Kondo screening in a Majorana metal

Abstract

Kondo impurities provide a nontrivial probe to unravel the character of the excitations of a quantum spin liquid. In the S = 1/2 Kitaev model on the honeycomb lattice, Kondo impurities embedded in the spin-liquid host can be screened by itinerant Majorana fermions via gauge-flux binding. Here, we report experimental signatures of metallic-like Kondo screening at intermediate temperatures in the Kitaev honeycomb material α-RuCl₃ with dilute Cr³⁺ (S = 3/2) impurities. The static magnetic susceptibility, the muon Knight shift, and the muon spin-relaxation rate all feature logarithmic divergences, a hallmark of a metallic Kondo effect. Concurrently, the linear coefficient of the magnetic specific heat is large in the same temperature regime, indicating the presence of a host Majorana metal. This observation opens new avenues for exploring uncharted Kondo physics in insulating quantum magnets.

Introduction

When a magnetic impurity is introduced into a metal, conduction electrons interact with the local magnetic moment. At temperatures below the so-called Kondo temperature, the impurity spin becomes effectively screened by the surrounding conduction electrons, creating a many-body entanglement cloud¹. This Kondo effect brings about a reduction in the magnetic moment of the impurity spins and a drastic increase in resistivity. Beyond normal metals, the purview of Kondo physics has expanded into various materials, including quantum dots, graphene, topological insulators, and Weyl semimetals^2,3,4,5,6. It is also envisioned that the Kondo effect may occur in quantum spin liquids (QSLs) that constitute highly entangled quantum states harboring fractionalized spinon excitations, an emergent gauge structure, and topological order^{7,8,9,10,11,12,13,14,15,16}. In addition, magnetic impurities incorporated into QSLs may be subject to RKKY-type interactions mediated by spinons or gauge fluctuations. In this context, Kondo impurities can act as in-situ probes for QSLs.

A S = 1/2 Kitaev model on the honeycomb lattice offers an archetypical platform for exploring unusual Kondo effects: While magnetic insulators often feature bosonic excitations, such as triplons or magnons, which cannot easily screen impurity spins, the fractionalized Kitaev QSL state hosts charge-neutral fermionic excitations, which can effectively screen impurity spins. At finite temperatures T, itinerant Majorana fermions (MFs) wander around thermally activated π-fluxes (W_p = −1)^17,18, emulating metallic behavior, whereas the fluxes freeze out at low T, resulting in a Majorana semimetal. When a spin-1/2 impurity is exchange-coupled to a Kitaev spin, a first-order transition takes place at low T as a function of the Kondo coupling between the weak-coupling flux-free phase and the strong-coupling impurity-flux phase. In the latter, each impurity moment binds a gauge flux in the enlarged impurity plaquette, thereby inducing locally metallic behavior of the MFs, in turn leading to Kondo screening^9,10,11.

A conspicuous candidate material for testing the proposed Kitaev Kondo effect is α-RuCl₃^19,20, as it is in close proximate to a Kitaev QSL. Its spin Hamiltonian is best described by the K-J-Γ-Γ’ model

$$H= \mathop{\sum }\limits_{ < {ij} > x}{S}_{i}^{x}\left(\begin{array}{ccc}J+K & {\varGamma }^{{\prime} } & {\varGamma }^{{\prime} }\\ {\varGamma }^{{\prime} } & J & \varGamma \\ {\varGamma }^{{\prime} } & \varGamma & J\end{array}\right){S}_{j}^{x}+\mathop{\sum }\limits_{ < {ij} > y}{S}_{i}^{y}\left(\begin{array}{ccc}J & {\varGamma }^{{\prime} } & \varGamma \\ {\varGamma }^{{\prime} } & J+K & {\varGamma }^{{\prime} }\\ \varGamma & {\varGamma }^{{\prime} } & J\end{array}\right){S}_{j}^{y}\\ +\mathop{\sum }\limits_{ < {ij} > z}{S}_{i}^{z}\left(\begin{array}{ccc}J & \varGamma & \varGamma {\prime} \\ \varGamma & J & \varGamma {\prime} \\ \varGamma {\prime} & \varGamma {\prime} & J+K\end{array}\right){S}_{j}^{z}$$

with dominant Kitaev interaction K = −5–10 meV over Heisenberg (J = −3 meV) and off-diagonal symmetric exchange interactions Γ = 2–3 meV and Γ’ = 0.1 meV^21,22,23. α-RuCl₃ shows the zigzag magnetic order below T_N = 6.5 K, preempting a Kitaev QSL. Although deviations from an ideal Kitaev model occur due to the presence of non-Kitaev terms, many independent experimental techniques suggest that Majorana and gauge degrees of freedom provide a good description of the α-RuCl₃ magnetism at elevated energies and temperatures (T > J, Γ, Γ’)^22,23,24,25. In addition, α-RuCl₃ benefits from the availability of its isostructural counterpart CrCl₃ (Cr: 3d³; S = 3/2)^26,27. CrCl₃ is a quasi-two-dimensional ferromagnet (FM) with consecutive FM and AFM orders at T_C = 17 K and T_N = 14 K, respectively. Taken together, mixed-metal trihalides α-Ru_1−xCr_xCl₃ with random Ru/Cr occupancies²⁸ constitute a suitable model system for studying a Kitaev Kondo problem, gaining a fundamental understanding of S = 3/2 impurities embedded in a Kitaev paramagnetic host.

