## Abstract

The observation of quantum criticality in diverse classes of strongly correlated electron systems has been instrumental in establishing ordering principles, discovering new phases, and identifying the relevant degrees of freedom and interactions. At focus so far have been insulators and metals. Semimetals, which are of great current interest as candidate phases with nontrivial topology, are much less explored in experiments. Here, we study the Kondo semimetal CeRu_{4}Sn_{6} by magnetic susceptibility, specific heat, and inelastic neutron scattering experiments. The power-law divergence of the magnetic Grünesien ratio reveals that, unexpectedly, this compound is quantum critical without tuning. The dynamical energy over temperature scaling in the neutron response throughout the Brillouin zone and the temperature dependence of the static uniform susceptibility, indicate that temperature is the only energy scale in the criticality. Such behavior, which has been associated with Kondo destruction quantum criticality in metallic systems, could be generic in the semimetal setting.

## INTRODUCTION

Quantum criticality is observed in many strongly correlated materials classes, with quantum spin systems (*1*), high-*T*_{c} cuprate (*2*) and iron pnictide (*3*) superconductors, and heavy fermion metals (*4*–*6*) being prominent examples. Among them, the quantum critical insulators are the best understood. For instance, in the insulating quantum magnet LiHoF_{4} (*7*), the experimentally detected quantum criticality is well described in terms of the Landau framework, i.e., by the critical fluctuations of the magnetic order parameter (*8*, *9*). In quantum critical metals, by contrast, the underlying physics is much richer. In some systems, the Landau description works well (*10*, *11*), but in others, it appears to fail (*5*, *6*, *12*). Theoretically, the charge carriers introduce additional nontrivial couplings to the order parameter or its underlying building blocks, which opens new possibilities for quantum criticality (*13*–*15*).

Here, we explore the case of the Kondo semimetal CeRu_{4}Sn_{6}. Its noncentrosymmetric crystal structure (Fig. 1A), strong spin-orbit coupling associated with the three large atomic-number elements, and the fact that the anisotropy in its electronic dispersion (*16*) cannot be attributed to a nodal ground-state wave function of the Ce^{3+}4*f*^{1} electron (*17*) have led to speculations (*17*) that the material may be topologically nontrivial. Density functional theory (DFT) calculation within the local density approximation (LDA) and the Gutzwiller scheme predict the material to host Weyl nodes (*18*). If the nodal excitations persist in a full treatment of the Kondo effect, then CeRu_{4}Sn_{6} will be a Weyl-Kondo semimetal (*19*–*21*). Our comprehensive investigation of the magnetization, Grüneisen parameter, and inelastic neutron scattering as functions of temperature and magnetic field reveals that this material is quantum critical without tuning, behavior that is only rarely observed (*22*). This raises the exciting question of whether a Weyl-Kondo semimetal phase may indeed be located nearby, nucleating out of the quantum critical fluctuations.

## RESULTS AND DISCUSSION

The semimetallic character of CeRu_{4}Sn_{6} (see the Supplementary Materials for information on crystal growth and characterization) is evidenced by the weak temperature dependence of the electrical resistivity below about 30 K and the small low-temperature charge carrier concentration (Fig. 1B). We have also measured the temperature dependence of the specific heat *C _{p}* (Fig. 1C) and the magnetization

*M*, the latter with different magnetic fields

*H*applied along the two main crystallographic directions (perpendicular and parallel to

*c*; Fig. 1, D and E, respectively). The high-temperature anisotropy was recently shown to be due to single-ion crystal-field effects on the Ce

^{3+}4

*f*

^{1}electrons (

*23*). Deviations from this behavior below room temperature indicate a partial gapping of the electronic density of states and/or the onset of Kondo screening (

*23*). Below 10 K, a strong field dependence is observed. The temperature dependence of the magnetic susceptibility χ =

*M*/

*H*∣

_{10 mT}is not of simple Curie-Weiss type [χ =

*C*/(

*T*− Θ)] but, instead, is well described by

*c*, respectively (Fig. 1, D and E, insets). This is inconsistent with quantum criticality involving a Lorentzian fluctuation spectrum, where α = 1 (Curie-Weiss law) is expected. The application of larger magnetic fields gradually restores Fermi liquid behavior, i.e., a temperature-independent low-temperature magnetization (Fig. 1, D and E, main panels).

