Introduction

The coupling between various degrees of freedom (spin, charge, lattice, and orbital) is an important paradigm in condensed matter physics. Such couplings are known to give rise to novel ground states and various emergent phenomena,1,2,3,4,5 including high TC superconductivity,2,3 colossal magnetoresistance,4 etc. Recently, there has been increasing interest in spin-phonon coupling (SPC) within the context of multiferroics and spintronics.5,6,7,8,9 In multiferroic materials, SPC has been invoked to explain intriguing phenomena such as the thermal Hall effect8 while it also determines the spin relaxation time in spintronics applications.9 Moreover, SPC presents the opportunity to engineer new functionality in transition metal oxides (TMOs). For example, it is possible to stabilize a specific magnetic ground state with epitaxial strain10 or generate magnetoelastic effects by changing the magnetic easy-axis.7 In this context, it has become increasingly important to develop a detailed understanding of SPC.

In general, we can describe the physics of most spin-related phenomena in solids using the spin Hamiltonian (Hspin). This can be written as a sum of the isotropic exchange (IE) interaction, the Dzyaloshinskii–Moriya (DM) interaction, the anisotropic exchange (AE) interaction, and the single-ion anisotropy (SIA):11,12,13

$$H_{{\mathrm{spin}}} = \mathop {\sum }\limits_{ij}^{{\mathrm{nn}}} \left[ {J_{{\mathrm{IE}}}\left( {{\mathbf{S}}_i \cdot {\mathbf{S}}_j} \right) + {\mathbf{D}}_{ij} \cdot \left( {{\mathbf{S}}_i \times {\mathbf{S}}_j} \right) + {\mathbf{S}}_i \cdot {\mathbf{\Gamma }}_{ij} \cdot {\mathbf{S}}_j} \right] + A\mathop {\sum }\limits_i \left( {{\mathbf{n}}_i \cdot {\mathbf{S}}_i} \right)^2,$$
(1)

where JIE, Dij, and Γij are the coefficients of the IE, DM, and AE interactions, respectively. A and ni are the coefficient of the SIA and the direction vector of corresponding easy-axis. Si and Sj are the spins at the nearest neighboring ith and jth sites.14,15 Spin interactions are mediated by the Coulomb interaction of electrons at neighboring sites16 and are highly dependent on the positions of the neighboring ions. Consequently, JIE, Dij, Γij and/or A terms can be dynamically modulated by a lattice vibration, resulting in SPC and a renormalization of the phonon frequencies.

A large number of studies have focused on 3d and 4d TMOs17,18,19,20,21,22,23,24,25,26,27,28,29 where the SPC may be explained in terms of dynamic modulation of the IE interaction. RMnO3 (R: rare-earth site ion), one of the widely investigated 3d TMOs, shows strong SPC phenomena which is closely linked with multiferroicity.17,18,19,20,21,22,23 Geometrically frustrated TMOs are also known to have strong SPC.24,25,26,27 For example, in 3d ACr2O4 (A: Cd, Zn), the coupling leads to spin-driven Jahn-Teller effects which relieve the magnetic frustration.24,25,26 Similarly, strong SPC could occur also in 4d Y2Ru2O7 and Y2Mo2O7 with spin-glass state.27,28 All of these SPC phenomena have been well explained by the dynamic modulation of the IE interaction.17,18,19,20,21,22,23,24,25,26,27,28,29

The Hspin in Eq. (1) indicates that other magnetic interactions could also induce SPC, but such possibilities have not been investigated to date. Recently, the DM interaction has attracted significant attention due to its critical role in multiferroics30,31 and topological materials.32 It is therefore highly desirable to explore the possibility of novel SPC mediated via the DM interaction. Equation (1) shows that such an alternative mechanism requires that both Dij and 〈Si × Sj〉 be large. It should be noted that the DM interaction term originates from spin-orbit coupling (SOC).14 The SOC is the relativistic interaction that increases roughly with Z2 (Z: atomic number).33 For example, the SOC of 5d TMOs is about 500 meV, much larger than that of 3d TMOs (typically <10 meV). Therefore, compounds containing heavy elements are better candidates to search for unconventional SPC based on strong SOC.

