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ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-28207-w Detailed theoretical considerations of narrow-gap insulators date back to the 1960s, when it was realized that if the energy required to form an electron-hole pair becomes negative, a phase transition into an excitonic insulator state can occur1–4. Unscreened electron-hole Coulomb attraction is perhaps the most obvious driving force behind this phase transition, and excitonic charge insulator states are indeed thought to occur in materials such as TmSe0.45Te0.55, 1T-TiSe2, and Ta2NiSe55–9. Although less intuitive, effective electron-hole attraction can also arise from on-site electron-electron Coulomb repulsion U via magnetic exchange interactions between the electron and hole10. In this case, the soft exciton is expected to be a spin-triplet, which passes through a quantum critical point (QCP) with increasing effective U. The condensation of the relevant triplet exciton at the QCP gives rise to an antiferromagnetic ground state hosting a well-defined excitonic longitudinal mode4, which coexists with transverse modes that are a generic feature of ordered antiferro- magnets. This longitudinal mode features excitonic character, in the sense that it modifies the local spin amplitude by creating electron-hole pairs4. In this work, we identify and study a long- itudinal mode in Sr3Ir2O7, the presence of which is the key experimental signature of an antiferromagnetic excitonic insulator. Results The formation of an antiferromagnetic excitonic insulator requires a very specific set of conditions. We need (i) a charge gap of similar magnitude to its magnetic energy scale and (ii) strong easy-axis anisotropy. Property (i) is a sign that the material is close to the excitonic QCP (see Fig. 1). Property (ii) is not a strict condition, but it facilitates the identification of an anti- ferromagnetic excitonic insulator, because the opening of a spin gap Δs protects the longitudinal mode from decay. This is because longitudinal fluctuations are often kinetically predisposed to decay into transverse modes generating a longitudinal continuum with no well-defined modes. This decay can be avoided when the energy of the longitudinal mode is lower than twice the spin gap. Iridates host strong spin-orbit coupling (SOC), which can help realize a large spin gap and bilayer Sr3Ir2O7 shown in Fig. 2a is known to have a narrow charge gap of order Δc ~ 150 meV11. The essential magnetic unit, with c-axis ordered moments, is shown in Fig. 2b12. In view of the antiferromagnetic order in Sr3Ir2O7, the material would be predicted to lie in the magnetically ordered region to the right of the QCP where the excitonic longitudinal mode is expected to appear. Because the exciton is predicted to have odd parity under exchange of the two Ir layers, we expect the excitonic longitudinal mode to be present at c-axis wavevectors corresponding to antisymmetric bilayer contributions and absent at the symmetric condition. We label these wavevectors qc = 0.5 and qc = 0, respectively. In contrast, transverse magnetic modes are expected to be present at all c-axis wavevectors, allowing the transverse and longitudinal modes to be readily distinguished. The excitation spectrum of Sr3Ir2O7 was studied with RIXS. Figure 2c–e displays energy-loss spectra at T = 20 K, well below the Néel temperature TN = 285 K and qc = 0, 0.25 and 0.5, cor- responding to L = 25.65, 26.95 and 28.25 in reciprocal lattice units (r.l.u.). These irrational L values arise because the bilayer separation d is not a rational fraction of the unit cell height c (see Methods section for details). The spectrum at qc = 0 is composed of a phonon-decorated quasi-elastic feature, a pronounced mag- netic excitation at~100 meV, which we later identified as the transverse mode, and a high-energy continuum. As explained above, changing qc is expected to isolate the anticipated excitonic mode. A longitudinal mode is indeed observed, reaching max- imum intensity at qc = 0.5, and is highlighted by red shading in Fig. 2d, e. In isolation, the presence of a longitudinal magnetic mode in this symmetry channel is a necessary but insufficient condition to establish an antiferromagnetic excitonic insulator, so we leverage the specific symmetry, decay, and temperature dependence of the longitudinal and transverse magnetic modes to establish the presence of the novel state. The only other candidate magnetic model that hosts a longitudinal mode of this type is a specific configuration of the bilayer Heisenberg Hamiltonian, in which the charge degrees of freedom are projected out. In particular, a model with a c-axis magnetic exchange Jc that is larger than, but Fig. 1 Antiferromagnetic excitonic insulator phase diagram. Charge excitations in paramagnetic band insulators consist of either electron-hole excitations across the insulating band gap (brown shaded area) or of bound electron-hole excitons below the particle-hole continuum [electrons (holes) are indicated with filled (empty) circles]. An antiferromagnetic excitonic insulator is established through the condensation of the predominately spin-triplet character exciton mode with spin quantum number Sz = 0. The excition is a superposition of an up-spin electron in the conduction band paired with an up-spin hole (equivalent to a down-spin electron) and a down-spin electron paired with a down-spin hole4. The other spin-triplet excitions Sz = ±1 feature an up-spin electron and a down-spin hole or a down-spin electron and an up-spin hole. Upon increasing Coulomb interaction U the Sz = 0 exciton condenses into the ground state at a QCP4, establishing magnetic order and leaving an excitonic longitudinal mode as the key signature of this state. 2 NATURE COMMUNICATIONS | (2022)13:913 | https://doi.org/10.1038/s41467-022-28207-w | www.nature.com/naturecommunications E QCP Particle-hole continuum Nonmagnetic band insulator Antiferromagnetic U excitonic insulator + , Charge Antiferromagnetic ground state Longitudinal mode Transverse modePDF Image | Antiferromagnetic excitonic insulator state in Sr3Ir2O7
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