Here, we find several key signatures of metallic Kondo screening in a Kitaev paramagnetic state: logarithmic singularities in magnetic susceptibility, the muon Knight shift, and the muon spin-relaxation rate. Along with these characteristic Kondo signatures, a substantial magnetic contribution to the specific heat, C_m/T, raises the possibility that the observed Kondo screening arises from a Majorana metal host.

Results

Fractionalized spin excitations and structural homogeneity

Figure 1a schematically illustrates the formation of impurity plaquettes (W_I = −1; gray polygons) with binding of a gauge flux in the three adjacent plaquettes when S = 1/2 magnetic impurities are introduced to a Kitaev spin system. In Fig. 1b, we plot the T–x phase diagram of α-Ru_1−xCr_xCl₃ (x = 0–0.07), which reveals a slight reduction in the magnetic ordering temperature to T_N ≈ 5 K. Additionally, within a Kitaev paramagnetic regime, there is an indication of a weak Kondo coupling, which is a central focus of this study.

Fig. 1: Schematic sketch of gauge-flux-driven Kondo screening, x-T phase diagram, and fractionalized excitations of α-Ru_1−xCr_xCl₃.

a (Top) A Kitaev paramagnetic state consists of coherently propagating Majorana fermions (black dots) and thermally populated π-fluxes (W_p = −1) out of the frozen Z₂ gauge fluxes (incarnadine hexagons; W_p = +1). (Bottom) Spin−1/2 impurities coupled strongly to individual host spins (blue spheres) engender impurity plaquettes (W_I = −1; gray polygons) by a gauge flux in the three adjacent plaquettes. In addition, distant magnetic impurities can interact via long-range interactions (orange arrows). b T–x phase diagram of α-Ru_1−xCr_xCl₃ (x = 0–0.07). The characteristic temperatures T_K^onset, T_K^end, and T_N are determined from the dc magnetic susceptibility, specific heat, and μSR measurements. The band edge energy D is evaluated from the logarithmic fits to the magnetic susceptibility. The black dashed curve is a guide to the eye. AFM stands for antiferromagnetically ordered phase. c As-measured Raman spectra at T = 5 K. The color shadings denote the broad magnetic continuum. The inset plots the normalized intensity of the magnetic continuum as function of the concentration of the Cr³⁺(S = 3/2) impurities.

We first confirmed the phase purity and composition of α-Ru_1−xCr_xCl₃ through EDX and X-ray diffraction (XRD) analyses, as presented in Supplementary Figs. 1–3. Subsequently, we examine their structural and magnetic excitations as a function of Cr³⁺ impurity concentration x to clarify the effects of the Cr-for-Ru substitution. Figure 1c shows the Raman spectra obtained at T = 5 K in in-plane polarization. For all the investigated x = 0–0.07, we observe a broad magnetic continuum (color shadings) with well-defined phonon peaks (Supplementary Fig. 4). In a Kitaev spin liquid, a magnetic Raman scattering process mainly involves the simultaneous creation or annihilation of pairs of MFs^29,30,31. The observed magnetic Raman response comprises both MF and incoherent magnetic excitations, consistent with previous Raman data^27,29. Remarkably, the magnetic continuum varies little with x in its spectral form and intensity (the inset of Fig. 1c). The robustness of fractionalized excitations against Cr³⁺ substitution indicates that a Kitaev paramagnetic state is hardly affected by the insertion of magnetic impurities. Moreover, the Cr³⁺ substitution for Ru³⁺ does not result in any essential changes in the frequency, FWHM, normalized intensity, and the asymmetry parameter 1/|q| of the A_g(1)+B_g(1) and A_g(2)+B_g(2) Fano resonance modes (Supplementary Figs. 4 and 5). Additionally, we could not detect any additional phonon peaks within the studied composition range. This observation, in conjunction with the absence of noticeable peak splitting in the single-crystal XRD data (Supplementary Figs. 2 and 3), strongly supports symmetry preservation, excluding the possibility of structural domains or phase segregation. These results suggest that the substituted Cr spins are randomly distributed throughout the lattice, although atomic-scale inhomogeneities cannot be entirely ruled out.

Magnetic impurity effects on a static magnetic response

The Cr³⁺-for-Ru³⁺ substitution modifies the K-J-Γ-Γ’ exchange parameters of the mother compound α-RuCl₃ by generating Heisenberg-type interactions on the Cr-Ru bonds. This is because Cr³⁺ ions in the high-spin d³ S = 3/2 configuration are orbitally inactive and, thus, are unable to provide multiple anisotropic and spin-dependent exchange paths required for K-Γ interactions. In the Kitaev paramagnet, this changes the local energetics of the fluxes and also leads to the scattering of the itinerant MFs.

