Unconventional polaronic ground state in superconducting LiTi_2O_4

Domain: physics
Published: 2026-01-06
Editorial outcome: minor_revision

Abstract
Geometrically frustrated lattices can display a range of correlated phenomena, ranging from spin frustration and charge order to dispersionless flat bands due to quantum interference. One particularly compelling family of such materials is the half-valence spinel LiB2O4 materials. On the B-site frustrated pyrochlore sublattice, the interplay of correlated metallic behavior and charge frustration leads to a superconducting state in LiTi2O4 and heavy fermion behavior in LiV2O4. To date, however, LiTi2O4 has primarily been understood as a conventional BCS superconductor despite a lattice structure that could host more exotic ground states. Here, we present a multimodal investigation of LiTi2O4, combining ARPES, RIXS, proximate magnetic probes, and ab-initio many-body theoretical calculations. Our data reveals a novel mobile polaronic ground state with spectroscopic signatures that underlie co-dominant electron-phonon coupling and electron-electron correlations also found in the lightly doped cuprates. The cooperation between the two interaction scales distinguishes LiTi2O4 from other superconducting titanates, suggesting an unconventional origin to superconductivity in LiTi2O4. Our work deepens our understanding of the rare interplay of electron-electron correlations and electron-phonon coupling in unconventional superconducting systems. In particular, our work identifies the geometrically frustrated, mixed-valence spinel family as an under-explored platform for discovering unconventional, correlated ground states.

Introduction
The interplay between electron–electron correlations and electron–phonon coupling and electron–electron interactions. Strong electron–phonon coupling has been of long-standing interest in correlations can further strengthen the electron-phonon coupling and understanding superconductivity. In traditional Bardeen-Cooper- enhance Tc. In certain cases, as explored for cuprates, the energy Schrieffer (BCS) superconductors, Coulomb repulsion between elec- scales of strong electron–electron correlations can overwhelm those of trons is thought to screen electron-phonon coupling and reduce the electron–phonon interactions, leading to charge and spin fluctuations superconducting transition temperature Tc. In unconventional super- that compete with or enhance superconductivity. Thus, understanding conductors, however, there can be a more complex interplay between how electronic correlations and electron–phonon coupling interact in superconductors is important in identifying new families of uncon- photoemission spectroscopy (ARPES). The combination of the element- ventional superconductors and ascertaining the mechanisms behind specific sensitivity to structural and local excitations of RIXS, together high-temperature superconductivity.

Although superconductivity was discovered in LiTi2O4 before the with the unique capability of ARPES to reveal energy and momentum cuprates, it remains a unique instance of a spinel superconductor and dependence of the quasiparticle self-energy, enables us to provide a with the highest Tc for a mixed-valence titanate. In this crystal struc- comprehensive description of the low-energy physics of LiTi2O4. We ture, d0.5 titanium atoms form a pyrochlore sublattice, composed of observe evidence of strong electronic correlations and signatures of alternating planes of Kagome and triangles (Fig. 1a, b). While this strong electron–phonon coupling. The cooperation between these two geometric construction has given rise to strong electron correlations interaction scales gives rise to a novel polaronic ground state—also in other systems, these interactions have been largely unexplored in found in weakly doped cuprates—that dynamically localizes titanium LiTi2O4. Indeed, superconductivity in LiTi2O4 has primarily been con- states in LiTi2O4. This interpretation is further supported by our theo- sidered as phonon-mediated BCS-like, supported by specific heat, retical calculations, which can reproduce some of the spectral features in tunneling spectroscopy, and muon spin rotation measurements. In our ARPES data when considering electron correlations or contrast, anomalous magnetotransport behavior and scanning tun- electron–phonon interactions separately, but are unable to reproduce neling spectroscopy experiments have suggested the presence of spin polaronic phenomena associated with the interplay of both interactions. fluctuations, orbital ordering, and pseudo-gap-like states in LiTi2O4, all Our work thus challenges the notion of phonon-dominated BCS super- of which are typically associated with unconventional super- conductivity in LiTi2O4 by revealing the complex correlations present in conductors and strongly correlated materials. Interestingly, pre- this material, hearkening comparisons to cuprate-like physics.

Results
Synthesis and characterization of superconducting LiTi2O4
Thin films of LiTi2O4 were grown via reactive oxide MBE on (111)- oriented MgAl2O4 substrates (see “Methods”). As shown in Fig. 1c, the films display a superconducting transition with Tc ~ 12.5 K and a resi- dual resistivity ratio (RRR) of ~ 5.85 (Fig. S3), consistent with previous reports of the highest quality LiTi2O4. Figure 1d, e show high-angle annular dark field scanning transmission electron microscopy (HAADF-STEM) images of LiTi2O4, attesting to the high structural quality of our films (also see Supplementary Fig. S1).

Additionally, we confirm the onset of superconductivity and probe the superconducting order parameter using scanning superconducting quantum interference device (SQUID) microscopy, which locally measures magnetic susceptibility (Fig. S4). The measured magnetic susceptibility can be used to extract the temperature dependence of the reduced London penetration depth (Supplementary Material Section S3), which can be well-fit by a fully gapped order parameter, consistent with s-wave-like pairing. The observation of the 2-D thin film limit of superconductivity, the high RRR, as well as a homogeneous diamagnetic response from scanning SQUID, demon- strates the high structural quality of our films, ensuring that further measurements probe the intrinsic behavior of LiTi2O4.

Electron–electron correlations in LiTi2O4
Poised with high-quality thin films, we first probe the electronic structure using elemental-specific RIXS measurements. Figure 2a shows the RIXS intensity of LiTi2O4 near the Ti-L edge. Our x-ray absorption spectroscopy (XAS) data (Fig. S16a) display resonant fea- tures associated with the mixed valence Ti4+ and Ti3+ states (for a detailed discussion on the mixed-valence XAS, see Supplementary Section S9). Given the lack of dd excitations expected for a trivial d0 state, we conclude that the broad Raman-like dd features centered at Eloss = 1.5 eV and Eloss = 4 eV, reflect a d1 titanium occupation state, which, given the lack of charge order in LiTi2O4, can only be dynami- cally populated. These Raman-like excitations, along with the sup- pressed fluorescence contribution, are unexpected, given that the itinerant charge carriers in our metallic and superconducting samples are proposed to have a dominant Ti-3d band character near the Fermi level. We note that our RIXS data resemble that of MgTi2O4 despite LiTi2O4 lacking large static local trigonal distortions of the TiO6 octahedra and Ti-Ti dimerization (Supplementary Figs. S1 and S22). The qualitative similarities between the RIXS energy maps in metallic LiTi2O4 and Mott-insulating systems like MgTi2O4 are suggestive of the presence of localized excitations in LiTi2O4.

In addition to the Raman-like excitations, we observe spectral weight associated with a fluorescence contribution with a linear dis- persion in Eloss that extends below Eloss ≤ 1.5 eV. This resembles the superposition of localized and delocalized excitations observed in a negative charge transfer insulators. Moreover, O K-edge RIXS, shown in Fig. 2b, also shows Raman-like and fluorescence contributions below the charge transfer gap, Δ ≈ 4.5 eV. This is the same energy as the intra- dd excitations at the Ti L3-edge, suggesting strong hybridization between titanium and oxygen carriers. While the hybridization between Ti-3d and O-2p states has been previously theorized, the RIXS data is the first instance of direct observation of strong titanium- oxygen hybridization. Moreover, the presence of fluorescence con- tributions at the O-K-edge near the elastic line (zero energy loss) sug- gests a larger contribution of oxygen carriers to electronic transport in LiTi2O4 than that expected from previous density of states calculations. Thus, our RIXS data suggest the presence of charge localization, Ti-3d and O-2p hybridization, and electron–electron cor- relations in LiTi2O4.

Electron–electron correlations are also observed in in-situ ARPES measurements on LiTi2O4. Figure 2c shows the Fermi surface of LiTi2O4 from our ARPES measurements. Here, kx and ky are chosen to lie along the [112] and [110] directions, respectively. The Fermi surface is char- acterized by large electron pockets with hexagonal symmetry centered at Γ, pushing toward the zone boundary in proximity to a Lifshitz transition, which has led to enhanced electronic interactions near the Fermi surface for other superconductors. A more detailed discussion of the Fermi surface and its kz dependence is given in Supplementary Figs. S5, S6 and in the Supplementary Material Section S4.

To analyze the electronic interactions, we look at high-symmetry cuts of the band structure of LiTi2O4. In Fig. 2d, e, we compare our experimental band structure along Γ–K to the momentum-resolved spectral function obtained from density-functional theory plus dyna- mical mean-field theory (DFT+DMFT). The non-interacting band dis- persion is shown in white in Fig. 2e for reference. The DFT+DMFT calculation reproduces the experimental dispersion near Γ, capturing the large effective mass renormalization compared to the bare DFT band—a clear indication of strong electron correlations in LiTi2O4. The calculated cyclotron mass (Supplementary Material Fig. S9) yields meff/mbare = 2.53 ± 0.4. This method of calculating the renormalization agrees with previous reports of the effective mass of LiTi2O4 by other optical probes. Additionally, we observe a broad Gaussian-like, intense incoherent spectral weight centered at EB ≈ 0.9 eV that resembles the lower Hubbard band of lightly doped cuprates. Similar to the cuprates, the spectral weight shows k dependence with a transition from a “champagne glass-like dispersion" to a “waterfall" feature in the data taken by Ne-Iα photon energy (16.85 eV) (Sup- plementary Fig. S6b). We note this feature cannot be reproduced by DMFT calculations but is consistent with a previous photoemission spectroscopy measurement of LiTi2O4, where it was interpreted as a signature of polaronic behavior. Nonetheless, our combined ARPES and RIXS data provide the first observation of significant electronic correlations in LiTi2O4.

Electron–phonon coupling in LiTi2O4
While electronic correlations play a key role in unconventional superconductivity, we also observe signatures of electron–phonon coupling in the electronic structure of LiTi2O4. Consistent with pre- vious ARPES reports, we observe a “kink" at EB ≈ 46 meV as shown in Fig. 3a. This feature is ascribed to an Eg oxygen phonon mode by tunneling spectroscopy measurements and inelastic neutron scattering, confirmed by calculations of the spectral function including the self-energy from only electron–phonon interactions (Supplementary Material Section S5). We extract a band renormaliza- tion, λtot = 1.80(1), three times higher than λe–ph = 0.65 determined by specific heat measurements and ab-initio calculations. The extracted band renormalization value from our ARPES data is comparable to that from electronic specific heat measurements, where the larger value is attributed to enhancement from electron–electron correlations or spin fluctuations. We note that λtot includes λe–ph as well as λe–e. While a precise determination of just λe–ph is non-trivial due to the interlinked nature of electron–electron correlations and electron–phonon coupling in this compound, we can approximate λe–ph ≈ 1.3 ± 0.5 by fitting the dispersion of the band at lower binding energies as the bare band, consistent with λe–ph extracted from other ARPES measurements on this compound. For a more detailed dis- cussion on the calculation of λtot and λe–ph as well as their temperature and momentum dependence, see Supplementary Material Section S4 and Fig. S7.