Among 5d TMOs, pyrochlore iridates are good candidates to search for new types of SPC. First, they have an intriguing ground state, called Jeff = 1/2 state,15,34 due to strong SOC. Second, pyrochlore iridates host an intriguing form of antiferromagnetism, namely the all-in-all-out (AIAO) magnetic order.32,35 As shown in Fig. 1a, all spins of one (neighboring) tetrahedron point inward (outward).36 This non-collinear spin configuration results in a large 〈Si × Sj〉 value in equation (1). Significant dynamic modulation of the DM interaction can therefore lead to a renormalization of the phonon frequencies which cannot be fully explained by the IE interaction alone.

Fig. 1
figure 1

Sample characterization of pyrochlore iridate Y2Ir2O7. a Lattice and magnetic structure of pyrochlore iridates. The pyrochlore lattice has a corner-sharing tetrahedral structure. The large green spheres show the positions of the Y ions, and Ir ions are located at the corner of each tetrahedron. O ion positions are not displayed, for simplicity. With the all-in-all-out (AIAO) antiferromagnetic order, all Ir spins at the corner of one tetrahedron point inward into the center of the tetrahedron, while those at the neighboring tetrahedra point outward. b The χm(T) graphs of the Y2Ir2O7 polycrystalline sample at H = 0.1T for field cooling (FC) and zero field cooling (ZFC). c Powder X-ray diffraction data for Y2Ir2O7. The blue bars indicate the positions of the Bragg peaks from the Rietveld refinement

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To investigate SPC with the AIAO magnetic order, we focused on polycrystalline samples of Y2Ir2O7. In Fig. 1b, the magnetic susceptibility of this compound clearly indicates antiferromagnetic ordering at Néel temperature TN ≈ 170 K. We verified the crystal structure of our sample using X-ray diffraction (XRD), as shown in Fig. 1c. The good agreement between experimental XRD results and the Rietveld refinement fitting confirmed the pyrochlore structure of our sample.

The phonon renormalization due to SPC can be directly measured by infrared (IR) spectroscopy.11 In general, SPC phenomena have been widely investigated by measuring anomalies in spin wave dispersion with either neutron or X-ray scattering experiments.37,38 However, in Y2Ir2O7 the constituent Ir ions strongly absorb neutrons, making neutron scattering experiments difficult. Resonant inelastic X-ray scattering experiments could be another option, but the precise analysis of anomalies due to SPC is complicated by a comparatively poor energy resolution (i.e., ~10 meV).39 In contrast, the high energy resolution (~0.5 meV) of IR spectroscopy allows us to investigate the details of SPC by precise measurement of the phonon frequency shifts across the TN.

Here, we report evidence for unconventional SPC via the DM interaction in Y2Ir2O7. To investigate DM-derived SPC, we monitored the temperature (T)-dependent phonon spectra in Y2Ir2O7 by infrared spectroscopy. Three IR-active phonons soften and sharpen below TN, implying the presence of strong SPC. Based on first-principles calculations and tight-binding model analysis, we show that the observed phonon frequency shifts originate from the modulation of the Ir–O–Ir bond angle and establish that the phonon renormalization in Y2Ir2O7 is predominantly driven by the DM interaction.

Results

Phonon spectra measurement and Fano–Lorentz model analysis

In the present work, we investigated the IR-active phonons of a Y2Ir2O7 polycrystalline sample. The photon energy range was 6 ≤ ℏω ≤ 5,000 meV and T ranged from 10 to 300 K. As presented in Fig. 2a, the σ1(ω) spectra show seven IR-active phonons in Y2Ir2O739,40,41 below 90 meV, as expected for the pyrochlore structure.40,41 The experimental σ1(ω) spectra also showed several weak peaks, possibly due to impurities,42 but they did not influence our phonon analysis of Y2Ir2O7. For later discussion, we label the seven strong phonon modes as Ph1 to Ph7 in order of increasing frequency, ω.