Figure 2a and Supplementary Figs. 6–8 exhibit the static magnetic susceptibilities χ(T) and magnetization of α-Ru_1−xCr_xCl₃ (x = 0–0.07) for B//ab and B//c, along with corresponding Curie-Weiss fits. The Curie-Weiss behavior is identified in the paramagnetic state above T = 100–180 K (indicated by the dashed lines in Supplementary Fig. 6), and the Curie-Weiss parameters are summarized in Supplementary Fig. 7. The in-plane χ_ab(T) shows a small variation with x: the Curie-Weiss temperature ${\Theta }_{{{{{{\rm{CW}}}}}}}^{{ab}}$ and the effective magnetic moment ${\mu }_{{{{{{\rm{eff}}}}}}}^{{ab}}$ hardly change with increasing Cr³⁺ impurities. The AFM ordering temperature is slightly reduced from T_N = 6.5 K at x = 0 to 5 K at x = 0.03–0.07 with no indications of spin-glass behavior down to 2 K. In sharp contrast to χ_ab(T), the out-of-plane χ_c(T) increases rapidly with increasing x. The large negative ${\Theta }_{{{{{{\rm{CW}}}}}}}^{c}$ is drastically repressed towards T = 0 K and ${\mu }_{{{{{{\rm{eff}}}}}}}^{c}$ = 3 μ_B is reduced to 2.3 μ_B as x increases up to 0.07 (Supplementary Fig. 7b, c). The drastic impact of the Cr³⁺ impurities on χ(T,x) is quantified by the magnetic anisotropy χ_ab(T, x)/χ_c(T, x), as shown in Fig. 2b. With increasing x, the XY-like magnetism becomes more isotropic, signaling that the Cr³⁺ substitution weakens the Γ-Γ’ terms while enhancing the Heisenberg interaction³². Noteworthy is that a non-monotonic T dependence of χ_ab/χ_c features a maximum at about T^* = 25–40 K above x = 0.01 (the vertical arrows in Fig. 2b). The decrease of χ_ab/χ_c below T^* alludes to the growth of isotropic magnetic correlations beyond the underlying K-J-Γ-Γ’ magnetism.

Fig. 2: Static magnetic susceptibility and magnetic anisotropy as a function of Cr content.

a Temperature dependence of dc magnetic susceptibility χ(T) of α-Ru_1−xCr_xCl₃ (x = 0–0.07) measured in an applied field of B = 0.1 T along the ab plane (full symbols) and the c-axis (open symbols). The out-of-plane χ_c(T) shows a drastic increase with increasing x, rendering the magnetism of α-Ru_1−xCr_xCl₃ isotropic. b Temperature and composition dependence of the magnetic anisotropy χ_ab/χ_c of α-Ru_1−xCr_xCl₃ measured in an applied field of B = 0.1 T. An XY-like magnetic anisotropy is systematically reduced with increasing Cr³⁺ concentration. The downward arrows indicate the broad maximum temperature T^* in χ_ac/χ_c. The inset plots T^* versus x.

Logarithmic singularities of static magnetic susceptibility

A number of theoretical predictions have been made for impurities in Kitaev QSLs^9,10,11,33, but most of them are valid in the limit of low temperatures only. Here, we are interested in a finite-T crossover regime where conventional metallic-like Kondo screening would lead to a logarithmic increase of χ(T)~ln(D/T), while the flux-binding mechanism in a semimetal would not lead to such logarithmic behavior¹¹.

To test the aforementioned scenarios, we plot χ_c(T) in Fig. 3a on a semilogarithmic scale, revealing a suggestive logarithmic behavior. To isolate the contribution induced by impurity spins, we present the difference of the static susceptibilities between the pristine and the Cr³⁺-substituted samples, Δχ_c(T) = χ_c(T)-χ_c(T; x = 0) in Fig. 3b and Supplementary Fig. 9. Remarkably, we observe that Δχ_c(T) follows a logarithmic dependence, ln(D/T), in the temperature interval between T_N and ~20 K. Within this range, we identify two characteristic temperatures, T_K^onset and T_K^end, which delineate the interval where the logarithmic temperature dependence of Δχ_c(T) appears. In the T = 30−100 K range, the logarithmic T dependence transits to an approximate power-law dependence χ(T) ~ T^α(T)−1 with α(T) ≈ −0.12–0.14 (Supplementary Figs. 9–11), which we interpret as a crossover to the high-T Curie-Weiss-like regime. The deviation from α = 0 is attributed to scatterings off of itinerant MFs by Cr³⁺ impurities. The fit parameter D is evaluated to be D = 23–67 K (the star symbols in Fig. 1b), which is comparable to the strength of the subdominant J-Γ-Γ’ interactions and roughly agrees with T^* in Fig. 2b. These results suggest that α-Ru_1−xCr_xCl₃ displays Kondo physics different from the flux-driven mechanism of ref. ¹¹. The out-of-plane χ_ab(T) data also hold logarithmic signatures, yet their weak x dependence (Supplementary Fig. 12a) disallows extracting reliable parameters. Further, we note that the Kondo temperature cannot be tracked as the logarithmic behavior is disrupted by the onset of AFM order. Furthermore, we attempted to analyze the Δχ_c(T) data in terms of the equivalent three-channel Kondo model³⁴. We observe a qualitative agreement within the temperature range of T_N < T<T_K^onset, but not extending to temperatures T_K^onset < T (Supplementary Fig. 10). Moreover, the derived Kondo temperature T_K is notably lower than T_K^onset. This discrepancy is related to the fact that Δχ_c(T) continues to increase upon cooling in the fitting range above T_N (see Supplementary Fig. 10f) and that the Cr impurity in α-Ru_1−xCr_xCl₃ is described by a S = 3/2 inequivalent three-channel Kondo model¹¹, as detailed in Supplementary Note 3. In addition, the remaining deviations may originate from inadequate fitting functions and the influence of vison dynamics.