Figure 3b shows an energy-momentum cut along the Γ–K–M direction (cut overlaid on the Brillouin zone is shown in Fig. S8a). Here we observe a non-dispersive feature at EB = 70.4 ± 6 meV at all k||, highlighting the accumulation of incoherent spectral weight below the main kink at E = 46 meV (Fig. 3a). To examine this flat feature, we present two energy dispersive cuts (EDC) in Fig. 3c, k which cuts EDC1 across momentum value devoid of bands and k which cuts across EDC2 the kF of the main LiTi2O4 band. The momentum position of these two cuts is indicated by two black arrows pointing to dotted white lines in Fig. 3b. We can see the peak position of this sub-band in the EDC of k as an abrupt and sharp increase in spectral intensity at EDC1 Ω = 70.4 ± 6 meV (green dot), also observed in k at the same energy EDC2 scale. Sub-bands like these are indicative of a highly interacting picture, in stark contrast to a non-interacting picture, where, in the absence of any bands, the expected EDC would not have any additional poles, unlike what is seen in k. The second derivative plot (Fig. 3d) EDC1 further highlight this feature and additionally shows a “spectral gap" where the non-dispersive feature interrupts the main dispersive elec- tronic band. We note that the flat feature, in addition to the spectral gap, can also be seen clearly in our raw data in Fig. 2d. These types of sub-bands due to strong electron-phonon coupling are spectral sig- nature that are consistent with the formation of intra-unit cell small polarons that have been observed in other correlated materials.

Our RIXS measurements also corroborate the presence of strong electron-phonon coupling and multiphonon processes in LiTi2O4. Figure 3e shows the RIXS spectra at the Ti L3- and O K-edges, respec- tively. As highlighted by black arrows in Fig. 3e, f, both RIXS spectra are characterized by prominent quasi-elastic peaks that are approximately equally spaced and monotonically decay with Eloss, thus resembling the harmonic progression of the multiphononic processes. The low- energy excitations shown in the titanium spectra are well fit by the Ament model, which considers a single non-dispersive phonon mode coupled to the electronic structure with strength g (See Supplemen- tary Material Section S6 for more details on the Ament model fit). At the Ti-L3 edge, the Ament model in conjunction with a broad Gaussian peak centered at Eloss = 90 meV provides a good fit for a mode centered at Ω = 47 ± 2 meV, in agreement with the oxygen Eg mode observed in ARPES. We interpret this broad Gaussian peak to represent multiple incoherent phononic excitations that cannot be resolved within the energy resolution of the instrument, also observed in MgTi2O4. The coupling strength, gTi = 9.1 ± 3.0, far exceeds those of other titanates, and is on par with some cuprates measured via RIXS, further supporting strong electron-phonon coupling in LiTi2O4. Moreover, the observation of a pure oxygen Eg mode at the titanium edge is indicative of strong hybridization between the O-2p and the Ti-3d orbitals. Similar to the Ti L3-edge, the three excitations at the O K-edge can also be well fit by the Ament model in conjunction with two broad Gaussian peaks. The fit gives us the first three harmonics of a phonon mode centered at Ω = 72 ± 1 meV with gO = 4.13 ± 2.76.

Methods
Synthesis of LiTi2O4 thin films via MBE
We used reactive oxide molecular beam epitaxy (Veeco GEN 10) to synthesize thin film LiTi2O4 on untreated MgAl2O4 (111) substrates (CrysTec GmbH). The lithium and titanium fluxes were matched and set to ~1.5 × 10^13 atoms/cm^2·s as measured by a quartz crystal micro- balance (QCM). We obtained the lithium flux by flowing O2 during the QCM process and forming Li2O, as the low atomic mass of elemental lithium yields small frequency changes on the QCM below the detec- tion sensitivity, except for at very high fluxes. For the deposition process, we heated the substrates with a 10.6 μm CO2 laser to 825 °C in a pressure of 1.0–2.0 × 10^−7 molecular O2 over a chamber background pressure of ~5.0 × 10^−8. We co-deposited the lithium and titanium sources for 1 h. After deposition, we shut off the O2 flow to prevent oxidation of titanium to the 4+ state and cooled at a rate of 100 °C/min.

Structural characterization
Annular dark field (ADF) STEM was performed on Thermofisher Sci- entific (TFS) Themis (operated at 200 keV, convergence semi-angle 18.9 mrad, ADF collection angle: 36–200 mrad). A small ADF inner collection angle was used to increase light element sensitivity. Electron-transparent TEM samples were prepared using TFS Helios DualBeam FIB/SEM.

Transport measurements
We performed all electrical transport measurements in a Hall bar geometry using evaporated Cr/Au (7 nm/100 nm) contacts with the Hall channel defined by a diamond scribe along the [11-2] crystal- lographic direction of the MgAl2O4 (111) substrate. We loaded the samples in 9 T Dynacool Physical Property Measurement System (PPMS) and used AC lock-in techniques at ~15 Hz.

Resonant inelastic X-ray scattering
The XAS and RIXS spectra were collected at the SIX beamline at NSLS-II and the I21 beamline of Diamond Light Source. The Ti L3-edge was measured at SIX and I21, while the O K-edge spectra were acquired at I21. The Ti L3-edge energy map is collected at 2θ = 150°, while the O K-edge energy map is collected at 2θ = 90°. All RIXS spectra are col- lected at 2θ = 90° at grazing incidence of θ = 20° unless indicated otherwise. The energy resolution was fixed between 20 to 25 meV at both SIX and Diamond, respectively. All data were measured at a base temperature of 22 K at both beamlines.

Angle-resolved photoemission spectroscopy
All in situ photoemission measurements were conducted by immedi- ately transferring the samples through a UHV manifold (P < 2 × 10^−9 Torr) to a measurement chamber immediately following film growth. ARPES measurements were taken with a Scienta Omicron DA30-L electron analyzer equipped with a Fermion Instruments BL1200s multi-gas discharge lamp using He-I photons at 21.2 eV, Ne-I photons at 16.85 eV, and He-II photons at 40.2 eV. The base pressure in the ARPES system is maintained during measurements at pressures lower than 5 × 10^−11 Torr. ARPES measurements were performed at a temperature of 7 K and a nominal experimental energy resolution of 10 meV unless otherwise indicated.

Muon spin relaxation measurements
Low-energy muon spin-relaxation (LE-μSR) measurements were done using the LEM instrument at the Swiss muon source. An applied field of 10 mT was applied transverse to the μ+ spin polarization direction. For more details on the μSR measurements, refer to Supplementary Material Section S7.

X-ray absorption spectroscopy
XAS of uncapped LiTi2O4 thin films was performed at 300 K at Beamline 6.3.1 at the Advanced Light Source at Lawrence Berkeley National Laboratory. XAS of capped LiTi2O4 thin films was performed at 300 K at ID-32 at Diamond Light Source. The spectra from Beamline 6.3.1 represent an average of 4 individual scans. All spectra are taken in the total electron yield configuration. The LiTi2O4 XAS spectra do not show any polarization dependence. For more details on the XAS, refer to Supplementary Material Section S9.

Data availability
All data used to obtain the conclusions in this paper are presented in the paper and/or the Supplementary Materials. This data may be requested from the authors. Please direct all inquiries to Z.H. (zhasan@g.harvard.edu).