The magnetic Grüneisen ratio Γ_{mag} = −(∂*M*/*∂T*)/*C _{p}* is expected to diverge at any quantum critical point (

*24*), as observed in a number of quantum critical heavy fermion metals (

*25*,

*26*). For CeRu

_{4}Sn

_{6}, from the low-field (10 mT) magnetization and specific heat data presented above, we find Γ

_{mag}∼

*T*

^{−ϵ}between about 0.4 K and 4 K, with ϵ = 1.43 ± 0.07 and 1.62 ± 0.11 for fields perpendicular and parallel to the

*c*axis, respectively (Fig. 2, A and B), providing strong evidence for quantum criticality without tuning in CeRu

_{4}Sn

_{6}.

The field and temperature scaling of the magnetization data underpins this assignment. In Fig. 2 (C and D), we plot −*∂( M/H)/∂T* ·

*H*

^{β}versus

*T*/

*H*

^{γ}over more than two orders of magnitude in

*T*/

*H*

^{γ}, for fields between 10 mT and 0.5 T both perpendicular and parallel to

*c*. We find a good data collapse with the exponents β being equal to α extracted from Fig. 1 (D and E) (insets) and γ = 0.35 ± 0.02 and 0.43 ± 0.02 for fields perpendicular and parallel to

*c*, respectively. The fitted exponents are internally consistent, as they satisfy a scaling relationship (see the Supplementary Materials). This kind of critical scaling, with a fractional exponent α < 1 (

*13*), indicates that the system is at, or very close to, a beyond-Landau quantum critical point and that magnetic field acts as a tuning parameter, with criticality at

*H*= 0.

This is corroborated by our neutron scattering investigation presented next. The inelastic neutron scattering intensity is dominated by features broad in momentum space, with no apparent energy scale, setting the lower bound for a gap in the spin excitations to less than 0.1 meV. In Fig. 3 (A to C), we show the intensity distribution at 1.5 K, integrated from 0.2 meV to 1.2 meV, in the (H0L), (HK0), and (HHL) planes, respectively. The energy profile along two high-symmetry directions, (00L) and (H00), integrated over ±0.2 reciprocal lattice units (r.l.u.) perpendicular to these directions, is presented in Fig. 3 (D and E). The strong peaks at H + K + L = even are tails of nonmagnetic elastic Bragg scattering. There is no apparent change in the momentum dependence with energy transfer, meaning that the **q** and ω dependence of the scattering cross section factorizes and there is no dispersion. This is clearly at odds with any description of the criticality in terms of the fluctuations of an incipient symmetry-breaking order parameter, where the scattering intensity should be increasingly peaked at the critical wave vector. The broadness of the features is especially pronounced along ^{3+}4*f*^{1} electrons. Scattering that is momentum independent along lines in the Brillouin zone is a sign of local (or Kondo destruction) quantum criticality (*13*) and has also been observed in quantum critical CeCu_{5.9}Au_{0.1} (*12*). The wave vector dependence of the scattering intensity away from these high-intensity lines is discussed below.

