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NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-28207-w ARTICLE a c db O Ir Sr z t2 tz(α) For Jc/Jab of order two, the bilayer Heisenberg Hamiltonian supports a longitudinal mode and for the current case of large easy-axis anisotropy, the transverse and longitudinal modes appear as well-defined modes throughout the Brillouin zone13–16. In fact, earlier reports have proposed this spin dimer model to explain RIXS measurements of the longitudinal ~ 170 meV fea- ture in Sr3Ir2O714, 17. Although prior and subsequent non- dimerized models have also been proposed to describe Sr3Ir2O7 as rival candidates12, 18–21. These models, however, do not support a longitudinal mode (a detailed comparison between the different models is given in Supplementary Information (SI) Section 1). We, therefore, map the in-plane dispersion relations at qc = 0 and 0.5 and show them in Fig. 3a, b. At qc = 0, where the longitudinal mode is suppressed by symmetry, we observe an excitation dis- persing from ~ 90 to 170 meV and a continuum at higher ener- gies. Simultaneously analyzing qc = 0.5 and qc = 0 for each in- plane reciprocal-lattice wavevector, while leveraging the distinct symmetry properties of the longitudinal and transverse modes, allows us to isolate the longitudinal mode (see Methods section). We plot the position and peak width of the longitudinal mode in green in Fig. 3b. The transverse mode, on the other hand, is symmetry-allowed at qc = 0.5 and qc = 0 and is shown in black on Fig. 3a, b. We find that the longitudinal mode is well-defined around (0, 0) (Figs. 2c–d, 3i), but decays into the high-energy continuum as it disperses away, becoming undetectable at (1/4, 1/ 4) (Fig. 3h). The longitudinal mode is also detectable as a shoulder feature on the transverse mode at (1/2, 1/2) before dis- persing upwards and broadening at neighboring momenta (Fig. 3e, g). The decay and merging of the longitudinal mode into the electron-hole continuum was not detected previously and suggests the realization of an antiferromagnetic excitonic insu- lator state because the longitudinal mode in this model has a bound electron-hole pair character and therefore will necessarily decay when it overlaps with the electron-hole continuum. This longitudinal mode decay is incompatible with a longitudinal mode arising from spin dimer excitations in a strongly isotropic bilayer Heisenberg model, which predicts well-defined modes throughout the Brillouin zone and projects out the high-energy particle-hole continuum13–16. Since optical conductivity, tunneling spectroscopy, and photo- emission studies all report charge gaps Δc on the same energy scale of magnetic excitations (100–200 meV)11, 22–25, we model the microscopic interactions within a Hubbard Hamiltonian that retains the charge degree of freedom. In particular, the crucial difference with the Heisenberg description is that the Hubbard model retains the electron-hole continuum, whose lower edge at ω = Δc is below the onset of the two-magnon continuum: Δc < 2Δs. We considered a half-filled bilayer, which includes a single “Jeff = 1/2” effective orbital for each of the two Ir sites in the unit cell, following methods developed in parallel with this experi- mental study26. The model contains an effective Coulomb repul- sion U, and three electron hopping parameters: nearest and next- nearest in-plane hopping terms tν (ν = 1, 2) within each Ir layer, and the spin dependent hopping strength tz(α) between Ir layers (Fig. 2b). tz(α) is composed of an amplitude ∣tz∣ and a phase α arising from the appreciable SOC in the material (further details are given in the Methods section)27. The model was solved using the random phase approximation (RPA) in the thermodynamic limit (SI Section 2), which is valid for intermediately correlated materials even at finite temperature28. We constrain tν and ∣tz∣ to values compatible with density functional theory and photo- emission measurements and consider the effective U, which is strongly influenced by screening, as the primary tuning parameter29. Figure 3c, d show the results of calculations with t1 =0.115 eV, t2 =0.012 eV, ∣tz∣=0.084 eV, α=1.41, and U = 0.325 eV. The small U is due to the extended Ir orbitals and y x t1 c Q = (0, 0, 25.65) qc = 0 d Q = (0, 0, 26.95) qc = 0.25 e Q = (0, 0, 28.25) qc = 0.5 2.4 1.2 0.0 2 1 0 1.6 0.8 0.0 0 100 200 Energy (meV) 300 Fig. 2 Isolating the excitonic longitudinal mode in Sr3Ir2O7. a Crystal structure of the bilayer material Sr3Ir2O7. b Ir-Ir bilayer with t1 the nearest- neighbor, t the next-nearest-neighbor and t (α) the interlayer hopping 2z terms. c–e RIXS spectra measured at T = 20 K and Q = (0, 0, L) with L = 25.65, 26.95 and 28.25 in reciprocal lattice units. The c-axis positions are also labeled in terms of the Ir-Ir interlayer reciprocal-lattice spacing qc = 0, 0.25 and 0.5. An additional mode appears around 170 meV with maximal intensity at qc = 0.5 (see shaded red area). The black circles represent the data and dotted lines outline the different components of the spectrum, which are summed to produce the grey line representing the total spectrum. Error bars are determined via Poissonian statistics. not dramatically larger than, the in-plane exchange Jab is needed to produce a longitudinal mode and large easy-axis magnetic anisotropy is required to reproduce the spin gap. If Jc ≪ Jab, the spectrum would show only a spin-wave-like in-plane dispersion contrary to the observed qc dependence in Fig. 2c–e, and in the Jc ≫ Jab limit the system would become a quantum paramagnet. NATURE COMMUNICATIONS | (2022)13:913 | https://doi.org/10.1038/s41467-022-28207-w | www.nature.com/naturecommunications 3 Intensity (arb. units)PDF Image | Antiferromagnetic excitonic insulator state in Sr3Ir2O7
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