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Hasan et. al report the experimental results on LiTi2O4 thin film using ARPES and RIXS. They reveal a complex interplay between electron-electron correlation and electron-phonon coupling, which result in a novel polaronic ground state in LiTi2O4. The samples grown by MBE are of high quality, both the RIXS and ARPES data show strong evidences of the existing electron-electron correlation and electron-phonon coupling. And the authors think the incoherent spectral weight ~ 800 meV in the ARPES spectra which is missing in the calculation refers to the novel mobile polaronic ground state of LiTi2O4. This observation emphasizes the importance of both electron-electron interaction and electron-phonon couplings in understanding the superconductivity in LiTi2O4, which also challenges the conventional BCS superconducting mechanism. The paper is well organized and I have several concerns in the following:\n1. As the authors claim an unconventional polaronic ground state of LiTi2O4, the experimental evidence of the existence of the polaron state is rather weak to me. First, the incoherent spectra weight in Fig. 2d at ~ 900 meV is missing in Fig. S6 (b). If this is a signature of a polaron, how can we interpret this behavior? If we take this as the band bottom of the electron bands, seems this can be consistent with the DFT calculations.\n2. The author mentions that the extract electron-phonon coupling strength is 1.8 in the main text. However, the author mention in the SM that they use the velocity from the calculation, and they show the experimentally extracted lamda in Fig. S7 which is much smaller than 1.8. I think it is not appropriate to use the extracted bare band from the calculation here, the experimental value will give a relevant reliable value, since there exist strong electron-electron correlations.\n3. In Fig.3c, the authors assign the weak flat band signature to the formation of intra-unit cell small polarons. I noticed that, in the reference 14 the author cited, they also observed a near dispersive band at the similar energy, which they think it is originated from a surface state.\n4. In line 189 and 194, the incoherent spectral weight around 46 meV is not correct. It should be somewhere ~ 80 meV.\n5. In line 164, the authors note EB~0.9 eV, while in line 272, they note E~ 800meV. This should be self-consistent.", "reviewer_2": "Report on “Unconventional polaronic ground state in superconducting LiTi2O4 “by Z. Hasan et al.\nThe authors report spectroscopic measurements of superconducting LiTi2O4 thin films using ARPES and RIXS. Based on comparisons with calculations using DFT and DMFT, they provide evidence for the interplay between electron-phonon coupling and electron-electron interactions. They also identify Eg and A1g phonon modes with energies of 47 and 71 meV, respectively. Additionally, the energy-dependent RIXS results, ARPES data, and DFT calculations—which fail to account for the broad incoherent spectral weight—lead the authors to propose a polaronic ground state in LiTi2O4. The manuscript is technically sound and well written; I support its publication, provided the following issues are addressed.\n1. The RIXS map shown in Fig. 2a is used to demonstrate the superposition of localized and delocalized excitations and to support the claim that the d1 Ti state is dynamically populated. However, the range of incident photon energies is too narrow to clearly show the coexistence of Raman-like and fluorescence excitations, as observed in MgTi2O4 (Fig. 4c of Ref. 24) Due to this limited energy range, the fluorescence contribution may be mistakenly interpreted as suppressed.\n2. Figures 3b, 3c, and 3d show that the dispersive electronic band interacts with the non-dispersive phonon bands at 70–80 meV, rather than at 46 meV as claimed by the authors (see line 188). Figure 3a indeed shows the strongest kink at 46 meV, and the sub-kinks appear to be distributed down to 70– 80 meV.\n4. RIXS data clearly show the presence of both Eg and A1g phonon modes. Specifically, Ti L3-edge RIXS detects only the Eg and phonons at an energy of 41 meV, while the A1g phonons appear in the O K-edge spectra. The authors are suggested to elaborate on why these two phonon modes are observed at different absorption edges.\n5. The intensity bars shown in Figs. 2 and 3 lack a scale. It is unclear whether the current intensity bars are linearly or logarithmically scaled.", "reviewer_3": "The authors present a spectroscopic study of LiTi2O4 thin films combining ARPES, RIXS, and theoretical calculations. They observe strong d1-like features in RIXS spectra despite the metallic d^0.5 electronic configuration, along with significant electron-phonon coupling and mass enhancement from ARPES measurements. Based on these observations, they propose that LiTi2O4 exhibits an unconventional mobile polaronic ground state arising from the cooperative interplay of electron-electron correlations and electron-phonon coupling, challenging the conventional BCS interpretation of superconductivity in this material. The authors suggest this represents a novel example of correlated physics in mixed-valence spinel oxides.\nWhile their study is important and interesting, there are several questions needed to be answered:\n1. The explanation of the Ti^4+ and Ti^3+ components in the XAS data shown in Fig. S13 is unclear. To support the claim that the spectrum reflects a mixture of Ti3+ and Ti4+, the authors should directly compare reference XAS spectra for Ti3+ and Ti4+ with the experimental data. Which specific spectral features correspond to Ti3+, and which are indicative of Ti4+? Please clarify.\n2. The authors state that the RIXS map in Fig. 2a reflects a Ti d¹ state, citing the “nearly featureless” spectrum expected for a d^0 configuration (as shown in Ref. 20). However, Ref. 20 reports several RIXS features between 0–200 meV for SrTiO3 (d0). Could the authors clarify whether “nearly featureless” implies that these d⁰-related peaks are significantly weaker compared to those from d¹ states? A quantitative comparison would help support this interpretation.\n3. In addition, is it possible that the RIXS peaks observed between 0–0.2 eV in Fig. 3e originate from Ti4+ (d0) states, similar to those reported in SrTiO (STO)? As noted earlier, STO exhibits several low-energy RIXS features in this range [Ref. 20: PRL 125, 126401 (2020)].\n4. Charge frustration between nominal Ti3+ and Ti3+ ions should lead to either (i) static charge order or (ii) rapid valence fluctuations that average to a homogeneous metallic state. Your Ti‑L RIXS spectra are interpreted as showing a robust d1 (Ti3+‑like) local configuration. Given that the core‑hole lifetime at the L edge is only a few femtoseconds, could the d1 signature simply reflect a snapshot taken on a timescale shorter than the electron‑hopping time between the two valence states? Authors need to compare the timescales of these processes. Could you estimate the hopping rate or activation energy of the polaron?\n5. DMFT calculation showed that LiTi2O4 is metallic, with average of Ti 3.5+. If there is \"charge localization\" in a short timescale, then should we also consider Ti3+ within DMFT, by adding additional electrons?\n6. Figs 1a-b shows the Kagome planes of LiTi2O4, but the Fermi surface appears large and dispersive . Could the authors comment on any flat‑band contribution near EF?\n7. Given the volatility of Li, even slight deviations from stoichiometry—or the presence of Li/Ti antisite defects—could induce polarons and significantly affect the electronic response. What is the measured Li:Ti ratio in your films, and how was it verified? Could modest off-stoichiometry or antisite disorder be responsible for the correlation or polaronic features observed in your data? In other words, is it possible that Li-related defects are the primary origin of the polaronic behavior?\n8. Following up on the previous question, the reported residual resistivity ratio (RRR = 5.3) is described as indicative of \"high-quality\" samples; however, bulk crystals often achieve RRR values of ~20. What types of defects are likely present in your samples, and how might these defects influence polaron formation or dynamics?"}, "author_response": "Response to reviewers for “Unconventional polaronic ground state in superconducting LiTi O thin films” (NCOMMS-25-00605-T)\n2 4\n(Dated: September 18, 2025)\nR1\nREVIEWER #1 (REMARKS TO THE AUTHOR):\nHasan et al. report the experimental results on LiTi O thin film using ARPES and RIXS.\n2 4\nThey reveal a complex interplay between electron-electron correlation and electron-phonon\ncoupling, which result in a novel polaronic ground state in LiTi O . The samples grown\n2 4\nby MBE are of high quality, both the RIXS and ARPES data show strong evidences of the\nexisting electron-electron correlation and electron-phonon coupling. And the authors think\nthe incoherent spectral weight ∼800 meV in the ARPES spectra which is missing in the cal-\nculation refers to the novel mobile polaronic ground state of LiTi O . This observation em-\n2 4\nphasizes the importance of both electron-electron interaction and electron-phonon couplings\nin understanding the superconductivity in LiTi O , which also challenges the conventional\n2 4\nBCS superconducting mechanism.\nOur response: We thank the reviewer for careful consideration of our experimental work\nand the positive comments regarding our ARPES and RIXS data that show strong electron-\nphonon coupling and electron-electron correlations. We hope to further clarify our work\nfrom our responses to the reviewer’s questions below.\n1. As the authors claim an unconventional polaronic ground state of LiTi O , the experimen-\n2 4\ntal evidence of the existence of the polaron state is rather weak to me. First, the incoherent\nspectra weight in Fig. 2d at ≈ 900 meV is missing in Fig. S6 (b). If this is a signature\nof a polaron, how can we interpret this behavior? If we take this as the band bottom of the\nelectron bands, seems this can be consistent with the DFT calculations.\nOur response: We appreciate the reviewer’s insightful question regarding the ARPES\nspectral features. We want to emphasize that the evidence for the existence of a polaronic\nstate is one possible interpretation of our results after investigation by multiple probes,\nwith the ARPES analysis providing only some but fundamental evidence for the proposed\npolaronic picture. We agree with the reviewer that the incoherent spectral weight in Fig.\n2d)atE ≈900meVissuppressedinFig. S6b). However, weremarkthatthedatapresented\nin Fig. S6b) is taken at a different k than the one in Fig. 2d. As a function of photon\nz\nR2\nenergy, the incoherent spectral weight in Fig. 2d) turns into a waterfall-like feature in Fig.\nS6b). As discussed in the main text, a similar k dependence of the incoherent spectral\nz\nweight below the Fermi level has been previously observed in other materials hosting strong\ncorrelations, including in electron-doped cuprates1. Nonetheless, we emphasize that while\nthe observed incoherent spectral weight at E ≈ 900 meV is not by itself a sufficient signature\nto claim polaronic physics in LiTi O , it showcases for the first time the presence of strong\n2 4\ncorrelation in this material. When combined with the rest of the experimental data,\ntheory\nand analysis, our comprehensive study of LiTi O promotes a possible polaronic ground\n2 4\nstate.\nTo clarify these points, we have made the following changes in the lines 164-172 in the\nmain text:\nMain text:\nAdditionally, we observe a broad Gaussian-like, intense incoherent spectral\nweight centered at E ≈ 0.9 eV that resembles the lower Hubbard band of\nB\nlightly doped cuprates2. Similar to the cuprates, the spectral weight shows k\nz\ndependence with a transition from a “champagne glass like dispersion” to a “wa-\nterfall”feature1 inthedatatakenbyNe-Iαphotonenergy(16.85eV)(Supp. Fig.