First, however, we investigate whether quantum critical scaling, as evidenced by the static uniform magnetic susceptibility χ = (*∂M*/*∂H*)_{H→0} = χ′(**q** = 0, ω = 0, *T*) (Fig. 1, D and E, insets), is also found in the dynamical spin susceptibility χ(**q**, ω, *T*). This would give rise to the form*f*(**q**) is an offset and α = 1 is expected for a Lorentzian fluctuation spectrum as prescribed by the Landau order parameter description (*8*, *9*), and a fractional exponent α < 1 points to its failure. The dynamic structure factor S(**q**, ω, *T*) measured by inelastic neutron scattering is related to the imaginary part χ′′ of χ(**q**, ω, *T*) by χ′′ = S · [1 − exp (− ℏω/*k*_{B}*T*)], where the second factor manifests the fluctuation-dissipation relation. Near the peak wave vector, where the singular susceptibility corresponds to a vanishing *f*(**q**), it takes the form*g* is a function that depends only on the ratio of ℏω and *k*_{B}*T*. We have determined S(ω, *T*) by integrating the neutron scattering intensity for momentum transfers **q** about the (100) wave vector (where the quasielastic intensity is maximal; see Fig. 3, A and B), and plot it for temperatures between 0.1 K and 10 K, and energy transfers between 0.1 meV and 1.3 meV as S · *T*^{α} versus ℏω/*k*_{B}*T* in Fig. 4A. The best data collapse is found for α = 0.6 ± 0.1 (inset of Fig. 4A). This is consistent with the α values determined from the static uniform magnetic susceptibility (α = 0.50 ± 0.01 for *H* ⊥ *c* and α = 0.78 ± 0.03 for *H* ∥ *c*), in particular, in view of the fact that the neutron scattering intensity is a directional average, taken around the (100) wave vector, where spin fluctuations polarized along and perpendicular to *c* contribute in equal measures. Thus, also the inelastic neutron scattering data evidence quantum criticality beyond the long-wavelength fluctuations of an order parameter (*8*, *9*), such as expected in the theory of Kondo destruction quantum criticality (*13*).

Last, we turn to the spatial profile of the critical fluctuations. To do so, we have fitted the experimental scattering intensity, corrected for the Ce form factor, in terms of Eq. 2. The resulting *f*(**q**) in the (H0L) and (HK0) planes are shown in Fig. 4 (B and C, respectively). For the rather featureless *f*(**q**) (see Fig. 4C). Likewise, in the (H0L) plane, it is seen that *f*(**q**) is vanishingly small for H + 0 + L = odd, where the dynamical spin susceptibility is also peaked. Along directions that move away from these ridges, the decreasing dynamical spin susceptibility corresponds to an increasing *f*(**q**). We can understand these features of *f*(**q**) in terms of the wave vector dependence of the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction

Here, *J*_{K} is the Kondo coupling strength and *L*(**q**, ω) the Lindhard function of the (uncorrelated) conduction electrons, which we have determined through LDA-DFT calculations, with one *4f* electron per Ce^{3+} ion placed in the ionic core. This “*f*-core” band structure, resembling that of LaRu_{4}Sn_{6}, characterizes the electronic structure of the *spd* conduction electrons that mediate the RKKY interaction between the 4*f* moments. The associated Fermi surface is shown in Fig. 3F, from which we can identify nesting wave vectors that correspond to the *M*) and between *X* points extending along (00L). At these wave vectors, collectively denoted as **Q**, the RKKY interaction is maximally antiferromagnetic. The Weiss temperature scale*f*(**q**), when the latter is raised to the power 1/α. In other words, the neutron scattering intensity is maximal where the Weiss temperature Θ(**q**) has a minimum, implying that the RKKY interaction defines the **q** space structure. Accordingly, the system may be near antiferromagnetic order mediated by the RKKY interaction, with an ordering wave vector at H + K + L = odd.

Beyond-Landau quantum criticality, as indicated by a fractional exponent α < 1, has been documented in a handful of other heavy fermion compounds (*12*, *22*, *27*–*29*). Except for the case of the heavy fermion metal CeCu_{5.9}Au_{0.1}, a determination of α from inelastic neutron scattering—as provided here—has, however, remained elusive. Our discovery of beyond-Landau quantum criticality, in both the inelastic neutron response and the static uniform magnetic susceptibility, in a genuinely quantum critical semimetal implicates the generality of the phenomenon. Because this constitutes the first observation of this phenomenon in a semimetal, it will be important to explore whether beyond-Landau quantum criticality is inherent to systems with reduced charge carrier concentration. Promising materials to explore in this context are the low-carrier density Kondo systems CeRhBi (*30*), CeNi_{2−δ}As_{2} (*31*), and Yb_{3}Ir_{4}Ge_{13} (*32*), as well as the pyrochlore iridate Pr_{2}Ir_{2}O_{7} (*33*, *34*), which all show thermodynamic signatures of quantum criticality.