Fig. 3: Thermodynamic signatures of Kondo screening.

a, b Temperature dependence of the static magnetic susceptibility χ_c(T) and the pristine-subtracted Δχ_c(T) = χ_c(T) –χ_c(T;x = 0) for α-Ru_1−xCr_xCl₃ (x = 0.01–0.07) in an applied field of B//c = 0.1 T. The solid lines are fittings to logarithmic divergence Δχ(T)~ln(D/T), where D is the band edge energy. c Comparison of the T-dependent magnetic specific heat C_m(T) between α-Ru_1−xCr_xCl₃ (x = 0.04) and the pristine material (x = 0). C_m(T) is obtained by subtracting a lattice contribution from the total specific heat (Supplementary Fig. 12). The solid lines indicate a T-linear dependence of C_m(T). The error bars represent one standard deviation of the three repeated specific-heat measurements. d Normalized magnetic entropy S_m/S_m^theory as a function of temperature evaluated by integrating C_m(T)/T in a semi-log scale. S_m^theory is Rln2 and 0.96Rln2 + 0.04Rln4 for x = 0.00 and 0.04, respectively. The solid and dashed lines denote a fit using three phenomenological functions (“Methods”).

Metallic behavior of Majorana fermions

To probe the Cr³⁺ substitution effect on low-energy excitations, we examine the magnetic specific heat C_m(T) obtained by subtracting lattice contributions from the total specific heat C_p(T) (Supplementary Figs. 11 and 12 and “Methods”). In Fig. 3c, we compare C_m(T) between α-Ru_1−xCr_xCl₃ (x = 0.04) and the pristine sample (x = 0). C_m(T) of the x = 0 sample shows a λ-like peak at T_N = 6.5 K, followed by a plateau in the temperature range of T = 15–50 K and a subsequent increase up to T_H ∼ 100 K. Upon introducing the Cr³⁺ impurities, two weak anomalies appear at T_N1 = 4.8 K and T_N2 = 10.4 K for x = 0.04, corresponding to the magnetic ordering of ABC- and AB-type stacking patterns (Supplementary Figs. 11 and 12). As evident from Supplementary Fig. 11b, the addition of 2% magnetic impurities induces a linearly increasing fraction of C_m in the intermediate T = 13–50 K plateau regime for x = 0. This trend is enhanced with increasing x up to 0.04. The emergence of a linear T contribution to C_m below T_H is a signature of metallic behavior of the itinerant MFs³⁵: Such effective metallicity arises from the presence of thermally populated π-fluxes (W_p = −1), as illustrated in Fig. 1a.

Shown in Fig. 3d is the magnetic entropy ${S}_{{{{{{\rm{m}}}}}}}\left(T\right)=\int {C}_{{{{{{\rm{m}}}}}}}/T{dT}$. We recall that in an ideal Kitaev system, each half of S_m (T) is released by itinerant and localized MFs³⁶. Unlike the x = 0 sample²³, the magnetic entropy of x = 0.04 is released in three steps with the weighting factors ρ₁ = 0.15Rln2, ρ₂ = 0.19Rln2, ρ₃ = 0.66Rln2 (R = ideal gas constant) and the crossover temperatures T₁ = 10.7(3) K, T₂ = 24(4) K, and T₃ = 70(7) K (“Methods”). T₁ and T₂ correspond to the end temperatures where the logarithmic behavior of χ_c(T) appears (Fig. 1b). On the other hand, the power-law dependence χ(T) ~ T^α(T)−1 is observed between T₂ and T₃ (Supplementary Fig. 9). We note that one Kondo S = 3/2 spin is coupled to the three adjacent S = 1/2 sites, leading to flux conservation in the Kitaev QSL only in the joint six-plaquette area surrounding to the impurity^9,10,11. Therefore, 4% Cr³⁺ substitution modifies 24% of the fluxes near the impurities. Qualitatively, the three-step entropy release is consistent with this picture.