\nS6b). We note this feature cannot be reproduced by DMFT calculations but is\nconsistent with a previous photoemission spectroscopy measurement of LiTi O ,\n2 4\nwhere it was interpreted as a signature of polaronic behavior3. Nonetheless, our\ncombined ARPES and RIXS data provide the first observation of significant\nelectronic correlations in LiTi O .\n2 4\n2. The author mentions that the extract electron-phonon coupling strength is 1.8 in the main\ntext. However, the author mention in the SM that they use the velocity from the calculation,\nand they show the experimentally extracted lamda in Fig. S7 which is much smaller than\n1.8. I think it is not appropriate to use the extracted bare band from the calculation here, the\nexperimental value will give a relevant reliable value, since there exist strong electron-electron\ncorrelations.\nOur response: We thank the reviewer for pointing out the discrepancy between the values\nR3\nreported in the main text (λ = 1.8) and the highest reported value in the supplement (λ ≈\n1.3). We agree that λ = 1.8 includes the band renormalization from strong electron-electron\ncorrelations as well and it is more appropriate to refer to this as a λ = λ + λ ,\ntot e−ph e−e\nwhich includes renormalization from both electron-phonon coupling and electron-electron\ncorrelations. To that end, we have made the following changes in lines 184-188 of the main\ntext:\nMain text:\nWe extract a band renormalization, λ = 1.80(1) three times higher than\ntot\nλ = 0.65determinedbyspecificheatmeasurements4 andab-initiocalculations5.\ne−ph\nThe extracted band renormalization value from our ARPES data is compa-\nrable to that from electronic specific heat measurements, where the larger\nvalue is attributed to enhancement from electron-electron correlations or spin\nfluctuations6–8. We note that λ includes λ as well as λ . While a pre-\ntot e−ph e−e\ncise determination of just λ is non-trivial due to the interlinked nature of\ne−ph\nelectron-electron correlations and electron-phonon coupling in this compound,\nwe can approximate λ ≈ 1.3 ± 0.5 by fitting the dispersion of the band at\ne−ph\nlower binding energies as the bare band, consistent with λ extracted from\ne−ph\nother ARPES measurements on this compound9. For a more detailed discussion\non the calculation of λ and λ as well as its temperature and momentum\ntot e−ph\ndependence, see Sec. Mater. Sec S4 and Fig. S7.\nThe following clarification has also been added in Supp. Sec. 4 of the supplement:\nSupplementary note 4: Angle-resolved photoemission spectroscopy\nNote that λ = 1.8 reported in the main text has been calculated the same way\ntot\nas described above, but with ν extracted from fitting the DFT-calculated bands\no\n¯ ¯\nat Γ - K for higher accuracy. This treatment includes the strength of electron-\nelectron correlations (λ ), as well as electron-phonon coupling (λ ).\ne−e e−ph\n3. In Fig.3c, the authors assign the weak flat band signature to the formation of intra-unit\ncell small polarons. I noticed that, in the reference 14 the author cited, they also observed a\nR4\nnear dispersive band at the similar energy, which they think it is originated from a surface\nstate.\nOur response: We thank the reviewer for their comment. We would like to clarify the ref-\nerences cited in favor of interpreting certain spectral features as signatures of small polarons.\nIn Fig. 3c, the weak flat band signature observed at E −E = 70.4± 6 meV, is similar to\nF\nsignatures observed in references 3510 and 3611 in the main text. In reference 143 there is\nno observation of a weak flat band signature or mention of states at an energy scale of ≈\n70.4 meV, but instead there is a spectral feature at 1 eV below E thought to arise from\nF\nthe formation of polarons. This 1 eV spectral feature is similar to the incoherent spectral\nweight we observe in Fig. 2d. In reference 143, the only mention of surface states is found\nwhere the oxygen bands live at E−E = 5 eV which is a significantly different energy scale\nF\nfrom the energy scale of the features we have highlighted in the main text. A visual guide\nto all the relevant energy scales discussed in reference 14 compared to our data is shown in\nFig. R1 below. Furthermore, we note that in reference 143, these surface states are thought\nto be due to surface oxygen vacancies, which increase upon annealing in vacuum to prepare\na clean surface. On the other hand, annealing is not necessary in our in-situ ARPES ex-\nperiment as the vacuum is never broken between MBE synthesis and ARPES measurement,\nthus maintaining a pristine surface.\nR5\nky (1/Å)\n)Ve(\nE-E F\ni ii\n0.0\nOxygen\n-0.1 non-dispersive\n2p surface\n-0.2\nfeature ~ 70 meV\nstates ~ 5 eV\n~ 1 eV -0.3\nstates -0.4\n-0.5\n-0.6\n-0.7\nIncoherent\n-0.8\nspectral weight\n-0.9\nat ~ 1 eV\n-1.0\n-0.50 -0.25 0.00 0.25 00..5500\n0 2 4 6 8 10 12\nFIG. R1: Comparison of energy scales in our data with Reference 14. i,\nAnnotated reproduction of Fig. 2 in reference 14. Relevant features are highlighted by\narrows. ii, Fig. 2d from the main text. The incoherent spectral weight at ≈ 1 eV is\nhighlighted as well as the non-dispersive spectral feature at ≈ 70 meV.\n4. In line 189 and 194, the incoherent spectral weight around 46 meV is not correct. It\nshould be somewhere ∼80 meV.\nOur response: We appreciate the reviewer for carefully reviewing the figures, pointing out\nthis possible point of confusion and catching our mistake in reporting the value of the replica\nband. Indeed, the kink occurs at E = 46 meV in Figure 3a, as expected for the oxygen E\ng\nmode. The discussion of the EDCs in Figure 3b highlights the existence of non-dispersive\nfeature below the main kink at around 70.4 meV, suggestive of small polaron formation as\nseen in Ref.11. We have changed lines 193-210 to correct for this:\nMain text:\n¯ ¯ ¯\nFigure 3b shows an energy-momentum cut along the Γ−K −M direction (cut\noverlaid on the Brillouin zone is shown in Fig. S8a). Here we observe a non-\ndispersive feature at E = 70.4±6 meV at all k , highlighting the accumulation\nB ||\nof incoherent spectral weight below the main kink at E = 46 meV (Fig. 3a).\nTo examine this flat feature, we present two energy dispersive cuts (EDC) in\nFig. 3c, k which cuts across momentum value devoid of bands and k\nEDC1 EDC2\nwhich cuts across the k of the main LiTi O band.\nThe momentum position of\nF 2 4\nR6\nthese two cuts are indicated by two black arrows pointing to dotted white lines\nin Fig. 3b. We can see the peak position of this sub-band in the EDC of k\nEDC1\nas an abrupt and sharp increase in spectral intensity at Ω = 70.4 ±6 meV (green\ndot) also observed in k at the same energy scale. Sub-bands like these are\nEDC2\nindicative of a highly interacting picture11, in stark contrast to a non-interacting\npicture where in the absence of any bands, the expected EDC would not have\nany additional poles unlike what is seen in k . The second derivative plot\nEDC1\n(Fig. 3d) further highlights this feature and additionally shows a “spectral gap”\nwhere the non-dispersive feature interrupts the main dispersive electronic band.\nWe note that the flat feature, in addition to the spectral gap, can also be seen\nclearly in our raw data in Fig. 2d. These types of sub-bands due to strong\nelectron-phonon coupling are spectral signatures that are consistent with the\nformation of intra-unit cell small polarons10 that have been observed in other\ncorrelated materials11,12.\n5. In line 164, the authors note EB∼0.9 eV, while in line 272, they note E∼800meV. This\nshould be self-consistent.\nOur response: We thank the reviewer for pointing out this error. It has now been fixed\nin the text as E ≈ 0.9 eV on line 272.\nB\nMain text:\nThird, we observe large incoherent spectral weight at E ≈ 0.9 eV, which cannot\nbe accounted by either of our density-functional perturbation theory (DFPT) or\nDMFT calculations.\nR7\nREVIEWER #2 (REMARKS TO THE AUTHOR):\nReport on “Unconventional polaronic ground state in superconducting LiTi O ” by Z. Hasan\n2 4\net al. The authors report spectroscopic measurements of superconducting LiTi O thin films\n2 4\nusing ARPES and RIXS. Based on comparisons with calculations using DFT and DMFT,\nthey provide evidence for the interplay between electron-phonon coupling and electron-\nelectron interactions. They also identify Eg and A1g phonon modes with energies of 47\nand 71 meV, respectively. Additionally, the energy-dependent RIXS results, ARPES data,\nand DFT calculations—which fail to account for the broad incoherent spectral weight—lead\nthe authors to propose a polaronic ground state in LiTi O . The manuscript is technically\n2 4\nsound and well written; I support its publication, provided the following issues are addressed.\nOur response: We thank the reviewer for their positive evaluation of our manuscript and\nfor their thoughtful summary of our RIXS, ARPES and theoretical results. We look forward\nto using their insights on our data to further clarify and strengthen our work.\n1. The RIXS map shown in Fig. 2a is used to demonstrate the superposition of localized and\ndelocalized excitations and to support the claim that the d1 Ti state is dynamically populated.\nHowever, the range of incident photon energies is too narrow to clearly show the coexistence\nof Raman-like and fluorescence excitations, as observed in MgTi O (Fig. 4c of Ref. 24)\n2 4\nDue to this limited energy range, the fluorescence contribution may be mistakenly interpreted\nas suppressed.\nOur response: We thank the reviewer for their astute observation. Indeed, while the\nMgTi O data (Fig. 4c of Ref. 2413) shows an energy map with both the L - and L -edges\n2 4 3 2\nof titanium, we have only measured the L -edge. It is generally customary to measure\n3\nthe L -edge for transition metal RIXS because magnetic and phononic signals are often\n3\nenhanced at this edge. However, one does not need both edges to see the superposition of\ndelocalized fluorescence features (diagonal feature in the energy map) and localized Raman\nfeatures (horizontal feature in the energy map highlighted by white dotted lines in Fig.\nR2). Measuring both edges will show a repeat of the features at each edge with an enhanced\nsignalattheL -edgeasisobservedinMgTi O . Below, weshowbothLiTi O andMgTi O\n3 2 4 2 4 2 4\nR8\n(cropped to the L -edge) to show the similarities between the two maps. Indeed, we can see\n3\nthe localized t states at ≈ 1 eV in both maps, with the e states at higher energy loss also\n2g g\nshowing non-dispersive Raman features in both LiTi O and MgTi O energy maps. A weak\n2 4 2 4\nfluorescence feature mixed with the Raman features can also be seen in both energy maps.\nNote that while the L -edge starts at 456 eV in LiTi O , it starts at 455 eV for MgTi O .\n3 2 4 2 4\nThe ≈ 1 eV difference in photon energy is due to known differences in the calibration of the\nincident energies at different beamlines.\ni\nMgTi O Ti - L edge RIXS\nLiTi O Ti - L edge RIXS\n2 4\n2 4\n5\n4\n3\n2\n1\n0\n455 456 457 458 459 460\nIncident Energy (eV)\n)Ve(\nssoL\nygrenE\nii\n5\n4\n3\n2\n1\n0\n455 456 457 458 459\nIncident Energy (eV)\n)Ve(\nssoL\nygrenE\nlocalized t\nlocalized t 2g\n2g states\nstates\nFIG. R2: Comparison of MgTi O RIXS data with LiTi O . The difference in photon\n2 4 2 4\nenergy in the two data sets is due to beamline calibration. i,ii, LiTi O Ti L-edge energy\n2 4\nmap and MgTi O Ti L-edge energy map (adapted from reference13).\n2 4\n2. Figures 3b, 3c, and 3d show that the dispersive electronic band interacts with the non-\ndispersive phonon bands at 70–80 meV, rather than at 46 meV as claimed by the authors (see\nline 188). Figure 3a indeed shows the strongest kink at 46 meV, and the sub-kinks appear to be distributed down to 70–80 meV.\nOur response: We thank the reviewer for the opportunity to further clarify our work.