Oxygen vacancy-driven orbital multichannel Kondo effect in Dirac nodal line metals IrO2 and RuO2

Strong electron correlations have long been recognized as driving the emergence of novel phases of matter. A well recognized example is high-temperature superconductivity which cannot be understood in terms of the standard weak-coupling theory. The exotic properties that accompany the formation of the two-channel Kondo (2CK) effect, including the emergence of an unconventional metallic state in the low-energy limit, also originate from strong electron interactions. Despite its paradigmatic role for the formation of non-standard metal behavior, the stringent conditions required for its emergence have made the observation of the nonmagnetic, orbital 2CK effect in real quantum materials difficult, if not impossible. We report the observation of orbital one- and two-channel Kondo physics in the symmetry-enforced Dirac nodal line (DNL) metals IrO2 and RuO2 nanowires and show that the symmetries that enforce the existence of DNLs also promote the formation of nonmagnetic Kondo correlations. Rutile oxide nanostructures thus form a versatile quantum matter platform to engineer and explore intrinsic, interacting topological states of matter.


Supplementary Note 1 Thermal annealing effect on the number density of oxygen vacancies n V O in M O 2 rutiles
The number density of oxygen vacancies n V O in a given sample can be tuned by thermal annealing. The annealing conditions for the M O 2 NWs studied in this work are listed below: • IrO 2 NW A was annealed at 300 • C in vacuum for 1 h.
• IrO 2 NW B1 was annealed at 450 • C in vacuum for 10 min.
• IrO 2 NW B2 was oxygenated in air for ∼ 5 months after the first measurement.
• RuO 2 NW A was annealed at 450 • C in argon for 5 min.
A control experiment of RuO 2 films. In addition to NWs, we have recently grown a series of 150-nm-thick RuO 2 films and carried out a control experiment of the annealing effect under various atmospheric conditions to extract the n V O values 1 . The n V O values in the samples have been inferred from both x-ray photoelectron spectroscopy and 1/f noise studies. These two independent methods give consistent results. Our 1/f noise measurements demonstrate the relation γ ∝ n V O .
Here γ is the Hooge parameter which characterizes the magnitude of the voltage noise power spectrum density S V (f ) = γV 2 /N c f , where V is the bias voltage, f is the frequency, and N c is the total number of charge carriers in the sample. Supplementary Fig. 1 is reproduced from figure 6(a) of Ref. 1 , which shows the variation of γ with T for four RuO 2 films that underwent different annealing conditions (at 500 • C), as indicated. We see that annealing for 25 min in O 2 greatly reduces the γ value (blue symbols), whereas an additional 5 min annealing in Ar significantly increases the γ value (pink symbols). Thus, annealing in O 2 reduces the n V O value in the sample. In contrast, annealing in Ar increases the n V O value in the sample.

Supplementary Note 2 Extraction of the anomalous contribution to the resistivity
Our measurements of ρ(T ) indicate that above ∼ a few (tens) degrees Kelvin, depending on the NWs, the variation of ρ with T in both IrO 2 and RuO 2 NWs obeys the Boltzmann transport equation, as previously established 2 . Thus, the residual resistivity ρ B0 (listed in Table 1) and the Debye temperature, θ D , of each NW can be extracted by the standard method. For RuO 2 NWs, an additional term due to the coupling of electrons with optical-mode phonons needs to be included to fully describe ρ(T ) curves 2 . The fitted values are θ D = 375 K and θ E = 790 K for all NWs, where k B θ E / is the optical-phonon frequency. Supplementary Fig. 2(b) shows the ρ(T ) curves for NWs F (diameter d = 21 nm) and G (d = 90 nm), which were aged in air. The solid curves are the Boltzmann transport predictions. In these two NWs, the low-T anomalies are barely seen. Note that in a number of RuO 2 NWs we have observed pronounced temporal resistance fluctuations, associated with an increase in the time-averaged resistivity with decreasing T at low T , as previously reported 3 . Thus, the averaged resistivity ρ(T ) = 1 t t+ t t dt ρ(T, t ) is our main focus, i.e., we have averaged the measured resistivities to smooth out any time-dependent fluctuations before extracting the low-T anomalies. The