Logarithmic singularities of the muon Knight shift and relaxation rate

To shine more light on the Kondo behavior, we carried out muon spin rotation/relaxation (μSR) measurements of α-Ru_1−xCr_xCl₃ (x = 0.04) in zero (ZF), longitudinal (LF), weak (wTF), and high (hTF) transverse fields. The wTF- and ZF-μSR data confirm the two successive magnetic transitions at T_N1 = 5 K and T_N2 = 12 K (Supplementary Figs. 13 and 14), in line with our magnetic and thermodynamic results.

As exhibited in Fig. 4a, the normalized fast Fourier transformed (FFT) amplitudes of the hTF-μSR spectra measured at T = 15 K show a Lorentzian shape with intriguing field evolution. Fittings reveal two Lorentzian relaxing cosine components (see Fig. 4b, c): (1) a sharp signal (yellow curve) and (2) a broad signal (green curve). The obtained fitting parameters are plotted in Fig. 4d–g and Supplementary Fig. 15. Given the fact that the field-induced crossover, involving the change of a magnetic domain structure, occurs across B ~ 1 T³⁷ (Supplementary Fig. 8), we chose the two representative fields B₀ = 0.5 and 3 T for detailed T-dependent studies.

Fig. 4: High transverse-field μSR data of α-Ru_1−xCr_xCl₃ (x = 0.04).

a Normalized FFT amplitudes of hTF-μSR in applied fields of B_ext//c = 0.2–3 T at T = 15 K. The data are vertically shifted for clarity. b, c Magnified views of normalized FFT amplitudes at B_ext = 0.5 and 3 T. The black solid lines denote the total fitting lines that are a sum of two Lorentzian damped cosines (yellow and green lines). d, e Temperature dependence of the muon Knight shift for the fast (K_f) and slow (K_s) relaxing components in applied fields of B_ext//c = 0.5 and 3 T. K_f(T) is described by power-law behaviors K_f ~ T⁻ⁿ (dashed lines), which deviates below T_N2 = 12 K, while K_s(T) exhibits a logarithmic dependence K_s~ln(D/T) (solid lines) predicted for a singlet vortex case above 10 K. Error bars represent one standard deviation. f, g Muon spin-relaxation rates for the fast (λ_f) and slow (λ_s) component as a function of temperature on a double logarithmic scale. λ_f(T) displays a power-law down to T_N2 (dashed lines), similar to K_f. On the other hand, λ_s(T) at B_ext = 3 T is well described by a logarithmic dependence λ_s ~ 1/T₁ ~ T[ln(D/T)]² (solid lines). Error bars of the muon Knight shift and the relaxation rat represent one standard deviation of the fit parameters.

The locally probed intrinsic magnetic susceptibility is reflected in the T-dependent muon Knight shifts K_f(T) and K_s(T). K_s(T) and K_f(T) scale well with ±χ_c(T) down to 2 K (Supplementary Fig. 15), indicating that the logarithmic dependence of χ(T) seen in the T = T_N −20 K range is little affected by extrinsic contributions. Notably, K_f(T) and K_s(T) clearly show distinct temperature dependences. K_s(T) displays a logarithmic dependence ln(D/T) above 10 K, while K_f(T) shows a power-law behavior T⁻ⁿ (see Fig. 4d, e). The extracted D = 30.8(6) K (27.9(7) K) for B = 0.5 T (3 T) is comparable to the value evaluated from the static χ_c(T) data shown in Fig. 3a, b. Furthermore, based on the relation λ ~ 1/T₁ ~ A²T[Imχ(T,ω)/ω]_ω→0, the muon relaxation rate could be expected to follow logarithmic behaviors of λ ~ T[ln(D/T)]² for a single vacancy or λ ~ 1/T [ln(D/T)]² for a pair of nearby vacancies on the same sublattice, respectively^9,10,11. We find that only the slow relaxation rate λ_s(T) for B = 3 T shows a logarithmic T dependence T[ln(D/T)]² with D = 45(1) K. On the other hand, λ_f(T) follows a power-law behavior T^−α with α = −0.98(1) for B = 3 T and α = −2.22(7) for B = 0.5 T above T = 8 K (see Fig. 4f). The concomitant power-law dependence of λ_f(T) and K_f(T) suggests that the fast component stems from correlated spins pertinent to defects and bond disorders, which inevitably occur due to stacking faults and local strains induced by the Cr³⁺-for-Ru³⁺ substitution. Actually, the static magnetic susceptibility follows an approximate power law χ(T) ~ T^α(T)−1 in the elevated temperatures of T = 30–100 K.

Discussion

Combining specific heat, magnetic susceptibility, and μSR probes, we find that mixed-metal trihalides α-Ru_1−xCr_xCl₃ offer a promising arena for exploring a Kitaev Kondo problem. The magnetism of α-Ru_1−xCr_xCl₃ is modeled by the K-J-Γ-Γ’ spin Hamiltonian³², where the strength of J relative to Γ-Γ’ increases with x. Our findings reveal several key points.