\nIndeed, Fig. 3a shows the strongest kink at E = 46 meV, and sub-kinks where we observe\nthe interaction of the electronic band with the non-dispersive phonon band occurs at E =\n70.4 ± 6 meV. To that end, we have made changes in lines 193-210 in the main text to\nR9\nclarify this distinction between the kink and the sub-kinks:\nMain text:\n¯ ¯ ¯\nFigure 3b shows an energy-momentum cut along the Γ−K −M direction (cut\noverlaid on the Brillouin zone is shown in Fig. S8a). Here we observe a non-\ndispersive feature at E = 70.4±6 meV at all k , highlighting the accumulation\nB ||\nof incoherent spectral weight below the main kink at E = 46 meV (Fig. 3a).\nTo examine this flat feature, we present two energy dispersive cuts (EDC) in\nFig. 3c, k which cuts across momentum value devoid of bands and k\nEDC1 EDC2\nwhich cuts across the k of the main LiTi O band.\nThe momentum position of\nF 2 4\nthese two cuts are indicated by two black arrows pointing to dotted white lines\nin Fig. 3b. We can see the peak position of this sub-band in the EDC of k\nEDC1\nas an abrupt and sharp increase in spectral intensity at Ω = 70.4 ±6 meV (green\ndot) also observed in k at the same energy scale. Sub-bands like these are\nEDC2\nindicative of a highly interacting picture11, in stark contrast to a non-interacting\npicture where in the absence of any bands, the expected EDC would not have\nany additional poles unlike what is seen in k . The second derivative plot\nEDC1\n(Fig. 3d) further highlights this feature and additionally shows a “spectral gap”\nwhere the non-dispersive feature interrupts the main dispersive electronic band.\nWe note that the flat feature, in addition to the spectral gap, can also be seen\nclearly in our raw data in Fig. 2d. These types of sub-bands due to strong\nelectron-phonon coupling are spectral signatures that are consistent with the\nformation of intra-unit cell small polarons10 that have been observed in other\ncorrelated materials11,12.\n3. RIXS data clearly show the presence of both Eg and A1g phonon modes. Specifically,\nTi L -edge RIXS detects only the Eg and phonons at an energy of 41 meV, while the A1g\n3\nphonons appear in the O K-edge spectra. The authors are suggested to elaborate on why\nthese two phonon modes are observed at different absorption edges.\nOur response: We thank the reviewer for their insight into the low-energy RIXS data. We\nwould also like to acknowledge that the coupling strength of the core-hole with a specific\nphonon mode is a very complex question that requires further theory to be fully addressed,\nR10\nwhich is outside the scope of our current work. Nevertheless, we can provide the following\narguments to explain the observed phenomenology.\nSince RIXS at the Ti L -edge is sensitive to changes in the local environment and our\nE mode at the Ti L -edge can be understood from a symmetry point of view. While an\ng 3\nE distortion reduces the D point group of the titanium atom, the A is a symmetry-\ng 3d 1g\nconserving breathing mode. We note that the observation of an oxygen phonon mode at the\nTi L-edge is a strong indication of polaronic physics, as it indicates a very strong coupling\nbetween the oxygen vibrations and the local electronic structure of the titanium atoms.\nRegarding the observation of phonon modes in the O K-edge, we agree with the referee\nthat the absence of the E mode requires a re-evaluation. As per the referee’s suggestion, we\ng\nnotice our fit does not fully capture the spectral intensity at ≈ 46 meV, the center energy of\nthe E mode. While the main feature remains the coherent resolvable harmonics of the A\ng 1g\nmode at ≈ 72 meV our data shows suggestions of the E mode, as captured by including a\ng\nGaussian peak in our fit to the data, shown in Fig. R3:.\n−0.10 0.00 0.10 0.20 0.30\nEnergy loss (eV)\n)stinu\n.bra(\nytisnetnI\nO - K edge RIXS\ndata\nbackground\nelastic\nA phonon\n1g\nincoherent phonons\nfit g = 6.87 ± 3.17\nE  71 meV\no\nE  142 meV\no\nE  213 meV\no\n−0.10 0.00 0.10 0.20 0.30\n)stinu\n.bra(\nytisnetnI\na b\nImproved fit O - K edge RIXS\ndata\nbackground\nelastic\nA phonon\n1g\nincoherent phonons\nE phonon g fit g = 4.13 ± 2.76\nE  72 meV\no\nE  144 meV\no\nE  216 meV\no\nEnergy loss (eV)\nFIG. R3: Comparing fitting of O K-edge RIXS. A new green gaussian centered at E ≈ 46\nmeV has been added in the new fit.\nOur new analysis suggests that the E mode also contributes to the O K-edge low energy\ng\nexcitations, but with less coherence than the A mode. This may be due to the high density\n1g\nR11\nof oxygen phonon modes in that energy range, as shown in Fig. 3h in the main text.\nTo\nsubstantiate the discussion of the different phonon modes at the different edges, Fig. 3f\nhas been updated in the main text, and the following discussion have been added to the\nsupplement and main text:\nMain text:\nThe coupling strength, g = 9.1±3.0, far exceeds those of other titanates14,15,\nTi\nand is on par with some cuprates measured via RIXS16 further supporting strong\nelectron-phononcouplinginLiTi O . Moreover,theobservationofapureoxygen\n2 4\nE mode at the titanium edge is indicative of strong hybridization between the\ng\nO-2p and the Ti-3d orbitals. Similar to the Ti L -edge, the three excitations at\n3\nthe O K-edge can also be well fit by the Ament model in conjunction with two\nbroad Gaussian peaks. The fit gives us the first three harmonics of a phonon\nmode centered at Ω = 72 ± 1 meV with g = 4.13 ± 2.76. We assign this mode\nO\nto the oxygen A phonon, which is known to be strongly coupled to the E\n1g g\nmode5. While the A mode is coherent at the O K-edge, the broad Gaussian\n1g\ncentered at E ≈ 46 meV, which encodes the coupling to the E mode also seen\ng\nat the Ti L-edge, indicates a reduced coherence of this mode at the O K-edge.\nThe absence of higher harmonics of the E mode at the O K-edge might be due\ng\nto the higher density of oxygen modes at that energy, as shown in Fig. 3h (For\na detailed discussion on the different phonon modes see Supp. Sec. 6). We\nnote that our DFT calculations of the phonon band structure (Fig. 3g) show\nthe modes around Ω ≈ 46 meV and ≈ 75 meV to be relatively nondispersive,\nsupporting our choice of Ament model fitting for the quasi-elastic peaks at the\nTi L - and O K-edges.\n0.00 0.00\n-0.05 -0.05\n-0.10 -0.10 -0.15 -0.15\n-0.20 -0.20\n-0.25 -0.25\n−0.10 0.00 0.10\n)stinu\n.bra(\nytisnetnI\ne f\nTi - L edge RIXS\n70\n60\n50\n40\n30\n20\n10\n0\n0.20 K Γ L X Γ\nEnergy loss (eV)\n)Vem(\nygrene\nnonohP\n-1.2-1.0-0.8-0.6-0.4-0.2\nk (1/Å) ||\nPhonon DOS (a.u)\n)Ve( E - E F\na b k\nEDC 1\n0.6\nk (1/Å) y\n)Ve(\nE - E F )Ve( E - E F\nk c d\nEDC 2\n0.00\n-0.05\n-0.10 -0.15\n-0.20 k EDC 1\nk\nEDC 2 ∂2E/∂k2\n-0.25 E  47 meV theory\nkink\n0.4 0.4 0.6 Intensity (a.u) -1.2-1.0-0.8-0.6-0.4-0.2\nk (1/Å) 0 2 Intens 4 ity (x103) 6 8 0 5 101 In 5 ten 2 s 0 ity 2 (x 5 10- 3 3) 03540 g 0 2 Intens 4 ity (x103) 6 8 -8 -6 In -4 tensity - ( 2 x103) 0 h 2 ||\nDFT phonon band structure\ndata\nbackground\nelastic\nE phonon g incoherent phonons\nfit g = 9.12 ± 3.0\nEo  47 meV\nE1  94 meV\nE2  141 meV\nOxygen\n−0.10 0.00 0.10 0.20 0.30\n)stinu\n.bra(\nytisnetnI\nO - K edge RIXS\ndata\nbackground\nelastic\nA phonon 1g incoherent phonons\nE phonon g fit g = 4.13 ± 2.76\nEo  72 meV\nEo  144 meV\nEo  216 meV\nEnergy loss (eV)\n¯ ¯\nFIG. R4: Electron-phonon coupling in LiTi O (a) Comparison of the Γ−K ARPES\n2 4\ndata to density-functional perturbation theory (DFPT) highlighting the kink feature at 47\n¯ ¯ ¯ ¯ ¯\nmeV at Γ−K at T = 7 K. b, Energy - momentum ARPES spectra along the Γ−K −M\ndirection. c, EDC cuts at two different k highlighted in b with dotted white lines. Black\n||\nand green dots correspond to the top of the two EDC poles. d, Second-derivative plot from\nb, highlighting a dispersionless spectral intensity at the second pole denoted by a black\narrow in c also observable in the raw data in b. A spectral gap is also prominent at the\nsame energy scale where the dispersionless feature interrupts the main band. e, Quasi-\nelastic RIXS excitations shown at E = 460 eV for the Ti-L edge. f, Quasi-elastic RIXS\ninc 3\nexcitations shown at E = 531.5 eV for the O K-edge. The total intensity is a fit (red\nsolid line) to a Voigt peak (grey shading), a background step function (blue dashed lines),\none (two) Gaussian(s) and an Ament model fit to the phonon spectra (maroon) in e and f\nrespectively. g and h, Calculated phonon band structure and total oxygen phonon density\nof states respectively. The purple boxes highlights the energy scale of the A and E modes,\n1g g\nrespectively, in both plots.\nSupplementary note 6: Resonant inelastic X-ray scattering:\nWe now turn our attention to the presence of different phonon modes at the\nO K-edge and the Ti L-edge. In the text below, we reason why the E mode\ng\ndominates at the Ti L-edge and the A mode at the O K-edge. Since RIXS at\n1g\nthe Ti L -edge is sensitive to changes in the local environment and our RIXS\n3\nat the O K-edge demonstrates strong O-Ti hybridization, the observation of the\nR13\noxygen E mode at the Ti L -edge can be understood from a symmetry point of\ng 3\nview. While an E distortion reduces the D point group of the titanium atom,\ng 3d\nthe A is a symmetry-conserving breathing mode. This can be seen in Fig. S18\n1g\n(R5). We note that the observation of an oxygen phonon mode at the Ti L-edge\nis a strong indication of polaronic physics, as it indicates a very strong coupling\nbetween the oxygen vibrations and the local electronic structure of the titanium\natoms.\nRegarding the lack of coherence of the E mode at the O K-edge and the dom-\ng\ninant intensity from the A mode, our data suggests that the E coherence at\n1g g\nthe O K-edge is impeded by the high density of oxygen phonon modes at ≈\n46meV, as shown in Fig. 3h of the main text. Instead, these phonons manifest\nas spectral intensity captured by a broad Gaussian centered at the E mode at\ng\nE≈ 46 meV. Thus, the O K-edge shows a dominant, coherent A mode and an\n1g\nincoherent background which includes the E mode among other unresolvable\ng\nphonon modes.\nA E\n1g g\nFIG. R5: Comparing the effect of E phonon mode with A phonon mode on an isolated\ng 1g\ntitanium octahedra.\n5. The intensity bars shown in Figs. 2 and 3 lack a scale. It is unclear whether the current\nintensity bars are linearly or logarithmically scaled.\nR14\nOur response: We thank the reviewer for giving us the opportunity to clarify our figures\nfurther. Fig. 2 and 3 have been revised with scale in the intensity bars as shown below and\nin the main text:\na b\nc d e\n)Å/1(\nyk\nkx (1/Å) ky (1/Å)\n)Ve(\nE-E F\nTi - L edge RIXS\n5\n4\n3\n2\n1\n0\n456 457 458 459 460\nIncident Energy (eV)\nHe-Iα ARPES 0.0\n-0.1\n0.5 -0.2\n-0.3\n-0.4\n0.0\n-0.5 Γ\n-0.6\n-0.7\n-0.5\nM -0.8\nM\nK -0.9\n-1.0 -1.0\n-1.0 -0.5 0.0 0.5 -0.50 -0.25 0.00 0.25 00..5500\n)Ve(\nssoL\nygrenE\nO - K edge RIXS\n5\nintegrated XAS\n4\n3\n2\n1\n0\n531 532 533 534 535\nIncident Energy (eV)\n)Ve(\nssoL\nygrenE\nintegrated XAS\n0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35\n0.0\n−0.1\n−0.2\n−0.3\n−0.4\n−0.5\n−0.6\n−0.7\n−0.8\n−0.9\n−1.0\n0.00 0.25 0.50\nky (1/Å)\n0 20 40 60 80 0 2 4 6 8 10 12\n)Ve(\nE-E F\n0 2 4 6 8 10\nFIG. R6: Electron-electron correlations in LiTi O a, RIXS intensity at the Ti L -\n2 4 3\nedge. b, RIXS intensity at the O K-edge. c, Iso-energy map at the Fermi level taken with\nHe-Iα showing a hexagonal Fermi surface. The white hexagon indicates the first Brillouin\n¯ ¯ ¯\nzone. d, Experimental band structure in the K − Γ − K direction. e, DMFT and DFT\n¯ ¯\n(white)comparisonalongΓ−K indicatingthemass renormalization due toelectron-electron\ncorrelations.