In heavy fermion systems, quantum criticality is generally observed at the *T* = 0 collapse of (antiferromagnetically) ordered phases. CeRu_{4}Sn_{6}, however, is quantum critical without tuning pressure, stoichiometry, or applied fields to a phase boundary. A natural question then arises: Is an antiferromagnetic phase nearby, as suggested by the minima of the Weiss temperature near possible antiferromagnetic ordering wave vectors, and can it be reached by tuning, for instance with pressure? And even more excitingly, do the quantum critical fluctuations give rise to emergent phases, maybe unconventional superconductivity in analogy to β-YbAlB_{4} (*35*) or, in view of all the necessary conditions being fulfilled, even a Weyl-Kondo semimetal phase akin to that recently discovered in Ce_{3}Bi_{4}Pd_{3} (*19*–*21*)? This will require further experiments at lower temperatures and as function of tuning parameters, which we hope this work will stimulate.

## MATERIALS AND METHODS

For our study, single crystals of CeRu_{4}Sn_{6} were grown from self-flux, using the floating zone melting technique with optical heating as reported previously (*36*) (see the Supplementary Materials for further details). The magnetic properties between 2 K and 300 K were measured in a SQUID magnetometer from Cryogenic Ltd., and the data between 0.3 K and 2 K used a ^{3}He insert. High-temperature magnetization measurements up to 1000 K were performed with a vibrating sample magnetometer in a Physical Properties Measurement System (PPMS) from Quantum Design. The ^{3}He PPMS option was used for specific heat measurements in the temperature range from 0.3 K to 20 K.

Neutron scattering was performed with the Multi Axis Crystal Spectrometer (MACS) at the National Institute of Standards and Technology Center for Neutron Research. For the momentum-space mapping, we co-aligned 3 g of single crystals and used incident energies from 3.7 meV to 5 meV and a fixed final energy of 3.7 meV, allowing for the use of Be and BeO filters before and after the sample, respectively, and providing an effective resolution better than 0.05 Å^{−1} and 0.15 meV. For the ω/*T* scaling measurements, 2 g of single crystals were accommodated within the restrictions of a dilution refrigerator. We used a fixed final energy of 2.5 meV and incident energies up to 3.7 meV. A magnetic field of 10 mT was applied along (0K0) to avoid superconductivity in the aluminum sample mount and maintain good thermal contact.

## SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/7/21/eabf9134/DC1

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is **not** for commercial advantage and provided the original work is properly cited.

## REFERENCES AND NOTES

**Acknowledgments:**W.T.F. is grateful to P. Nikolic for fruitful discussions.

**Funding:**The work at IQM was supported as part of the Institute for Quantum Matter, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under award no. DE-SC0019331. C.L.B. was supported by the Gordon and Betty Moore Foundation through GBMF9456. S.P. and A.S. acknowledge financial support from the European Community (H2020 project no. 824109) and from the Austrian Science Fund (FWF projects P29296-N27, 29279-N27, and W1243). Q.S. was supported by the NSF (grant no. DMR-1920740) and the Robert A. Welch Foundation (grant no. C-1411). Q.S. and S.P. acknowledge the hospitality of the Aspen Center for Physics, which is supported by the NSF (grant no. PHY-1607611). W.T.F. is grateful to the ARCS Foundation, Lockheed Martin, KPMG, and the Schmidt Science Fellows program in partnership with the Rhodes Trust for the partial support of this work.

**Author contributions:**H.W. and A.P. synthesized and characterized the material. A.S., J.H., and H.W. performed the thermodynamic and transport experiments under the supervision of S.P. W.T.F., J.H., J.A.R.-R., and Y.Q. performed the neutron scattering experiments under the supervision of C.L.B., and P.B. performed the DFT calculations. W.T.F., A.S., Q.S., C.L.B., and S.P. analyzed the data and interpreted the results. W.T.F., A.S., and S.P. wrote the manuscript, with contributions from all authors.

**Competing interests:**The authors declare that they have no competing interests.

**Data and materials availability:**All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

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