Fig. 2
figure 2

T-dependent infrared (IR)-active phonons in Y2Ir2O7. a The σ1(ω) spectra at T = 300 K (orange diamonds) and 10 K (blue circles). The experimental σ1(ω) were fitted using the Fano–Lorentz model (black solid lines), identifying seven IR-active phonon modes. The σ1(ω) for each T has a vertical offset for clarity. b Evolution of all phonons as T varies from 300 K to 10 K. Vertical offsets are included for clarity. As T decreases, some phonons (Ph4, Ph5, and Ph6) exhibit a non-monotonic T-evolution of the peak energy (red arrows). The other phonons (Ph1, Ph2, Ph3, and Ph7) exhibit monotonic behavior (blue arrows).27,43 TN is approximately 170 K

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Three of the IR-active phonons exhibit an anomalous T-dependence. Figure 2b shows the T-dependent evolution of each phonon. The blue arrows show that Ph1, Ph2, Ph3, and Ph7 blueshift (i.e., harden) monotonically with decreasing T. On the other hand, the red arrows indicate that Ph4, Ph5, and Ph6 redshift (i.e., soften) abruptly. To perform a more quantitative analysis, we fitted the σ1(ω) spectra with a Fano–Lorentz model. Figure 3a–c show that the ω0(T) and Γ(T) of Ph4, Ph5, and Ph6 show anomalous T-dependent changes near TN. This behavior is in contrast with the smooth T-dependence of phonon parameters for Ph7 shown in Fig. 3d. (The parameters of other phonons are shown in Supplementary Note 1.)

Fig. 3
figure 3

T-dependence of phonon parameters, determined by the Fano–Lorentz model analysis. a–d T-dependent frequencies ω0 and linewidths Γ from 10 to 300 K are shown. Blue dashed lines indicate TN. Red solid lines indicate the ω0 calculated from anharmonic phonon-phonon scattering theory (ω0th in the main text).27,43 The anharmonic theory is parameterized so as to match the data at T > TN. For three phonons (Ph4, Ph5, and Ph6), ω0 deviates significantly from the red solid lines below TN. The Γ of these same modes also reveal anomalies around TN (for the parameters of other phonon modes, see Supplementary Note 1). On the other hand, the parameters of phonon (Ph7) do not show any anomalies and are well-described by the anharmonic phonon scattering theory

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A comparison with the phonon-phonon scattering theory can provide an insight into the phonon renormalization of Y2Ir2O7. The red lines in Fig. 3 indicate the theoretical phonon frequencies, ω0th, expected based on the simple phonon-phonon scattering theory.27,43 The ω0(T) of Ph7 is well-described by the theory. On the other hand, the ω0(T) of Ph4, Ph5, and Ph6 show abrupt redshifts at TN ≈ 170 K that are not captured by the theory. For later discussion, we define the frequency difference, Δω0(T) ≡ ω0th(T) – ω0(T) at T = 10 K. In addition, the Γ(T) of Ph4, Ph5, and Ph6 also show kink-like features near TN, as shown in the lower panels of Fig. 3a–c. The phonon linewidth, the inverse of the lifetime, can become smaller in the ordered state which is away from the phase transition. However, earlier XRD and neutron diffraction experiments showed that AIAO magnetic ordering in pyrochlore iridates is not accompanied by a structural transition.44,45,46 Therefore, we can conclude that the anomalous T-dependence of Ph4, Ph5, and Ph6 should be coupled to the AIAO magnetic order.

Phonon analysis with density functional theory (DFT)

DFT calculations were performed to determine the phonon frequencies and corresponding eigenmodes of Y2Ir2O7 (see Methods). The AIAO spin configuration of the pyrochlore structure was used for the calculations. These spins are aligned along the local Ir [111] axis in the Ir tetrahedra. Figure 4a shows the spin-canting angle \(\theta _{\mathrm{spin}} = {\mathrm{cos} ^{-1}}\left({-\frac{1}{3}}\right){\approx} 109.5^\circ\) in the Ir–O–Ir plane,47 which is marked as a blue area. Here, we choose the xo- and zo-axes of the local coordinates for O ions in the Ir–O–Ir plane, and the yo-axis perpendicular to the plane. Our DFT calculations suggest that the Ir–O–Ir bond angle θ0 is about 126.9°. This is close to the experimental values from 127.5° to 128.5° measured by neutron diffraction44,48 and XRD,49 respectively. As shown in Table 1, the DFT-calculated phonon frequencies agree well with the experimental values and therefore support the validity of our calculations.