\nR15\n0.00 0.00\n-0.05 -0.05\n-0.10 -0.10 -0.15 -0.15\n-0.20 -0.20\n-0.25 -0.25\n−0.10 0.00 0.10\n)stinu\n.bra(\nytisnetnI\ne f\nTi - L edge RIXS\n70\n60\n50\n40\n30\n20\n10\n0\n0.20 K Γ L X Γ\nEnergy loss (eV)\n)Vem(\nygrene\nnonohP\n-1.2-1.0-0.8-0.6-0.4-0.2\nk (1/Å) ||\nPhonon DOS (a.u)\n)Ve( E - E F\na b k\nEDC 1\n0.6\nk (1/Å) y\n)Ve(\nE - E F )Ve( E - E F\nk c d\nEDC 2\n0.00\n-0.05\n-0.10 -0.15\n-0.20 k EDC 1\nk\nEDC 2 ∂2E/∂k2\n-0.25 E  47 meV theory\nkink\n0.4 0.4 0.6 Intensity (a.u) -1.2-1.0-0.8-0.6-0.4-0.2\nk (1/Å) 0 2 Intens 4 ity (x103) 6 8 0 5 101 In 5 ten 2 s 0 ity 2 (x 5 10- 3 3) 03540 g 0 2 Intens 4 ity (x103) 6 8 -8 -6 In -4 tensity - ( 2 x103) 0 h 2 ||\nDFT phonon band structure\ndata\nbackground\nelastic\nE phonon g incoherent phonons\nfit g = 9.12 ± 3.0\nEo  47 meV\nE1  94 meV\nE2  141 meV\nOxygen\n−0.10 0.00 0.10 0.20 0.30\n)stinu\n.bra(\nytisnetnI\nO - K edge RIXS\ndata\nbackground\nelastic\nA phonon 1g incoherent phonons\nE phonon g fit g = 4.13 ± 2.76\nEo  72 meV\nEo  144 meV\nEo  216 meV\nEnergy loss (eV)\n¯ ¯\nFIG. R7: Electron-phonon coupling in LiTi O (a) Comparison of the Γ−K ARPES\n2 4\ndata to density-functional perturbation theory (DFPT) highlighting the kink feature at 47\n¯ ¯ ¯ ¯ ¯\nmeV at Γ−K at T = 7 K. b, Energy - momentum ARPES spectra along the Γ−K −M\ndirection. c, EDC cuts at two different k highlighted in b with dotted white lines. Black\n||\nand green dots correspond to the top of the two EDC poles. d, Second-derivative plot from\nb, highlighting a dispersionless spectral intensity at the second pole denoted by a black\narrow in c also observable in the raw data in b. A spectral gap is also prominent at the\nsame energy scale where the dispersionless feature interrupts the main band. e, Quasi-\nelastic RIXS excitations shown at E = 460 eV for the Ti L -edge. f, Quasi-elastic RIXS\ninc 3\nexcitations shown at E = 531.5 eV for the O K-edge. The total intensity is a fit (red solid\nline) to a Voigt peak (grey shedding), a background step function (blue dashed lines), and\nAment model fit to the phonon spectra (maroon) in e and f. g and h, Calculated phonon\nband structure and total oxygen phonon density of states respectively. The purple boxs\nhighlights the energy scale of the A and E modes, respectively, in both plots.\n1g g\nR16\nREVIEWER #3 (REMARKS TO THE AUTHOR):\nThe authors present a spectroscopic study of LiTi O thin films combining ARPES, RIXS,\n2 4\nand theoretical calculations. They observe strong d1-like features in RIXS spectra despite the\nmetallic d0.5 electronic configuration, along with significant electron-phonon coupling and\nmass enhancement from ARPES measurements. Based on these observations, they propose\nthat LiTi O exhibits an unconventional mobile polaronic ground state arising from the coop-\n2 4\nerative interplay of electron-electron correlations and electron-phonon coupling, challenging\nthe conventional BCS interpretation of superconductivity in this material. The authors sug-\ngest this represents a novel example of correlated physics in mixed-valence spinel oxides.\nWhile their study is important and interesting, there are several questions needed to be an-\nswered:\nOur response: We thank the reviewer for acknowledging the importance of our study\nand for their thoughtful summary of our main findings. We appreciate their constructive\nquestions and look forward to engaging with them to further strengthen our work.\n1. The explanation of the Ti4+ and Ti3+ components in the XAS data shown in Fig. S13\nis unclear. To support the claim that the spectrum reflects a mixture of Ti3+ and Ti4+,\nthe\nauthors should directly compare reference XAS spectra for Ti3+ and Ti4+ with the experi-\nmental data. Which specific spectral features correspond to Ti3+, and which are indicative\nof Ti4+? Please clarify.\nOur response: We thank the reviewer for their comment. We agree that a comparison\nof LiTi O XAS spectra with a reference Ti3+ and Ti4+ spectra would be helpful. To that\n2 4\nend we have compared our capped and uncapped LiTi O thin films with stoichiometric and\n2 4\nun-stoichiometric LiTi O powders17, as well as PLD grown thin films18. To tease out the\n2 4\nrelevant Ti3+ and Ti4+ features, we have also compared it with SrTiO (Ti4+) and YTiO\n3 3\n(Ti3+) XAS spectra19. As both SrTiO and YTiO have octahedral coordination of the\n3 3\ntitanium atom like in LiTi O , they make good comparisons for our LiTi O data set. The\n2 4 2 4\nfollowing section has been added to the supplement with references to the new section in\nthe main text:\nR17\nMain text:\nOur x-ray absorption spectroscopy (XAS) data (Fig. S16a), displays resonant\nfeatures associated with the mixed valence Ti4+ and Ti3+ states (for a detailed\ndiscussion on the mixed-valence XAS, see Supp. Sec. 9).\nMethods:\nX-ray absorption spectroscopy\nXAS of uncapped LiTi O thin films was performed at 300 K at Beamline 6.3.1\n2 4\nat the Advanced Light Source at Lawrence Berkeley National Laboratory. XAS\nof capped LiTi O thin films was performed at 300 K at ID-32 at Diamond\n2 4\nLight Source. The spectra from Beamline 6.3.1 represents an average of 4 in-\ndividual scans. All spectra are taken in the total electron yield configuration.\nThe LiTi O XAS spectra do not show any polarization dependence. For more\n2 4\ndetails on the XAS, refer to Supp. Sec. 9\nSupplementary note 9: X-ray absorption spectroscopy:\nFigure S26i)(R7i) compares the Ti L-edge spectra to a reference Ti4+ spectra\n(SrTiO )19 and a reference Ti3+ spectra (YTiO )19, as well as another LiTi O\n3 3 2 4\nthin film18. The Ti L-edge spectra have been aligned to two dotted blue lines\ncorresponding to the pre-edge feature labeled L and the first t peak (labeled\npre 2g\nt ). With this alignment, we can see that our uncapped LiTi O thin film re-\n2g 2 4\nsembles the PLD grown LiTi O thin film18 as well as other LiTi O powder\n2 4 2 4\nspectra17 (not shown here). The analysis in reference18, assigns the qualitative\nfeatures such as the intensity of the pre-peak as well as the merging of the t\n2g\nand e peaks as a signature of the mixing of Ti3+ and Ti4+ oxidation states. The\ng\nreference LiTi O thin film spectra is extremely similar to our uncapped LiTi O\n2 4 2 4\nthin film spectra, showing that our uncapped results are consistent with liter-\nature where LiTi O is also uncapped. However, LiTi O is known to be air\n2 4 2 4\nand moisture sensitive and prone to “aging”, as discussed in references20,21. The\nresult of such a degradation process would lead to surface oxidation with excess\nTi4+ on the surface of any LiTi O crystal or powder. Since the spectra in Fig.\n2 4\nS26 (R6) are measured in the surface sensitive total electron yield (TEY) config-\nuration, all the uncapped LiTi O thin films show a higher concentration of Ti4+\n2 4\nR18\nthan would be native to LiTi O . Thus, our LiTi O capped thin film represents\n2 4 2 4\na more pristine surface which is more representative of the native oxidation state\nin LiTi O . We can see that the capped LiTi O thin film has a higher Ti3+\n2 4 2 4\ncontent, with a pronounced pre-edge feature as well as further merging of the\nt and e states highlighting its Ti3+ content.\n2g g\nSince the comparison data in Fig. S26 (R8) were collected at different beamlines\nand times with varying incident-energy calibrations, the observed energy shifts\ncannot be used to quantitatively determine the titanium oxidation state in our\nfilms. In addition, because the intensity axis was already offset in the spectra\ntaken from Ref.17–19, the relative intensities of specific features cannot be mean-\ningfully compared across spectra. Nevertheless, we can put bounds on what\nthe titanium oxidation can be by looking at the O K-edge. Figure S26ii) (R6ii)\nshows the O K-edge XAS spectra of LiTi O powders and thin films for com-\n2 4\nparison with our capped and uncapped data. We can see that our capped data\nshows the first two peaks which are indicative of oxygen-titanium hybridization\nfeatures, also observed in the uncapped data and the reference literature17,\nafter\nwhich the SiO spectrum dominates the O K-edge. However, the first two pre-\npeaks can be used to bound the titanium oxidation state in LiTi O . Looking at\n2 4\nthe LiTi O powder17 for comparison, we can see that as titanium moves more\n2 4\ntowards Ti3+ with excess lithium, the two pre-edge hybridization features in the\nO K-edge seem to merge, which is inconsistent with our LiTi O thin film data.\n2 4\nThe stoichiometry of our samples is further corroborated with our T of ≈ 12 K\nc\nand lattice constant of a = 8.40 ˚A, which is within the stoichiometric bounds\nthat give this T and lattice constant as shown in reference22—any higher than\nc\na 5 percent excess of lithium would decrease the superconducting T and the\nc\nlattice constant. Furthermore, lithium deficiency is outside of the spinel homo-\ngeneity range and results in (Li, Ti)O secondary phases23 which are absent from\nx\nour x-ray diffraction (XRD) data which shows phase pure LiTi O (Fig. S1).\n2 4\nThus, while small uncertainties cannot be absolutely ruled out, our XRD data,\nlattice constant, XAS data and superconducting transition temperature are all\nconsistent with having stoichiometric LiTi O thin films with a mixed-valence\n2 4\nTi3.5+.\nR19\nTi - L edge O - K edge\nLi TiO powder\n1.2 2 4\ncomparison\nLi TiO powder\n1.1 2 4\ncomparison\nSrTiO comparison LiTiO powder\n3 2 4 comparison\nLiTiO PLD film\n2 4\nLiTiO capped comparison 2 4\nLiTiO\n2 4\nuncapped\nLiTiO\n2 4\nuncapped\nLiTiO PLD film\ncom 2 p 4 a rison LiTiO capped\n2 4\nYTiO comparison\n3\n)stinu\n.bra(\nytisnetnI\n)stinu\n.bra(\nytisnetnI\na) b)\nL t e\npre 2g g\n452 454 456 458 460 462 464 466 530 535 540 545\nEnergy (eV)\nEnergy (eV)\nFIG. R8: X-ray absorption spectroscopy of reference spectra for comparison to LiTi O .\n2 4\nOur work is labelled as “LiTi O capped” denoting a LiTi O /MgAl O (111) thin film\n2 4 2 4 2 4\nwith a ≈ 5 nm SiO cap and and “LiTi O uncapped” which is without the SiO cap. a)\n2 4\nTi L-edge spectra. SrTiO and YTiO have been adapted from19 whereas the LiTi O thin\n3 3 2 4\nfilm comparison is from18. b) O K-edge spectra. Stoichiometric and excess lithium LiTi O\n2 4\npowders have been adapted from17, whereas the thin film comparison is from18.\n2. The authors state that the RIXS map in Fig. 2a reflects a Ti d1 state, citing the “nearly\nfeatureless” spectrum expected for a d0 configuration (as shown in Ref. 20). However,\nRef. 20 reports several RIXS features between 0–200 meV for SrTiO3 (d0). Could the au-\nthors clarify whether “nearly featureless” implies that these d0-related peaks are significantly\nweaker compared to those from d1 states? A quantitative comparison would help support this\ninterpretation.\nOur response: We thank the reviewer for giving us the opportunity to further clarify our\ntext. While quasi-elastic low-energy features like phonons do not reflect the valence state\nR20\nof the resonant ion, the dd excitations depend on the presence of valence electrons that can\nhave intra-band transitions. Ref. 20 shows several RIXS features between 0-200 meV for\nSrTiO , which has a nominal d0 state. However, Ref. 20 also reports the existence of Ti3+ in\n3\ntheirsamples. Theddfeaturearound2.5eV,showsresonantbehaviorattheTi3+ absorption\nresonance, and as such, it is attributed to a crystal field excitation of the d1 state. This is\nfurther supported by the observed dd excitations in anatase TiO where oxygen vacancies\n2\ndope carriers into the bottom of the conduction band14 as well as vanadium doped LiTi O\n2 4\nwhere increasing the titanium oxidation and thus reducing the d1 states, decreased the\nintensity of the dd excitations24. We also note that a fully empty d shell has a trivial RIXS\nspectrum as there are no possible multiplet excitations in the intermediate state. Thus, as\ndiscussed in Ref. 20 the low-energy features in SrTiO are either phonons that are possible\n3\nin either a d0 or d1 ground state, or Ti3+ dd excitations dressed by phonons.