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NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-28207-w ARTICLE Sr3Ir2O7 suggests that it would be a narrow-gap band insulator or semi-metal even when Coulomb repulsion is neglected29. This 300 occurs due to bonding-antibonding band splitting arising from the bilayer hopping alongside SOC, generating a minimum of the conduction band dispersion near the Brillouin zone center and a maximum in the valence band dispersion near the anti- ferromagnetic zone center. A finite value of U in a quasi-two- dimensional bilayer structure such as Sr3Ir2O7 produces an attractive particle-hole interaction in the triplet channel because of the well-known direct-exchange mechanism. In turn, particle- hole pairs at wavevectors favored by the band structure form bound states, i.e., excitons, in the magnetic channel appearing at qc = 0.5, because of the odd parity of the exciton under exchange of the two layers. The spin anisotropy arising from SOC splits the exciton triplet into a low energy state with c-axis spin quantum number Sz = 0 and higher energy Sz = ± 1 states. Strictly speak- ing, SOC means that total spin is not a good quantum number, but we retain the singlet-triplet labels for clarity. As shown in the schematic representation in Fig. 1, the Sz = 0 exciton condenses to form magnetic order at a wavevector of (0.5, 0.5) (qc = 0.5). The corresponding QCP, which exists at U = Uc = 0.27 eV (for t1 = 0.115 eV), then signals the onset of the antiferromagnetic excitonic insulator state in Sr3Ir2O7. Within the ordered state, what was a gapless Sz = 0 exciton mode at U = Uc becomes a 0 gapped excitonic longitudinal mode for U > Uc. The existence and relatively low energy of this mode implies that U in Sr3Ir2O7 is 180 only slightly above Uc. This property, together with the suffi- ciently large transverse mode gap Δs, protects the excitonic 120 longitudinal mode from decay into pairs of transverse modes. The longitudinal mode’s bound electron-hole pair nature is especially vividly illustrated by its smooth merging with the particle-hole continuum away from (0.5,0.5) and (0,0). We plot the layer- resolved charge structure of the exciton in SI Section 3. q 6 1 6 1 1 0.1 a (0, 0), qc=0 b x0.1, (1/2, 1/2), c=0 200 100 0 0 150 300 0 150 300 Temperature (K) Temperature (K) c (0, 0), qc=0.5 d x0.1, (1/2, 1/2), qc=0.5 300 200 100 60 0 e qc=0 qc=0.5 (0, 0) f (1/2, 1/2) When heating an antiferromagnetic excitonic insulator, ther- mal fluctuations modify the magnetic properties via two different processes. The first one corresponds to the destruction of Néel order via softening of the longitudinal mode. This softening signals the exciton condensation below T=TN. The second process, that takes place at a higher temperature T*, corresponds to thermal breaking of the excitons (unbinding of particle-hole pairs). A RIXS temperature series designed to test this idea at different high symmetry locations is plotted in Fig. 4a–d (linecuts at selected temperatures are shown in Supplementary Fig. S2). As expected, heating up from base temperature towards TN enhances the decay of the modes into the electron-hole continuum broadening the spectra and making it difficult to isolate the two modes in a single spectrum. We can, however, leverage the symmetry properties of the modes at different reciprocal space points to clarify the soft mode phenomenology. Since the trans- verse mode occurs at the same energy independent of qc, and the longitudinal mode is present at qc = 0.5 and absent at qc = 0, the transverse mode temperature dependence can be studied in iso- lation at qc = 0 (Fig. 4a, b). We observe that this mode has only minimal detectable softening, which is expected in view of the Ising nature of magnetism. In contrast, a substantial softening is seen at (0.5, 0.5) in Fig. 4d. Although both modes are present at qc = 0.5, we know from qc = 0 measurements that the transverse mode displays only minimal softening. Thus the longitudinal mode must play a major role in the softening to form the anti- ferromagnetic state. Our observed phenomenology is only cap- tured with the intermediate coupling regime (U/t1 = 2.83) that we conclude is relevant for Sr3Ir2O7. The strong coupling limit (U/t1 ≫ 1) would require a charge gap much larger than the observed values of 100–200 meV and, to our knowledge, it has not been able to predict any aspects of the temperature- dependent phenomenology of Sr3Ir2O7. The excitonic insulator 100 300 500 Temperature (K) 100 300 500 Temperature (K) NATURE COMMUNICATIONS | (2022)13:913 | https://doi.org/10.1038/s41467-022-28207-w | www.nature.com/naturecommunications 5 Fig. 4 Excitonic mode condensation at the Néel temperature. a–d Temperature dependence of the Sr3Ir2O7 excitation spectrum at (0, 0) and (0.5, 0.5) for qc = 0 and 0.5 (RIXS spectra at selected temperatures are shown in Supplementary Fig. S2). The intensity at (0.5, 0.5) has been scaled for comparison reasons. The dashed lines show temperature- dependent calculations of our model (the full theoretical predictions are plotted in Supplementary Fig. S5). Based on the qc behavior of the modes, we know that panels a, b show only the transverse mode, while c, d show both the transverse and longitudinal mode. e, f Quasi-elastic intensity as function of temperature for qc = 0 and 0.5 in blue and red, respectively. The non-monotonic enhancement at qc = 0.5 in e provides additional support that the condensation of the excitonic longitudinal mode establishes the magnetic long-range order in Sr3Ir2O7. Panel f also shows the anomalous temperature dependence of the electric resisitivity ρ (taken from30), which shows a change in gradient at TN further indicating that charge fluctuations are involved in the transition. model is also supported by our temperature-dependent calcula- tions, which are shown as dashed lines in Fig. 4c, d. Full calcu- lations are shown in Supplementary Fig. S5 and explained in SI Section 2. Theory shows that exciton formation takes place at T* ≈ 2TN, controlled by the exciton binding energy, which is of order the charge gap minus the longitudinal mode energy at the ordering wavevector. The mean-field transition temperature prediction is TN = 424 K, which is not too far above the measured TN = 285 K and which is expected since fluctuations are expected to reduce TN below the mean-field prediction. The predictions in Fig. 4c, d are shown with temperatures re-normalized to the experimental TN. Intensity (arb. units) Intensity (arb. units) ρ (Ω∙cm) Elastic line amplitude (arb. units) Energy (meV) Energy (meV)PDF Image | Antiferromagnetic excitonic insulator state in Sr3Ir2O7
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