solid curves in the main panel of Supplementary Fig. 2(c) show the Boltzmann transport predictions for RuO 2 NWs B and C. The symbols are the measured ρ(T ) . The inset shows ρ(T ) on a lin-log scale to demonstrate the logarithmic behavior of the Kondo effect for T at and above T K . (We also would like to remark that in IrO 2 NWs and in a good number of other metallic NWs, such as ITO NWs 4 and heavily indium-doped ZnO NWs 5 , we have not found any noticeable temporal resistance fluctuations at low T .) Supplementary Fig. 3 shows that the relation between our extracted ρ B0 and the concentration of orbital Kondo scatterers n V O , with the exception of NW B, indicates an approximately linear relation for RuO 2 NWs (cf. Supplementary Note 6 for the extraction of n V O and Table 1  We note that, due to the presence of high levels of point defects in the M O 2 NWs studied in this work, our measured resistance ratios are small, i.e., R(300 K)/R(10 K) < 2, in all samples, implying that the contribution of the electron-phonon scattering to the measured ρ(T ) below several tens degrees Kelvin is minute. This specific material property turns out to be a great advantage in the separation of the anomalous resistivity from ρ B0 . Quantitatively, the electron-phonon scattering contributes less than a few percent to our measured anomalous resistivity in every NW.

Supplementary Note 3 Electron-electron interaction and weak-localization effects
Here we expand on the arguments briefly presented in the main text indicating the negligible contribution of EEI effect and weak localization (WL) corrections to the low-T transport properties. In weakly disordered metals, the EEI effect generically dominates over the WL effect in causing characteristic resistivity increases at low T , which depends sensitively on the effective di-mensionality of the metal 6 . For isotropic 3D systems like the M O 2 rutiles, this EEI correction, ρ(T ) = ρ(T ) − ρ(T 0 ), possesses a √ T dependence and is given by 7 where T 0 is a reference temperature, e is the electronic charge, 2π is the Planck constant,F is the screening factor which parameterizes the degree of screening of the EEI effect, and D is the diffusion constant. Empirically, it has been established that, typically,F 0.1 in weakly disordered metals and alloys [8][9][10] . It thus appears as if the EEI correction may offer an alternative explanation for the observed √ T transport anomaly found in IrO 2 NWs. Indeed, the explanation of a √ T resistivity anomaly at low T in terms of a nonmagnetic 2CK effect driven by a dynamical Jahn-Teller effect 11 has recently been challenged in favor of an explanation based on the Altshuler-Aronov correction [12][13][14] . (The samples studied in Ref. 11 were the layered compound ZrAs 1.58 Se 0.39 .) It is thus important to irrefutably connect the behavior observed in IrO 2 NWs to the orbital 2CK effect. We therefore stress that the observed resistivity rise at low T cannot be due to the EEI effect and the even smaller WL correction. These two effects would only cause a resistivity correction which would be approximately one order of magnitude smaller than the low-T anomalies observed in our measurements on IrO 2 and RuO 2 NWs 3 . For example, for the IrO 2 NW B1 with ρ(10 K) = 74 µΩ cm, D 6.2 cm 2 /s, and by takingF = 0, Supplementary Eq. (1) predicts a largest possible resistance increase of ρ/ρ 2.8 × 10 −4 as T is cooled from 20 to 1 K. In reality, we have observed a much larger resistance increase of 5.1 × 10 −3 (Fig. 2). The characteristic thermal diffusion length in the EEI effect is calculated to be T nm (where T is in K), justifying that the NW is 3D with regard to the EEI effect.
The anomalous resistivities we observe in Figs. 2 and 3 are intrinsic properties of IrO 2 and RuO 2 rutiles, independent of sample geometry. However, in practice, the resistance of a bulk single crystal (e.g., of a volume ∼ 0.1×0.1×0.5 mm 3 ) will be too small for the relative resistance change due to the orbital Kondo effect to be detected unless a large current is applied, which is deemed to cause electron heating. On the other hand, the orbital Kondo resistance in a patterned IrO 2 or RuO 2 film (e.g., of a volume ∼ 0.1×1×10 µm 3 ) shall be readily detectable by using a small current to avoid electron overheating. Experiments on several types of M O 2 films are in progress.