\nWe note that our assignment, in Fig. 2a of the main text, of the broad Raman features in\nLiTi O centered at E = 1.5 eV and E = 4 eV as d1 dd excitations is further supported\n2 4 loss loss\nby our exact diagonalization calculations in Supp. Note 6 (Fig. S16). To make this point\nclearer, we have replaced lines 120-128 in the main text with the following changes:\nMain text:\nOur x-ray absorption spectroscopy (XAS) data (Fig. S13a), displays resonant\nfeatures associated with the mixed valence Ti4+ and Ti3+ states (for a detailed\ndiscussion on the mixed-valence XAS, see Supp. sec. 9). Given the lack of dd\nexcitations expected for a trivial d0 state, we conclude that the, broad Raman-\nlike dd features centered at E = 1.5 eV and E = 4 eV, reflect a d1 titanium\nloss loss\noccupation state, which given the lack of charge order in LiTi O can only be\n2 4\ndynamically populated25. These Raman excitations, along with the suppressed\nfluorescencecontribution,areunexpected,giventhattheitinerantchargecarriers\nin our metallic and superconducting samples are proposed to have a dominant\nTi-3d band character near the Fermi level7,26.\n3. In addition, is it possible that the RIXS peaks observed between 0–0.2≈eV in Fig. 3e\noriginate from Ti4+ (d0) states, similar to those reported in SrTiO (STO)? As noted earlier,\n(2020)].\nR21\nOur response: We thank the reviewer for their question. Following up on the previous\nanswer, we would like to emphasize the distinction between electronic excitations, which\nare present in a d1 state and absent for a d0 state, versus quasi-elastic phononic excitations\nwhich do not reflect the Ti oxidation state and can be accessed from d0 and d1 ground states.\nThe low-energy RIXS features in Fig. 3e and 3f, are attributed to phononic excitations. To\nprovide further evidence of the phononic nature of these features, we are including the\nfollowing data and text in the supplement:\nSupplementary note 6: Resonant inelastic x-ray scattering To corrobo-\nrate the quasi-elastic phononic nature of the excitations in Fig. 3e, we present\nlinear vertical (LV) and linear horizontal (LH) polarization data for the Ti L-\nedge. In this geometric configuration (2θ = 90°, θ = 45°), charge excitations\nare suppressed in the LH polarization due to Thompson scattering. As shown\nbelow, we observe a near complete suppression of the phonon modes in the LH\ndata compared to the LV data, further corroborating that these excitations are\nphononic in nature.\nR22\n)stinu\n.bra(\nytisnetnI\nTi - L edge\n-100 -50 0 50 100 150 200 250\nEnergy Loss (meV)\nFIG. R9: Ti L-edge RIXS spectra in linear horizontal (LH) and linear vertical (LV) polar-\nization in 2θ = 90°, θ = 45° configuration at 22 K. LH shows a complete suppression as is\nexpected for phonon excitations.\n4. Charge frustration between nominal Ti4+ and Ti3+ ions should lead to either (i) static\ncharge order or (ii) rapid valence fluctuations that average to a homogeneous metallic state.\nYour Ti-L RIXS spectra are interpreted as showing a robust d1 (Ti3+-like) local configuration.\nGiven that the core-hole lifetime at the L edge is only a few femtoseconds, could the d1 sig-\nnature simply reflect a snapshot taken on a timescale shorter than the electron-hopping time\nbetween the two valence states? Authors need to compare the timescales of these processes.\nCould you estimate the hopping rate or activation energy of the polaron?\nOur response: We thank the reviewer for their perceptive questions regarding the RIXS\nprocess and the polaron hopping rate. While a single-shot experiment is an interesting idea\nfor a follow-up study, the RIXS measurement itself is not a single-shot experiment—the\ncore hole lifetime in RIXS is a few femtoseconds, but the RIXS measurements are typically\nR23\nten to fifteen minutes, which is an average of all possible snapshots within that timeframe.\nThus, the d1 signature is not a reflection of a snapshot taken on a timescale shorter than the\nelectron-hopping time but is a reflection of the inelastic RIXS process itself where a d0 state\nis a “trivial” state where RIXS features like dd excitations are not allowed. Therefore, in a\nsense the RIXS process ”selects” out the d1 state and shows us inelastic features related to\nthat state.\nThe hopping rate or activation energy of the polaron is an interesting question that\nassumes there is some thermal barrier to overcome for conduction to take place. However,\nstoichiometric LiTi O displays band-like metallic transport where no activation energy\n2 4\nwould be required and hopping would be allowed even at room temperature. The proposed\nmechanism for such a transport is shown in Fig. 4 of the main text and is supported by\na similar mechanism proposed for conductivity in Li Ti O 27. While native Li Ti O is\n4 5 12 4 5 12\ninsulating, theintroductionofoxygendefectsinduceTi3+ stateswhichmixedwiththenative\nTi4+ oxidation state leads to conducting behavior with mobile small polaronic transport.\nReference27 proposes that due to the relatively small barriers between the transition from\nTi3+ to Ti4+ states in Li Ti O , charge hopping dynamics can already occur at room\n4 5 12\ntemperature without needing a significant activation energy. This transport mechanism\nis consistent with our data for LiTi O where band-like metallic transport coexists with\n2 4\nlocalized Ti-sites. Furthermore, this mixing of Ti3+ and Ti4+ states needed for polaronic\ntransport happens natively in LiTi O due to the charge balance in this mixed-valence\n2 4\ncompound. We have added this explanation and Ref.27 to the main text lines 265-274.\nMain text:\nWespeculatethatinLiTi O withanoccupationd0.5 intheabsenceofchargeor-\n2 4\nder, two titanium ions share an electron dynamically, such that their occupation\nstate fluctuates between d1 and d0. In this scenario, the transiently occupied d1\nsite has an associated local symmetry reduction due to dynamic lattice fluctua-\ntions of the E and A oxygen phonons, as seen in MgTi O 28. As the electron\ng 1g 2 4\nmoves to the unoccupied d0 site, it drags the local distortion with it, leading\nto the formation of a polaronic state in which charge motion and lattice distor-\ntions are coupled (Fig. 4b). Notably, this polaronic mechanism for Ti3+ defect\ninduced carriers in a Ti4+ system has already been proposed for Li Ti O 27,\n4 5 12\nR24\nsuggesting a similar mechanism for polaronic transport in LiTi O where the\n2 4\nmixed-valence is native to the material.\n5. DMFT calculation showed that LiTi O is metallic, with average of Ti3.5+. If there is\n2 4\n“charge localization” in a short timescale, then should we also consider Ti3+ within DMFT,\nby adding additional electrons?\nOur response: Wethankthereviewerfortheirquestionaboutincorporatingmixed-valence\nwithin DMFT. We note that while polaron formation can be explored within first-principles\nelectronic structure methods via stabilizing a polaron in a large supercell within a well-\nestablished DFT context, extending it beyond DFT methods is an interesting research di-\nrection but beyond the scope of the current work. However, DMFT does have access to\nall the atomic multiplets of the atomic impurity. Thus, we have access to the probability\nof our Ti impurity being found in a specific atomic configuration. To clarify how DMFT\nincorporates the mixed-valence nature of tilanium in LiTi O , the following text and figure\n2 4\nhas been added to Supplementary note 5.\nSupplementary note 5: DFT, DFT+DMFT, and electron-phonon cal-\nculations. Computational details and additional data\nThe DMFT calculation incorporates all local quantum fluctuations for the Ti-\nt impurity that we consider here. This means that we have access to all of\n2g\nthe atomic multiplets and the probability of the impurity being found in a given\nmultiplet. Figure S13 (R9) shows a histogram of the atomic multiplets for the Ti\nimpurity, where N denotes the total number of electrons in the 3 t -like orbitals.\n2g\nThe histogram reflects the mixed-valence nature of the spinel oxide with about\nequal probability for the N = 0 (d0) and N = 1 (d1) atomic configurations.\nR25\n0.5\nS(S+1)\n0.4 0.0\n0.75\n0.3\n2.0\n0.2 3.75\n0.1\n0.0\n0 1 2 3 4 5 6\nN\nFIG. R10: Atomic histogram for the t considered for LiTi O obtained directly from\n2g 2 4\nmeasuring the density matrix within the continuous-time quantum Monte Carlo impurity\nsolver.\n6. Figs 1a-b shows the Kagome planes of LiTi O , but the Fermi surface appears large and\n2 4\ndispersive . Could the authors comment on any flat-band contribution near E ?\nF\nOur response: We thank the reviewer for their insightful question regarding the absence of\nobvious flat band physics in LiTi O . While an ideal tight-binding kagome lattice yields flat\n2 4\nbands near E , LiTi O deviates from the ideal picture with a three dimensional kagome\nF 2 4\nnetwork of titanium atoms. This sub-lattice is not electronically isolated, as Ti-O bonds are\ndispersed throughout the 3D-kagome network. Significant hybridization between titanium\nand oxygen as shown in Fig. S10b) and evidenced by RIXS in Fig. 2a and 2b also compli-\ncates the picture as mixing with itinerant oxygen states can broaden the strong localization\nneeded for flat-band physics to emerge. However, while LiTi O does not have the extreme\n2 4\nlocalization needed for emergent flat bands, the geometric frustration from the kagome lat-\ntice does lend itself to localization of titanium sites as seen in Ti L-edge RIXS map (Fig. 2a).\nThe geometric frustration may also prevent static charge order that has observed in other\nmixed-valence materials like Fe O . To clarify the role of the kagome lattice in LiTi O , the\n3 4 2 4\nfollowing changes have been made to lines 63-68:\nMain text:\nAlthough superconductivity was discovered in LiTi O before the cuprates23,\n2 4\nR26\nit remains a unique instance of a spinel superconductor and with the highest\nT for a mixed-valence titanate. In this crystal structure, d0.5 titanium atoms\nc\nform a pyrochlore sublattice, composed of alternating planes of Kagome and\ntriangles (Fig. 1a and 1b). While this geometric construction has given rise\nto strong electron correlations in other systems29, these interactions have been\nlargely unexplored in LiTi O .\n2 4\n7. Given the volatility of Li, even slight deviations from stoichiometry—or the presence of\nLi/Ti antisite defects—could induce polarons and significantly affect the electronic response.\nWhat is the measured Li:Ti ratio in your films, and how was it verified? Could modest\noff-stoichiometry or antisite disorder be responsible for the correlation or polaronic features\nobserved in your data? In other words, is it possible that Li-related defects are the primary\norigin of the polaronic behavior?\nOur response: We would like to thank the reviewer for their interesting point about defect-\ninduced polarons. We answer this question in two parts. First we will discuss the presence\nof potential Li/Ti antisite defects in our film and then discuss the likelihood of Li-related\ndefects being the primary origin of polaronic behavior.\nWe first note that conventional methods of establishing stoichiometric ratio of Li/Ti like\nRutherford Back Scattering (RBS) cannot be employed because lithium is a light element\nwith minimal scattering signal.