First, we observe that the Cr³⁺ substitution exerts no significant impact on fractionalized excitations at intermediate T (Fig. 1c and Supplementary Fig. 4) despite the Heisenberg-type interaction J_Ru-Cr perturbs the original K-J-Γ-Γ’ exchange interactions. Second, as evident from the rapid suppression of XY-like magnetic anisotropy in Fig. 2b, the inclusion of the spin-$\frac{3}{2}$ impurities diminishes the Γ-Γ’ terms, while augmenting the isotropic Heisenberg interaction. Third, C_m(T) and S_m(T), tracking thermal fractionalization of spins into itinerant MFs and Z₂ fluxes, demonstrate that the addition of magnetic impurities expands the Kitaev paramagnetic state down to T_N, which is much lower than ~50 K of α-RuCl₃. The sizeable linear term in C_m, a hallmark of the metallic density of states, negates a paramagnon scenario. This expanded Majorana-metal regime can be rationalized by noting that the impurities both increase the fluctuations of the gauge fluxes and, at the same time, scatter the itinerant MFs, thereby inducing low-energy Majorana states. Fourth, both static and dynamic magnetic probes commonly feature logarithmic singularities of the conventional Kondo effect. Finally, the three-step release of S_m(T), the three-step evolution of χ(T), and the magnetic anisotropy (χ_ab/χ_c) anomaly at T^* ≈ 25–40 K equivocally evidence the emergence of magnetic correlations induced by a few percentages of magnetic impurities.

This together with the large Kondo energy of ~30 K suggests that the scenario¹¹ of low-T gauge-flux-driven Kondo screening in a Majorana semimetal is not applicable to α-Ru_1−xCr_xCl₃. Instead, at elevated temperatures, a strongly fluctuating flux (or vison) background produces a Majorana metal host. In this situation, no explicit binding of fluxes to impurities is required for Kondo screening. Rather, the global presence of thermally excited gauge fluxes provides a natural mechanism for a metallic Kondo effect with logarithmic signatures, here for S = 3/2 moments with three inequivalent screening channels¹¹, here for S = 3/2 moments with three screening channels. At larger x, this Kondo physics will compete against the fluctuation-mediated inter-impurity interactions. We recall that the Kondo effect in a magnetic insulator has recently been reported in the Zn-brochantite ZnCu₃(OH)₆SO₄, a Kagome antiferromagnet that holds a proximate QSL⁸. In this case, magnetic impurities originating from Cu-Zn intersite disorders act as Kondo spins that may be screened by spinon-spinon interactions, but the precise mechanism has not been clarified. Thanks to its analytical solvability, however, an impurity-doped Kitaev system enables the exploration of uncharted territory including multi-channel Kondo physics and its interplay with gauge fluctuations.

To conclude, we have showcased metallic-like Kondo behavior in the Kitaev candidate material α-Ru_1−xCr_xCl₃ containing S = 3/2 magnetic impurities, demonstrating the presence of a host Majorana metal. Multiple Kondo impurities and their interplay may bring about a new species of Kondo and ordering phenomena. Extending the present phenomena to low temperatures in a material without magnetic ordering would give access to the regime of flux binding by impurities¹¹, then raising the prospect of braiding impurity fluxes via impurity manipulation toward the implementation of quantum computation^17,18.

Methods

Sample preparation

Single crystals of α-Ru_1−xCr_xCl₃ (x = 0–0.07) were synthesized by a vacuum sublimation method. A commercial compound of RuCl₃ (Alfa Aesar) was ground and dried in a quartz tube under vacuum until it was completely dehydrated. The resulting powder was then sealed in an evacuated quartz ampule, which was placed in a temperature gradient furnace. The ampule was heated at 1080 °C for 24 h and then slowly cooled down to 600 °C at a rate of 2 °C/h. The obtained single crystals have typical sizes of about 5 × 5 × 1 mm³ with a shiny black surface.

Structural and thermodynamic measurements

The crystal structure of α-Ru_1−xCr_xCl₃ was determined by X-ray diffraction measurements using Cu Kα radiation (the Bruker D8-advance model). The phase purity and stoichiometry of the single crystals were confirmed by energy dispersive X-ray spectroscopy (EDX). The actual Ru:Cr ratio was evaluated by scanning a dozen spots of 50 μm size (Supplementary Fig. 1). The standard deviation from the mean value is evaluated to be ~1 mol% Cr for all crystals. We measured dc magnetic susceptibility and magnetization with a SQUID (Quantum Design MPMS) and Physical Property Measurements System (Quantum Design PPMS Dynacool) for B//ab and B//c in the temperature range T = 2–300 K. High-field magnetization measurements were conducted at the Dresden High Magnetic Field Laboratory with a pulsed-field magnet (25 ms duration) using an induction method with a pickup coil device at T = 2 K. Specific heat experiments were carried out under applied fields of B//c = 0, 0.5, and 3 T in the temperature range of T = 2–200 K with a thermal relaxation method using a commercial set-up of Physical Property Measurements System.