Fig. 4
figure 4

All-in-all-out (AIAO) spin configurations (gray arrows), Ir–O–Ir bond of Y2Ir2O7, and classification of phonon modes based on the O ion motion. a O ions (red spheres) are bonded to the nearest neighboring Ir ions (blue spheres) forming the Ir–O–Ir bond. In the AIAO magnetic order, the Ir spin-canting angle is about 109.5°. From first-principles calculations, the Ir–O–Ir bond angle θ0 was estimated to be about 126.9°. b The IR-active phonons of Y2Ir2O7 cause three types of O displacement. For the in-plane stretching mode, the O ion displacement is mostly along the xo-axis, parallel to the Ir-Ir bond. For the out-of-plane bending mode, the O ion displacement is mostly along yo-axis, perpendicular to the Ir–O–Ir plane. For the in-plane stretching mode, the O ion displacement is mostly along the zo-axis in the Ir–O–Ir plane

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Table 1 IR-active phonons determined via optical conductivity measurements at T = 10 K and DFT calculations
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To obtain further insight into the phonon renormalization, let us focus on the Ir–O–Ir bond. Note that O is the lightest element in Y2Ir2O7, so its vibration amplitude should be largest. In this context, the motion of the O ion between the neighboring Ir spins plays an important role in the electron hopping along the Ir–O–Ir path and in Hspin.13 The seven IR-active phonon modes can be classified into three categories based on the dominant motion of the O ions along the xo-, yo-, and zo-directions (see Supplementary Note 2). The classification is shown in Fig. 4b: Ir–O–Ir stretching (Ph1 and Ph7), out-of-plane bending (Ph2, Ph3, and Ph4), and in-plane bending (Ph5 and Ph6) motion, respectively (see Supplementary Note 3). For the in-plane bending motion, the O displacement is parallel to the zo-direction, making the associated θ change about 100 times larger than the other motions (see Supplementary Note 2). It should be noted that there are two kinds of O ions: one participating in the Ir–O–Ir bond and the other between Y ions (see Supplementary Note 3). In Ph5 and Ph6, O ions vibrate along the zo-direction, as shown in Fig. 4b. For Ph6, all of the O ions in Ir–O–Ir bond participate in the in-plane vibrational motion. However, for Ph5, only one-third of O ions in the Ir–O–Ir bond participate. It makes the difference in the average change of angle θ between two distinct eigenmodes.

Based on the DFT results for the phonon eigenmodes, we estimated the change in the bond angle θ for each phonon mode. We assumed that the magnitude of the ionic displacements associated with each eigenmode is ~0.01 Å. Then, we calculated the concomitant bond angle change, Δθ ≡ θ − θ0 = θ − 126.9°. The calculated Δθ/θ0 ratios are plotted as a bar graph in Fig. 5. Note that Δθ/θ0 becomes the largest for Ph6. The experimental Δω0/ω0 ratios are also plotted as black squares. We found that Δω0/ω0 and Δθ/θ0 have similar trends. This behavior suggests that the unusual phonon renormalization in Y2Ir2O7 is closely related to the modulation of the Ir–O–Ir bond angle.

Fig. 5
figure 5

Comparison between the phonon renormalization and calculation results. Relationship between the Ir–O–Ir bond angle change Δθ/θ0 and the experimental phonon renormalization Δω0/ω0 at 10 K. We calculated Δθ/θ0 (red bars) using the ionic displacements obtained by DFT. We obtained the experimental Δω0/ω0 (solid squares) from Fig. 3

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For conventional SPC, the bond length is the most critical parameter for modulating the IE interaction.27,29,50 Change in the bond length may affect the overlap integral significantly, resulting in a sizable change in JIE.51 In Y2Ir2O7, however, the Ir–O bond length change Δl has only a slight effect on Δω0. We calculated the length changes Δl/l0 associated with each phonon mode and found that they deviate markedly from the experimental results, especially for Ph4 and Ph7 (see Supplementary Note 4). Additionally, if Δl is indeed the critical parameter for the phonon renormalization, the stretching modes would be expected to show the largest frequency shift. However, the stretching mode Ph7 shows no frequency shift near TN. These observations indicate that Δl cannot play important roles in ∆ω0 at TN.