\nWhile RBS cannot be effective at determining the exact ratio of Li/Ti in our LiTi O thin\n2 4\nfilms,wecanneverthelessattemptto: 1)useHAADF-STEMimagingtocharacterizedefects;\n2) define the bounds of the Li/Ti ratio by looking at discussions on XAS spectra in response\n1 (Supp. Sec. 9); and 3) assess the consistency of features across ARPES, RHEED patterns,\nand electrical transport across many different samples, as discussed below. By looking at\nall these metrics, we can see that our samples lie within the bounds of a stoichiometric film\nof LiTi O . The following text has been added to the supplement:\n2 4\nSupplementary note 8: Scanning transmission electron microscopy\nWe also attempt to characterize the presence of Li/Ti antisite defects via STEM\nimaging. Our zoomed out HAADF-STEM image in Fig. 1d shows uniform crys-\nR27\ntallinity over a large area of the film. Additional STEM imaging in Fig. S25\n(R10) below also shows that within a large field of view, the film is relatively\ndefect free. The quantitative ratio of the presence of defects and the Li/Ti ratio\ncannot be established due to the beam sensitivity of the light lithium atoms\nas shown in Fig. S25b and S25c (R10b and R10c). Nevertheless, our quali-\ntative conclusion of a relatively defect free, stoichiometric LiTi O thin film is\n2 4\nbacked up by several observations and comparisons to literature. Firstly, lithium\ndeficiency where x = -0.1 and -0.2 in Li Ti O are outside of the spinel ho-\n1+x 2−x 4\nmogeneity range and would be evident in an X-ray diffraction as a secondary\n(Ti,Li)O phase23. Considering our smooth RHEED patterns, as well as phase\nx\npure X-ray diffraction patterns (Fig. S1a), we can rule out Li-deficiency in our\nthinfilms. Forlithiumexcess, reference23 andreference22 notethatLi Ti O\n1+x 2−x 4\nshows higher room temperature resistivity as well as semiconducting behavior\nas x is increased from 0 ≤ x ≤ 1. Not only is the room temperature resistiv-\n3\nity of our films consistent with the room temperature resistivity of other high\nquality thin films30,31, our films are also metallic all the way down to a sharp\nsuperconducting transition which is consistent with having high quality stoichio-\nmetric thin films with x = 0 in Li Ti O 22,23. Finally, our lattice constants\n1+x 2−x 4\nas established by reciprocal space maps and X-ray diffraction patterns (a =\n8.40 ˚A) fall within the stoichiometric ratio expected for bulk LiTi O : devia-\n2 4\ntions from stoichiometry would result in a smaller lattice constant as shown in\nFig. 4 in reference22,23. Thus, we can conclude that within the possible ways\nof establishing stoichiometric ratio of Li/Ti in LiTi O , our films fall within the\n2 4\nstoichiometric ratio expected for LiTi O .\n2 4\nR28\na) b) Pre-exposure\n1 nm\nc) Post-exposure\n4 nm 1 nm\nFIG. R11: Radiation sensitivity of LiTi O thin film. a) Large-area TEM image of the high\n2 4\nquality as-grown film, showing a uniform, defect-free microstructure. b, c) High-resolution\nTEM images taken from the same region before (b) and after (c) electron beam exposure.\nThe crystal structure undergoes significant rearrangement under irradiation, likely driven\nby knock-on damage to the light lithium atoms. Images were acquired at 200 keV with a\nprobe current of 40 pA, a dwell time of 200 ns per pixel, and a scan resolution of 1024 x 1024\npixels. (b) and (c) correspond to the 1st and 40th frames, respectively, during continuous\nscanning.\nHaving established that there is a low likelihood of Li-related defects in our thin films, we\nadditionally consider how small deviations in stoichiometry, outside of the detection limit of\nconventional probes like x-ray, RHEED, transport and STEM imaging can impact polaron\nformation in LiTi O . We first note that defect-induced polaron formation typically requires\n2 4\na large activation energy, often leading to semiconducting behavior such as in TiO 32,\n2−x\nwhich is unlike the metallic transport of LiTi O . Furthermore, our ARPES data have\n2 4\nconsistently shown spectral features consistent with polarons (dispersionless spectral weight\nat E ≈ 70 meV and incoherent spectral weight from E ≈ 0.6 - 0.9 eV) across different\nsamples. SincethesesamplesshowstoichiometricbehaviorbybulkprobeslikeRHEED(Fig.\nR29\nR11) and transport, as well as show consistent features across ARPES samples, we conclude\nthat these spectral features cannot be due to small, un-detectable variations in lithium\ncontent but due to the intrinsic polaronic nature of LiTi O . The following discussion has\n2 4\nbeen added to the supplement:\nSupplementary note 4: Angle-resolved photoemission spectroscopy\nTo establish the consistency of certain ARPES spectral features, we show below\nARPES measurements of samples synthesized at different times. We conclude\nthat even if there are un-detectable changes in lithium content that are not\ncaptured by our x-ray, RHEED, transport or STEM imaging data, they do not\naccount for spectral features observed in our ARPES data. Figure S10 (R11)\nshows the ARPES band structure of the different samples, all showing incoher-\nent spectral weight that emerges at E ≈ 0.6 eV, and extends down to E ≈ 0.9\neV (Fig. 2d) and a dispersionless spectral feature at E ≈ 70 meV. The consis-\ntency in our ARPES data across many different samples indicates that if there\nare slight deviations from stoichiometry in our samples that are outside of our\nmeasurable range, they do not contribute to the ARPES spectral features dis-\ncussed above and instead it is likely that these features are due to the intrinsic\npolaronic nature of LiTi O .\n2 4\nTo establish the high crystalline quality and lack of secondary phases of all the\nsamples in Fig. S10 (R11), we have included RHEED images of the samples\ntaken right before the in-situ transfer to ARPES (Fig. S11 (R12)). The differ-\nent growth conditions of the samples are included in the Table S1 (R1) below.\nWenotethatthehighvariationinthegrowthtemperaturebetweensample1and\nthe other samples is due to the difference in the substrate preparation. Sample\n1 was synthesized on a native single-side polished (111)-oriented MgAl2O sub-\n4\nstrate whereas samples 2-4 were synthesized on substrates which previously were\nbackside coated with platinum. These backside coatings were partially removed\nprior to synthesis but nevertheless these substrates required higher power from\nthesubstratelaserheatercomparedtoanuncoatedsubstrate,producingahigher\nnominal substrate temperature. The variance in the growth temperature of the\nlithium cell and current/voltage of the Ti-ball™is due to a decrease in flux rate\nR30\nas the source material depletes over the course of several growths. The oxygen\npressure is kept constant throughout. Even though the samples were grown at\ndifferent times, they consistently show high crystallinity and a RHEED pattern\nthat is expected for a homogenous spinel phase, without secondary phases.\nSample Date Li cell (°C) Ti-ball™(A/V) Growth temperature (°C) Oxygen pressure (Torr)\n1 7/4/2024 660 27.23/20.73 780 2E-2\n2 12/8/2022 675 29.8/20.69 825 2E-2\n3 12/7/2022 675 29.15/19.99 825 2E-2\n4 12/5/2022 675 29/19.92 825 2E-2\nTABLE R1: Growth conditions for different samples across two years.\nR31\nSample 1 Sample 2\n14\n10\n12\n8 10\n6 8\n6\n4\n4\n2\n2\n0.0\n-0.1\n-0.2\n-0.3\n-0.4\n-0.5\n-0.6\n)Ve(\nygrenE\n0.0\n-0.1\n-0.2\n-0.3\n-0.4\n-0.5\n-0.6\nSample 3 Sample 4\n8 5\n4\n6\n3\n4\n2\n2\n1\n0\n-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6\nky (1/Å)\n)Ve(\nygrenE\n0.0\n-0.1\n-0.2\n-0.3\n-0.4\n-0.5\n-0.6\n-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6\nky (1/Å) ky (1/Å)\n-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6\nky (1/Å)\n)Ve(\nygrenE\nx103 x103\n0.0\n-0.1\n-0.2\n-0.3\n-0.4\n-0.5\n-0.6\n)Ve(\nygrenE\na b\nc d\nx103 x103\nFIG. R12: a-d)ARPES band structure of different samples synthesized over the period of\ntwo years. All samples show consistent spectral features like the flat dispersionless feature\nat E ≈ 70 meV and the incoherent spectral weight around E ≈ 0.6 eV extending all the\nway down to 1 eV (not pictured here but shown in Fig. 2d of the main text).\nR32\na b\nSample 1\nSample 2\nc d\nSample 3 Sample 4\nFIG. R13: a-d) RHEED images corresponding to samples in Fig. R10. All images were\ntaken after growth and before in-situ transfer to ARPES. All samples show pristine surface\nquality with no visible secondary phases or signatures of surface reconstruction due to excess\nor deficient lithium or oxygen.\n8. Following up on the previous question, the reported residual resistivity ratio (RRR≈ 5.3)\nis described as indicative of “high-quality” samples; however, bulk crystals often achieve RRR\nvalues of 20. What types of defects are likely present in your samples, and how might these\ndefects influence polaron formation or dynamics?\nOur response: We thank the reviewer for your question and the opportunity to clarify.\nR33\nWhile an extended discussion on the defects and the effect of defects on polaron formation\nhas been provided in the previous question, we would like to elaborate on the RRR value of\nour thin films. The RRR is not 5.3 as previously written in the main text but is closer to 6,\nwith RRR ≈ 5.85 being a better estimation as reported in the supplement. This is revised\nin the main text lines 100-102.\nMain text:\nThin films of LiTi O were grown via reactive oxide MBE on (111)-oriented\n2 4\nMgAl O substrates (see Methods). As shown in Fig. 1c, the films display\n2 4\na superconducting transition with T ∼ 12.5 K and a residual resistivity ratio\nc\n(RRR)of∼ 5.85(Fig. S3)consistentwithpreviousreportsofthehighestquality\nLTO23,30.\nFinally,wewouldalsoliketonotethatwhilewehavereportedaRRRof≈5.85,consistent\nwith the data presented in Fig. S3a), the highest RRR of our grown samples is ≈ 6.3 as\nshown below.\n600\n500\n400\n300\n200\n100\n0\n0 50 100 150 200 250 300\nTemperature (K)\n)mc\nµ(\nytivitsiseR\nFIG. R14: Resistivity vs temperature of another LiTi O thin film, with a RRR of ≈ 6.3\n2 4\nOur samples have consistently shown RRR ranging from 5.5 - 6.3, comparable with the\nRRR of the highest quality thin films in literature. To compare these values with other high\nR34\nquality films and powders, we have made a table of growth direction and RRR of our films\ncompared with the some of the best ones reported in literature. For a fair comparison, we\nestimate the RRR to be the ratio of the resistivity at room temperature (300 K) to the\nresitivity before the superconducting transition at T ≈ 12 K. We take the highest RRR of\nc\nthe best sample if multiple samples are reported in the same paper. We also include the\nresistivity at 300 K as well as the resistivity at 20 K as another test for the quality of the\nsamples.\nType Reference RRR Growth direction R µΩ cm R µΩ cm\n300K 20K\nFilm Our work 6.33 [111] 646.5 111.7\nFilm 30 6.25 [001] 517.9 171.4\nPowder 23 2.00 – – –\nFilm 33 3.09 [001] 467 152\nFilm 34 3.10 [111] 1039 337\nFilm 35 5.00 [001] – –\nTABLE R2: Sample table with Type, Reference, RRR, growth direction, and resistances\nat 300K and 20K.\nAs shown in the Table R2, the RRR of our samples are consistent with some of the best\ngrown LiTi O in literature. Amongst the samples grown in the [111] direction, our films\n2 4\ndemonstrate one of the highest RRR. 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Scientific Reports, 8(1):3995, 2018.\nR39"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have addressed all my concerns and I would recommend its publication.", "reviewer_4": "My comments:\n==========================================================================================================================\nThe authors have replied thoroughly and clearly to almost all of the referee’s questions.\n1. The additional XAS measurements clearly demonstrate the presence of both Ti³⁺ and Ti⁴⁺ features.\n2. The expanded discussion of the d–d features satisfactorily addresses the third referee’s question.\n3. The additional explanation regarding the RIXS peaks also adequately answers the third referee’s inquiry.\n4. The authors’ clarification of the RIXS measurements fully resolves the referee’s concern.\n5. Although the authors’ response does not directly answer the third referee’s question in full detail, I agree with their assessment that this issue lies beyond the scope of the present work. Considering the probability of Ti site occupation is sufficient for the current study. As noted by the authors, Fig. R10 clearly illustrates the mixed-valence nature.\n6. The authors’ response satisfactorily addresses the referee’s question.\n7–8. I agree that the film exhibiting the highest RRR corresponds to the highest sample quality. While a small amount of Li defects could still induce polarons, as the referee previously noted, the authors have convincingly shown that the overall sample quality is very high in the additional discussion.\nOverall, the authors have provided clear and comprehensive responses, and I recommend the manuscript for publication"}, "author_response": ""}]

Decision letter