The magnetic specific heat of α-Ru_1−xCr_xCl₃ was obtained by subtracting the specific heat of the isostructural nonmagnetic counterpart ScCl₃. Using the Bouvier method³⁸, we scaled the specific heat data of ScCl₃ by the molecular mass and Debye temperature and then used this scaled specific heat data to evaluate the magnetic specific heat of the Cr-doped RuCl₃. In doing that, we assumed that the Debye temperature does not vary significantly with the small Cr concentration (Supplementary Figs. 11 and 12). The magnetic specific heat was fitted using a sum of two phenomenological functions³⁹, ${S}_{m}={\sum }_{i={{{{\mathrm{1,3}}}}}}{S}_{{m}_{i}}={\sum }_{i={{{{\mathrm{1,3}}}}}}\tfrac{{\rho }_{i}/2}{1+\exp \left[\left(\tfrac{{\beta }_{i}+{\gamma }_{i}{T}_{i}/T}{1+{T}_{i}/T}\right){{{{\mathrm{ln}}}}}(\tfrac{{T}_{i}}{T})\right]}$. Here, ρ_i is the weighting factor with a scaled constraint of ρ₁ + ρ₂ + ρ₃ = 2.14 and T_i is the crossover temperature. β_i and γ_i are the power exponents at high and low temperatures, respectively. The fitting parameters are evaluated to be ρ₁ = 0.32(1), β₁ = 2.9(2), γ₁ = 5.18(9), T₁ = 10.7(3) K, ρ₂ = 0.41(5), β₂ = 5.13(7), γ₂ = 1.6(3), T₂ = 24(4) K, ρ₃ = 1.41(3), β₃ = 3.88(9), γ₃ = 0.7(2), and T₃ = 70(7) K.

Raman scattering

Raman scattering experiments were conducted in backscattering geometry with the excitation line λ = 532 nm of the DPSS SLM laser. The Raman scattering spectra were collected using a micro-Raman spectrometer (XperRam200VN, NanoBase) equipped with an air-cooled charge-coupled device (Andor iVac Camera). We employed a notch filter to reject Rayleigh scattering at low frequencies below 15 cm⁻¹. The laser beam with P = 80 μW was focused on a few-micrometer-diameter spot on the surface of the crystals using a ×40 magnification microscope objectives. The samples were mounted onto a ⁴He continuous flow cryostat by varying a temperature T = 4–300 K.

Phonon excitations below 200 cm⁻¹ were fitted using an asymmetric Fano profile $I\left(\omega \right)={I}_{0}\frac{{\left(q+\epsilon \right)}^{2}}{(1+{\epsilon }^{2})}$, where $\epsilon=\left(\omega -{\omega }_{0}\right)/\varGamma$ and $\varGamma$ is the full width at half maximum (FWHM) in case of strong coupling between spin and lattice degree of freedom. $1/\left|q\right|$ provides a measure of the coupling strength between a magnetic continuum and optical phonons or conveys information about Majorana excitations.

Muon spin relaxation/rotation

Muon spin-relaxation/rotation (μSR) measurements were conducted on the GPS⁴⁰ and the HAL-9500 spectrometers at the Paul Scherrer Institute (Villigen, Switzerland). For the GPS spectrometer measurements, a mosaic of a-axis coaligned single crystals (~0.5 g) was packed in an aluminum foil and attached to a sample holder. The Veto mode was activated to minimize the background signal. ZF- and TF-μSR experiments on the GPS spectrometer were performed in the spin-rotated mode, where the initial muon spins were rotated by 45° from the muon momentum direction (c-axis). It should be noted that α-RuCl₃ shows anisotropic 2D XY-like magnetism, resulting in weaker spin correlations along the c-axis compared to those in the ab-plane. This makes it difficult to detect changes in the muon spin relaxation when the muon spins are directed along the c-axis. To minimize the contribution of spin correlations along the c-axis, up and down detectors were utilized in this spin-rotated mode. On the other hand, LF-μSR measurements on the GPS spectrometer were carried out in the longitudinal mode, where the initial muon spins were parallel to the c-axis. For the HAL-9500 experiments, a single piece of large single crystal (8 × 8 × 1 mm³, ~150 mg) was wrapped with a Ag foil and attached to a silver sample holder using GE varnish. All the measurements were carried out in the spin-rotated mode that the initial muon spins were rotated by 90° and lie in the ab-plane. The transverse fields (B = 0–3 T) were applied along the c-axis.

All obtained μSR spectra were analyzed with the software package MUSRFIT with GPU acceleration support^41,42,43,44. The weak transverse-field (wTF) μSR spectra were fitted with a sum of an exponentially decaying cosine and a simple exponential function, ${P}_{z}\left(t\right)=f\cos (2\pi {\nu }_{{{{{{\rm{s}}}}}}}t+{\phi }_{{{{{{\rm{s}}}}}}})\exp (-{\lambda }_{{{{{{\rm{s}}}}}}}t)+\left(1-f\right)\exp \left(-{\lambda }_{{{{{{\rm{f}}}}}}}t\right)$, where f is the slow relaxing fraction, ν_s is the muon spin precession frequency, ϕ_s is a phase, and λ_s (λ_f) is the muon spin-relaxation rate for the slow (fast) decaying component.