Importance of the DM interaction for the phonon renormalization

To elucidate the microscopic origin of SPC in Y2Ir2O7, we meticulously examined the relation between Δω0 and Δθ. The bond angle change induced by lattice vibration can affect the Hspin in Y2Ir2O7, given in equation (1). Due to the strong SOC in pyrochlore iridates, the low energy magnetic degrees of freedom may be represented by pseudospins, Jeff = 1/2.34,52 Due to the 1/2 value of Jeff, the SIA term becomes negligible.53 Previous studies have also concluded that the AE interaction is at least an order of magnitude smaller than the other terms in Hspin.13,54,55,56 Consequently, for Y2Ir2O7, Δω0 may be largely attributed to the IE and DM interactions:

$$\Delta \omega _0 \cong - \frac{2}{{I\omega _0}}\left( {\mathop {\sum}\limits_{i,j}^{{\mathrm{nn}}} {\frac{{\partial ^2J_{{\mathrm{IE}}ij}\left( {\mathrm{\theta }} \right)}}{{\partial {\mathrm{\theta }}^2}}} \left\langle {{\mathbf{S}}_i \cdot {\mathbf{S}}_j} \right\rangle + \mathop {\sum}\limits_{i,j}^{{\mathrm{nn}}} {\frac{{\partial ^2{\mathbf{D}}_{ij}\left( {\mathrm{\theta }} \right)}}{{\partial {\mathrm{\theta }}^2}}} \cdot \left\langle {{\mathbf{S}}_i \times {\mathbf{S}}_j} \right\rangle } \right),$$
(2)

where I is the rotational inertia of the given phonon57 (see Supplementary Note 5 for derivation). The positive or negative sign of Δω0 determines whether the corresponding phonon softens or hardens.

To compare the contributions of the IE and DM interactions quantitatively, we considered a tight-binding model based on O-mediated Ir-Ir hopping.13 Previous tight-binding models for pyrochlore iridates assumed that Ir–O hopping could occur along the local axes.13,54,55,56,58 However, the actual lattice structure includes a strong trigonal distortion of the O octahedra even though the global lattice symmetry is maintained. This distortion moves the O ion position away from the local axes of Ir ions (i.e., along the zL and yL axes in the Fig. 6a). Figure 6a show the changes of Ir–O–Ir bond during the trigonal distortion. Note that when the Ir–O–Ir bond angle changes to θ0, the Ir–O hopping is no longer aligned to the local axes. With proper consideration of the ionic position changes, we analytically obtained the θ-dependent Ir–O–Ir hopping. Then, using the Slater-Koster tables, we derived the bond angle-dependent coefficients JIE(θ) and D(θ) (=‖Dij(θ)‖) in Hspin54,55,56,58,59 (see Supplementary Note 5 for derivation).

Fig. 6
figure 6

Comparison between the IE and DM interaction contributions for SPC. a O-mediated hopping between the Ir sites. While maintaining the global lattice symmetry, the trigonal distortion of octahedra displaces the O ions from one of the Ir local axes. This O ion displacement direction is parallel to the in-plane bending motion. b, c The coefficients of the IE and DM interactions (JIE and D), and their second derivatives of EIE = JIEij(θ)〈Si∙Sj〉 and EDM = Dij(θ)∙〈Si × Sj〉. Here we anchored the JIE and D values at 130° using values from quantum chemistry calculations (see Supplementary Note 5).60 Note that near θ = 130° the value of ∂2EDM/∂θ2 is an order of magnitude larger than that of ∂2EIE/∂θ2, implying the importance of the DM interaction for SPC

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To make a quantitative comparison between JIE(θ) and D(θ) for Y2Ir2O7, we need to anchor their values for at least one angle. Unfortunately, experimental values have not yet been reported for Y2Ir2O7. We therefore rely on a previous quantum chemistry calculation (QCC) study that computed JIE(θ) (= 18.8 meV) and D(θ) (= 2.6 meV) at the experimentally determined angle θ0QCC (= 130°).60,61 Using these QCC values, we quantitatively determined JIE(θ) and D(θ), which are displayed in Fig. 6b.