Ruling out surface defect scattering scenario. In this work, the low-T resistivity anomaly measured in a given NW is a bulk property of the NW. It originates from the coupling of conduction electrons with those V O 's distributing across the radius of the NW. If the dynamic defect scattering were dominantly taking place near the surface of a NW, one would expect quasi-two-dimensional (quasi-2D), instead of 3D EEI effect to play a role. The resistance increase due to the quasi-2D EEI effect is given by 5, 10 where R is the sheet resistance. For concreteness, assume the thickness of an active surface defect layer, if any exists, to be of the order of the electron mean free path ∼ 3 nm ( L T ). For a residual resistivity ρ B0 ∼ 100 µΩ cm, R = ρ B0 / ∼ 300 Ω. Then, takingF = 0-0.1, Supplementary Eq. (2) predicts a resistance increase R (T )/R (T 0 ) ∼ 1% as T is cooled from 20 to 1 K. However, in practice, this hypothetical surface layer must be shunted by the much larger current conduction through the bulk of the NW which has a diameter d ∼ 100 nm. Thus, this estimate should be corrected by a factor ∼ /(d/2) ∼ 1/20, leading to a reduced relative sheet resistance change of 1% × (1/20) ∼ 0.05%. This value is one order of magnitude smaller than what is observed in Fig. 2. Furthermore, this quasi-2D effect predicts a ln T dependence, distinct from the measured √ T characteristic. The quasi-2D WL effect will also cause a (smaller) ln T dependence which should be suppressed by a magnetic field of ∼ a few T (Refs. 5,10 ). These features are definitely not seen in Fig. 2. Thus, our measured low-T anomalies cannot be ascribed to a surface defect scattering scenario.
Other scenarios compatible with power-law in temperature behavior. A characteristic feature of Kondo physics in metals is the increasing scattering rate as T is lowered in contrast to, e.g., phonon scattering. Processes like dynamical Coulomb blockade and Luttinger liquid behavior are also associated with power-law behavior in T . These can be safely ruled out as underlying cause in our IrO 2 and RuO 2 NWs as their diameters are much larger than the electron mean free paths in these systems which renders electron transport 3D. Moreover, dynamical Coulomb blockade requires contacts with small tunneling conductances to observe the resulting power-law behavior in T and bias voltage across the junction. The resulting power-law exponent is given by the resistance in units of the quantum of resistance h/(2e 2 ) (Ref. 15 ). Changes in the diameter of a quasi-1D conductor affect the number of transmission channels coupled to the contact and thus the dynamical Coulomb blockade exponent in contrast to the universal exponent we observe. The 3D nature of our NWs also rules out impurities in Luttinger liquids as a possible cause for the observed power-law behavior. In addition, the T dependence of ρ in this case would have a complicated and impurity concentration dependent structure 16 in contrast to our observations. Classical magnetoresistance. For a (3D) normal metal placed in a magnetic field, the classical magnetoresistance due to the Lorentz force is given by ∆R(B)/R(0) = (µB) 2 = (B/n e eρ B0 ) 2 , where µ is the mobility. In our NWs with charge carrier concentration n e ∼ 1 × 10 28 m −3 and ρ B0 ∼ 100 µΩ cm, ∆R(B)/R(0) ∼ 4 × 10 −7 (3 × 10 −5 ) in B = 1 (9) T, which can be completely ignored. For comparison, in IrO 2 and RuO 2 bulk single crystals 2 with small ρ B0 1 µΩ cm, the classical magnetoresistance is pronounced.

Supplementary Note 4 Atomic arrangement around an oxygen vacancy in M O 2 rutiles
Here we provide detailed information for the lattice parameters around the oxygen vacancy V O1 depicted in Fig. 1 Table 1. isosceles triangle. (b) The four oxygen ions, labeled O4 , O1 , O5 and O3 , surrounding the M 3 ion form a (planar) rectangle. (c) The four oxygen ions, labeled O4, O9, O10 and O8, surrounding the M 2 ion form an almost perfect (planar) square. That is, r 1 differs slightly from r 2 , and θ 5 and θ 6 differ slightly from 90 • . These values are calculated from density functional theory and given in Refs. 17 Table 1: Lattice parameters around the oxygen vacancy V O1 . The interionic distances r i (i = 1, ..., 7) are inÅ, and the angles θ i (i = 1, ..., 6) are in degree.

