harmonic oscillator, depending on whether or not it was resolution limited. (4) The magnetic continuum was reproduced using a broad damped harmonic oscillator multiplied by an error function to mimic its onset. The excitonic longitudinal mode is strongly qc dependent, whereas the transverse magnetic mode and the magnetic continuum vary very weakly with qc. Thus, we analyzed the spectra measured at qc = 0 and qc = 0.5 simultaneously to disentangle the excitonic contribution from the other components. The positions and lineshapes of the transverse magnetic mode and the electron-hole magnetic continuum were constrained to be independent of qc, i.e., only the amplitudes were varied. The extra peaks at qc = 0.5 give information about the excitonic longitudinal mode. During the procedure, the elastic energy was allowed to vary to correct for small fluctuations of the incident energy. Theoretical model. Sr3Ir2O7 hosts Ir4+ ions, which have 5 electrons in the active Ir 5d5 valance band. The dominant splitting of this band comes from the close-to- cubic crystal field leaving empty eg states and 5 electrons in the t2g states. SOC further splits the t2g manifold into a full Jeff = 3/2 orbital a half-filled Jeff = 1/2 orbital at the Fermi level34. Our model involves projecting the band structure onto this Jeff = 1/2 doublet. The basic structural unit, shown in Fig. 2b, contains two Ir atoms, so the experimental data were interpreted using a half-filled bilayer Hub- bardmodelH=−HK+HIwithHI=U∑rnr↑nr↓and HK 1⁄4 ∑ tνcyrcrþδ þ∑cyðr ;1ÞtzðαÞcðr ;2Þ þH:c:; ð1Þ r;δ νr?? ? ν where tν (ν = 1, 2) are the nearest- and next-nearest-neighbor hopping amplitudes within the square lattice of each Ir-layer, and tz ðαÞ 1⁄4 jtz jeiα2εr σz , with σz the Pauli matrix describes the Jeff spin dependent hopping strength between layers. The overall phase was chosen to gauge away the phase for tν. The operator cyr = [cy";r, cy#;r ] creates the Nambu spinor of the electron field at r = (r⊥, l) with l = 1, 2 denoting the layer index and r⊥ = r1a1 + r2a2. Here, the primitive in-plane lattice magnetic ordering. Sr3Ir2O7, however, lacks strong Fermi surface 23–25, 29 nesting and is in the intermediately correlated (t1 ~ U) rather than the weakly correlated (t1 ≫ U) regime, so the Slater mechanism is expected to have minimal relevance. Discussion In summary, we have isolated and characterized a longitudinal magnetic mode in Sr3Ir2O7, which merges with the electron-hole continuum at certain points in the Brillouin zone, and which softens upon heating concurrent with a decrease in the material’s resistivity. These properties are consistent with those of an anti- ferromagnetic excitonic insulator state4. We substantiate this via calculations of a bilayer Hubbard model, in which electron-hole pairs are bound by magnetic exchange interactions between the electron and hole. This consistently explains all the electronic and magnetic properties of Sr3Ir2O7 based on only one free parameter U, since all other parameters are strongly constrained by the electronic band structure of the material. The totality of these results identifies Sr3Ir2O7 as a compelling candidate for the long- sought-after antiferromagnetic excitonic insulator. Looking to the future, the intrinsically coupled spin and charge degrees of freedom in this state could have the potential for realizing new functionalities31, and suitably tuned material and/or laser-based approaches could realize methods to photo-excited these modes32. Further research on the topic may also include efforts to identify materials closer to the QCP, which in our study occurs at U/t1 = 2.35. This could extend the reciprocal space regions where the excitonic longitudinal mode exists. Another interesting direction would involve identifying excitonic easy- plane, rather than easy-axis, bilayer systems. These would host a different kind of soft excitonic longitudinal mode, often called “Higgs” mode, and could be used to study Higgs decay and renormalization effects in the presence of strong charge fluctua- tions. Careful selection of materials with multiple active orbitals could realize orbitally-ordered excitonic insulator states. Experi- mental realizations using chemical substitutions, strained thin films, high pressure, or different bilayer materials, including ruthenates, osmates, and other iridates, may help to answer some of these intriguing questions. 6 NATURE COMMUNICATIONS | (2022)13:913 | https://doi.org/10.1038/s41467-022-28207-w | www.nature.com/naturecommunicationsPDF Image | Antiferromagnetic excitonic insulator state in Sr3Ir2O7
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