Using the JIE(θ) and D(θ) values, we then estimated the IE and DM interaction energies EIE = JIEij(θ)〈Si∙Sj〉 and EDM = Dij(θ)∙〈Si × Sj〉. By taking the second derivatives, i.e., the ∂2EDM/∂θ2 and ∂2EIE/∂θ2 curves shown in Fig. 6c, we can compare both the IE and DM contributions to Δω0 using equation (22). The results suggest ∂2EDM/∂θ2 ≈ 1.00 eV/radians2 and ∂2EIE/∂θ2 ≈ −0.09 eV/radians2 at θ = 130°. The contribution to phonon renormalization of the DM interaction (∂2EDM/∂θ2) is therefore about an order of magnitude greater than that of the IE interaction (∂2EIE/∂θ2). The dominance of the DM interaction is robust even when the anchoring values of JIE(θ) and D(θ) were varied significantly. The DM interaction therefore plays a key role in the unconventional SPC observed in Y2Ir2O7.

Discussion

We now discuss the importance of the bond angle, as this determines the dominant SPC mechanism. For the IE interaction, we can empirically write \(J_{{\mathrm{IE}}}\left( {\mathrm{\theta }} \right) = J_{90^\circ }\sin ^2{\mathrm{\theta }} + J_{180^\circ }\cos ^2{\mathrm{\theta }}\), where J90° and J180° represent the IE interaction terms from the bond angles 90° and 180°, respectively.62 Using a Taylor expansion, JIE(θ) becomes an approximately linear function of θ around 135°. And as a result, the SPC due to the IE interaction (i.e., ∂2EIE/∂θ2) is strongly suppressed with θ ~ 135°. In contrast, the DM interaction is expected to vary as sin(2θ),63 meaning ∂2EDM/∂θ2 is zero at 180° and a maximum near 135°. Therefore, the DM interaction can play a dominant role in driving the SPC when θ is close to 135°.

To obtain a better perspective, we now compare the phonon renormalization of Y2Ir2O7 to that of two other 5d TMOs: NaOsO3 and Ca2Os2O7. The relevant physical parameters are summarized in Table 2.11,50,60,61,64,65 Recently, Calder et al. reported that the breathing phonon mode of NaOsO3 has Δω0~5.0 meV, the largest observed in solids.50 This large renormalization was attributed to the strong modulation of the IE interaction by the Os–O-Os bond stretching motion. Indeed, NaOsO3 has a distorted perovskite structure and θ is close to 180°. Following our earlier arguments regarding θ, the IE interaction is expected to be the leading contributor to the SPC. The SPC mechanism in NaOsO3 may therefore be distinguished from that of Y2Ir2O7 due to different bond angles in the two compounds.

Table 2 Physical parameters for 5d transition metal oxides: iridate and osmates
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We now turn to the SPC occurring in Cd2Os2O7, a compound that shares the same crystal structure and magnetic ground state as Y2Ir2O7. Sohn et al. reported a huge phonon hardening below TN in Cd2Os2O711 and attributed it to the SPC via the SIA. The SIA energy can change rapidly with the trigonal distortion of the OsO6 octahedra and so is especially sensitive to phonons that modulate this distortion.11,65 In contrast, in Y2Ir2O7, we can neglect the SIA term because of Jeff = 1/2, as shown in equation (22). It is also of interest to note that, despite their structural and magnetic similarities, Cd2Os2O7 and Y2Ir2O7 reveal opposite signs of Δω0 due to the different origins of the SPC.

In summary, we have established DM-mediated strong SPC in Y2Ir2O7 using both IR spectroscopy and first-principles calculations. Remarkable variations in the phonon parameters (frequency and linewidth) were observed near TN. These changes can be understood in terms of the change in Ir–O–Ir bond angle induced by the relevant phonon. We also explained that the dynamic modulation of the DM interaction is maximized in materials with large SOC and Ir–O–Ir angle close to 135°. We expect that similar phenomena should be observed in other materials with large SOC and highly non-collinear spin ground states.