Supplementary Note 5 Vacancy driven 2CK and 1CK effects and dynamical large-N calculations
This section expands the discussion of orbital 1CK and 2CK physics in the DNL metals RuO 2 and IrO 2 due to V O 's in the rutile structure.
Vacancy driven orbital 2CK and 1CK effects. Our starting point is that each V O will generate two defect electrons. We refer to the corresponding basis functions as |Γ 2dim 1 and |Γ 2dim 2 which can be taken as the components of a pseudospin variable. For the 5d electrons of IrO 2 (and 4d electrons of RuO 2 ), the C 4ν at M 2 ensures the degeneracy of d xz and d yz , where the symmetry axis is parallel to theẑ axis. In the rutile structure, the oxygen ions above and below the square centered around M 2, i.e., along theẑ-axis, are minimally (less than half of their ionic radius) set off from the C 4 rotation axis. A V O thus enhances the symmetry around the M 2 site, see Fig. 1. An orthogonality theorem ensures that the conduction electrons wavefunction at the M 2, |Ψ , can be decomposed in terms of the local basis functions as where |Γ m i denotes the i th basis function of the m th irreducible representation. The pseudospin associated with the two-dimensional representation is locally conserved.
In IrO 2 , the valence of Ir is +IV so that the Ir ion at position M 2 results in a [Xe]5d 6 electron configuration. Arguments reminiscent of those used by Cox in the context of UBe 13 yield an effective Kondo model where the role of the nonmagnetic Γ 3 is replaced by the two-dimensional irreducible representation of C 4ν (Ref. 22 ).
where τ d and τ c are vectors of Pauli matrices for d and conduction electrons in orbital space, and J K is a pseudospin exchange coupling constant. The conduction electron spin projection in Supplementary Eq. (4) distinguishes the two degenerate screening channels which are necessary for obtaining an overscreened Kondo effect. If the degeneracy between the two screening channels is lifted an orbital 1CK effect ensues. This situation occurs in RuO 2 . RuO 2 is an antiferromagnet with a high Néel temperature 23,24 . In RuO 2 the electron configuration of the M 2 Ru ion is [Kr]4d 5 . Band structure calculations show that the two t 2g orbitals are half-filled 24 . As a result, spin excitations of the defect electron are gapped and only orbital fluctuations are possible. These results are in line with our model for IrO 2 : For M 2 sites in the vicinity of V O 's, the orbitals d xz and d yz , defined with respect to theẑ-axis shown in Fig. 1d, are half-filled and form the ground state for the defect electron. As in IrO 2 , V O 's in RuO 2 trigger an orbital Kondo effect but due to the lack of time-reversal symmetry, an orbital 1CK effect occurs as demonstrated by the scaling plot of Fig.  4a.
The approximate nature of the almost perfect C 4ν symmetry implies that a low-energy scale exists below which the orbital degeneracy is lifted. We identify this scale with the deviations from the 2CK an issue for RuO 2 NWs due to the more localized nature of the 4d vs. the 5d orbitals, in line with the experimental observation.
In our arguments we focused on the defect electron localizing at site M 2. If the second defect electron localizes at site M 1, one may expect two-impurity Kondo physics to occur. Due to non-symmorphic rutile structure, however, theẑ axis with respect to which we defined d xz and d yz are not parallel. This fact, together with the local nature of the expansion Supplementary Eq. (3), favors local orbital Kondo screening over an orbital dimer.