The zero-field (ZF) μSR data were well described by a sum of the Gaussian-broadened Gaussian (GbG) function with a simple exponential decay and a simple exponential function,

$${P}_{z}\left(t\right)=f{P}_{{{{{{\rm{GbG}}}}}}}\left(t{{{{{\rm{;}}}}}}\,{\Delta }_{0},\, W\right)\exp (-{\lambda }_{{{{{{\rm{s}}}}}}}t)+\left(1-f\right)\exp (-{\lambda }_{{{{{{\rm{f}}}}}}}t)$$

The GbG depolarization function is defined as a convolution of the Gaussian Kubo-Toyabe function, characterizing a broader field distribution than the Gaussian field distribution,

$${P}_{{{{{{\rm{GbG}}}}}}}\left(t\right)= a+\left(1-a\right){\left(\frac{1}{1+{R}^{2}{\Delta }_{0}^{2}{t}^{2}}\right)}^{3/2}\left(1-\frac{{\Delta }_{0}^{2}{t}^{2}}{1+{R}^{2}{\Delta }_{0}^{2}{t}^{2}}\right)\\ \exp \left[-\frac{{\Delta }_{0}^{2}{t}^{2}}{2(1+{R}^{2}{\Delta }_{0}^{2}{t}^{2})}\right].$$

Here, a is the tail fraction, 1-a is the damped relaxing fraction, Δ₀ is the mean value, W is the Gaussian width, and R (=W/Δ₀) is the relative Gaussian width of the Gaussian distribution, respectively. The GbG function well accounts for inhomogeneous static magnetic moments with short-range correlations^45,46,47,48. Note that the ZF-μSR results of the nonmagnetic Ir³⁺(J_eff = 0) substituted α-Ru_1−xIr_xCl₃ are also well described by the identical model, suggesting the similar effects of magnetic (Cr³⁺; S = 3/2) and nonmagnetic impurities on the Kitaev quantum spin system α-RuCl₃⁴⁸.

The longitudinal-field (LF) μSR data were fitted by a sum of the static and the dynamic Gaussian Kubo-Toyabe functions in longitudinal fields,

$${P}_{z}\left(t\right)=f{P}_{{{{{{\rm{SGKT}}}}}}}\left(t,\, {\Delta }_{{{{{{\rm{s}}}}}}},\, {B}_{{{{{{\rm{LF}}}}}}}\right)+\left(1-f\right){P}_{{{{{{\rm{DGKT}}}}}}}\left(t,\, {\Delta }_{{{{{{\rm{f}}}}}}},\, {\varGamma }_{{{{{{\rm{f}}}}}}},\, {B}_{{{{{{\rm{LF}}}}}}}\right),$$

where, P_SGKT (P_DGKT) are the dynamic (static) Gaussian Kubo-Toyabe function, Γ_f is the local field fluctuation rate, B_LF is the applied LF, and Δ_f (Δ_s) is the local-field width at the muon interstitial sites. The internal field is evaluated to be <B_loc> ~16.88 mT (Supplementary Fig. 16).

High transverse-field (hTF) μSR results were analyzed by the single histogram fit method. The positron histogram of the i-th detector N_i(t) is given by ${N}_{i}\left(t\right)={N}_{0,i}{e}^{-t/{\tau }_{\mu }}\left[1+{A}_{0,i}{P}_{i}\left(t\right)\right]+{N}_{{{{{{\rm{bkg}}}}}},i}$, where N_0,i is the total muon decay events at t = 0, τ_μ is the mean lifetime of the muon (~2.2 μs), A_0,i is the intrinsic asymmetry of the i-th detector, P_i(t) is the time-dependent muon spin polarization, and N_bkg,i is background events. We employed a sum of two Gaussian damped cosines for fittings, ${P}_{i}\left(t\right)=f\cos (2\pi {\nu }_{{{{{{\rm{s}}}}}}}t+{\phi }_{{{{{{\rm{s}}}}}}})\exp [-{{{{{{\rm{\lambda }}}}}}}_{{{{{{\rm{s}}}}}}}t]+\left(1-f\right)\cos \left(2\pi {\nu }_{{{{{{\rm{f}}}}}}}t+{\phi }_{{{{{{\rm{f}}}}}}}\right)\exp [-{{{{{{\rm{\lambda }}}}}}}_{f}t]$, where f is the relaxing fraction.

In general, to calculate the Knight shift, the narrow peak arising from the Ag sample holder is used as an internal reference. However, as shown in Fig. 4, the FFT spectra of α-Ru_1−xCr_xCl₃ (x = 0.04) display the overlap of the background and the intrinsic sample signals at slightly higher than the applied field B_ext. Therefore, we used the peak position of the sharp signal at T = 30 K that was obtained from the analysis as the reference field for evaluating the Knight shift.