Methods

Synthesis

We obtained a high-quality polycrystalline Y2Ir2O7 sample using Y2O3 (99.999%; Alfa Aesar) and IrO2 (99.9%; Sigma Aldrich) powders. As IrO2 evaporates easily during sintering, we used 4–10% more IrO2 than suggested by the stoichiometric ratio, Y2O3: IrO2 = 1:2. The mixture was pelletized and calcinated in a box furnace at 760 °C for 48 h. After re-grinding and re-pelletizing, several annealing cycles were performed at temperatures between 900 °C and 1030 °C, each for 48 h. High-density targets were prepared for optical experiments by pelletizing Y2Ir2O7 powder using a high-pressure anvil (NCAP-035; Nanki Engineering) operating at 4.3 GPa. Samples with densities >87% were polished to render the surfaces shiny and flat (see Supplementary Note 6 for details).

X-ray diffraction measurement

XRD was used to verify the quality of our Y2Ir2O7 crystals, as outlined below. Powder XRD data were obtained using a high-resolution powder diffractometer (D8 Discover; Bruker) operating at wavelengths of 1.54590 and 1.544310 Å (Kα1 and Kα2, respectively).

Optical conductivity fitting

We measured the near-normal reflectance spectra R(ω) by Fourier transform infrared spectroscopy (VERTEX 80 v; Bruker) and spectroscopic ellipsometry (V-VASE; J.A. Woollam). To compensate for the effects of rough surface, we normalized R(ω) relative to those measured after in-situ gold evaporation.66 From R(ω), we calculated the optical conductivity spectra, \(\tilde \sigma\)(ω) = σ1(ω) + iσ2(ω) using the Kramers-Kronig transformation. To track the spectral components, we fitted σ1(ω) to a Fano–Lorentz line shape, given by

$$\sigma _1\left( \omega \right) = \frac{{{\mathrm{SW}}}}{{{\mathrm{4\pi \Gamma }}}}\frac{{q^2{\mathrm{ + 2}}q\epsilon - {\mathrm{1}}}}{{q^2\left( {{\mathrm{1}} + \epsilon ^2} \right)}},$$
(3)

where ω is the angular frequency of the photon, ϵ = (ω − ω0)/Γ is a dimensionless parameter for reduced energy, and q is the Fano asymmetry parameter. Here, ω0, Γ, and SW = \(8{\int} {\sigma _1\left( {\omega ^\prime } \right){\rm{d}}\omega ^\prime }\) are the resonance frequency, linewidth, and spectral weight of the phonon, respectively. For the data measured at 10 and 300 K, the fitted curves are presented as black solid lines in Fig. 2a (for more detailed information, see Supplementary Note 1).

DFT calculations

To obtain the phonon eigenmodes of Y2Ir2O7, we performed DFT+U calculations with the PBEsol exchange correlation function implemented in VASP.67 We chose the effective U = 1.5 eV to simulate the local Hubbard interaction of Ir,35 and used the non-collinear DFT formalism to express the AIAO magnetic configuration. We fully optimized the lattice structure and obtained Ir–O–Ir angle θ0 ≈ 126.9°, as shown in Fig. 4b. We calculated the zone-center phonon modes using the frozen-phonon method. The resulting phonon eigenmodes are summarized in Supplementary Note 3.

Tight-binding model analysis

Using a tight-binding model, we derived JIE(θ) and D(θ), which were required to calculate Δω0 in equation (22). We derived the effective hopping Hamiltonian, based on the model of Pesin and Balents.13 However, we applied the effects of strong trigonal distortion of octahedra. We considered two types of electron hopping between the nearest neighbor Ir ions: namely, O-mediated Ir-Ir hopping toxy and direct Ir-Ir hopping tσ. With t(θ) = a(θ) toxy + b tσ and v(θ) = a’(θ) toxy + b’ tσ, we could derive the IE and DM interaction: \(J_{{\mathrm{IE}}} = \frac{4}{U}\left( {t^2\left( {\mathrm{\theta }} \right) - \frac{1}{3}v^2\left( {\mathrm{\theta }} \right)} \right)\) and \(D = \frac{8}{U}t\left( {\mathrm{\theta }} \right)v\left( {\mathrm{\theta }} \right)\). The coefficients a(θ), b, a’(θ), and b’ were obtained using the Slater-Koster method59 (see Supplementary Note 5). We estimated the values of toxy and tσ by anchoring with the reported values (JIE = 18.8 meV and D = 2.6 meV at θ0QCC = 130°) obtained from quantum chemistry calculation (QCC).60 The calculated JIE(θ) and D(θ) are presented in Fig. 6b.