Dynamical large-N calculations. The T dependence of the transport lifetime τ in our IrO 2 NWs due to orbital 2CK scatterers can be obtained, e.g., via a dynamical large-N approach. We assume that the concentration of V O 's (n V O ) is small enough so that inter-vacancy effects can be ignored. The lifetime τ is related to T -matrix T (ω, T ) via τ (k) −1 = 2 c 2CK Im k|T (ω, T )|k , where c 2CK is the concentration of spherically symmetric dynamic scattering centers, which we identify with n V O , and |k is a plane wave state. T (ω, T ) can be obtained from a dynamical large-N method which is known to give reliable results for multichannel Kondo, including orbital 2CK problems 22,[25][26][27] . In this approach, the symmetry group is extended to N spin and M orbital channels, resulting in an SU(N )×SU(M ) invariant Hamiltonian of the form where n f (n B ) is the fermionic (bosonic) distribution function, and A f (A B ) is the spectral function associated with the fermionic pseudoparticle representation (bosonic decoupling field) 25,27 . For κ = 1, one finds with A < 0 as shown in Fig. 4d. The large-N analog of the 1CK case where κ = 1/2 is shown in Supplementary Fig. 5 where m ∼ 1.6 m 0 is the effective electron mass (m 0 is the free electron mass) 29 , n e ∼ 1 × 10 28 m −3 is the carrier density 30 , Λ a ∼ 1 × 10 −29 m −3 is the atomic volume, and ImT (ω = 0, T ) is the imaginary part of the T -matrix calculated from the dynamical large-N method. The n V O value of every IrO 2 NW has been extracted by fitting the measured slope of the ρ ∝ √ T curve in Fig.  2 to Supplementary Eq. (8). The extracted value is in good agreement with that inferred from the conformal-field-theory calculations 31 .
As the low-T resistivities of RuO 2 NWs conform to the 1CK scaling form, the ρ K0 value in every NW can be inferred. In this unitary limit, the Kondo resistivity is given by 32 where m ∼ 1.4 m 0 (Ref. 33 ), n e ∼ 1 × 10 28 m −3 (Ref. 34 ), and the density of states per spin orientation g 0 ∼ 1 × 10 47 J −1 m −3 (Ref. 33 ). The n V O value in every RuO 2 NW has been calculated from the extracted ρ K0 value, according to Supplementary Eq. (9).

Supplementary Note 7 Kondo temperature distribution
Our experimental findings are compatible with a single energy scale, termed Kondo temperature T K , that characterizes the low-T transport in each NW. This implies that the distribution of Kondo temperatures in each NW is sharply peaked around a single T K . This is demonstrated in Supplementary Fig. 6 where the measured ρ K (T ) of RuO 2 NW C is plotted against NRG results for the T -matrix T (T ) of the 1CK Kondo model, with the least-squares fitted T K = 70 K (solid curve). In addition, we plot T (T ) with four selected T K values (dashed curves) for comparison. Obviously, the theoretical curves with T K = 35 K and T K = 105 K already deviate significantly from the experimental data.
Supplementary Figure 6: Kondo temperature extraction. NRG calculations for the 1CK model are fitted to ρ K (T ) of RuO 2 NW C with the Kondo temperature T K as the only adjustable parameter. Solid and dashed curves are characterized by different T K values, as indicated.
The Kondo energy scale T K thus seems to be unaffected by the disorder generated through variations accompanying the oxygen defects. Similar observations have been reported in a variety of systems, including metal nanoconstrictions 35 , dilute magnetic noble metals 36 , and other Kondo systems 37,38 . This insensitivity of T K is not entirely unexpected. The exchange coupling constant J K , Supplementary Eq. (4), reflects the wavefunction overlap between the local basis functions of the dynamic scattering center and the conduction electrons. J K is therefore not affected by background variations due to nearby V O 's or the presence of the second defect electron on a nearby M ion site. This form of disorder will instead lead to local variations of the background potential. Thus, the situation resembles that considered by Chakravarty and Nayak, who have shown that for weak disorder the distribution of the local density of states is very narrow 39 . Furthermore, the thermodynamics is unaffected near the strong-coupling limit. While T K is unaffected by the background variations caused by the weak disorder, it does change from NW to NW. These variations arise from changes in ρ(E F ) as a result of the stress that exists in each NW, where ρ(E F ) is the local density of states of conduction electrons coupling to the orbital degree. As the bulk and shear modulus of RuO 2 is less than those of IrO 2 (Refs. 17,18 ), it appears natural that the correspond-