Quantum Fisher information in a strange metal

MainStrange metal behaviour refers to a linear temperature dependence of the electrical resistivity at low temperatures instead of the square-in-temperature Fermi liquid form. First recognized in cuprate high-temperature superconductors, strange metals are being identified in an increasing number of materials classes, from heavy-fermion, pnictide and organic compounds to frustrated-hopping and moiré flat-band systems1. Heavy-fermion compounds have played an important role in the search for other salient features of strange metallicity2, and a Fermi volume jump3, dynamical scaling in the spin response4 and charge (or current) response5, and the suppression of shot noise6 have been evidenced. All of them are consistent with the static Kondo screening transitioning, in the zero-temperature limit, from being in place to being absent7,8,9, a scenario that is actively pursued with various theoretical techniques10,11,12,13. As evidence for these features is accumulating in other strange metal platforms1 and theoretical efforts are made14,15,16,17 to understand these systems in Kondo-based frameworks, it is possible that the Kondo destruction (or breakdown) scenario is pertinent beyond the heavy-fermion setting. However, very different scenarios are also considered18,19,20 and a unified understanding is still lacking.Here we explore the potential of a quantum information-inspired probe—the quantum Fisher information (QFI)—to make progress. As recently shown theoretically21, the QFI can be defined for condensed-matter systems in thermal equilibrium via a Kubo response function$${f}_{{\rm{Q}}}(T)=\frac{4}{\pi }{\int }_{0}^{\infty }\tanh \left(\frac{\hslash \omega }{2{k}_{{\rm{B}}}T}\right){\chi }^{{\prime\prime} }(\omega ,T){\rm{d}}(\hslash \omega )$$ (1) involving the imaginary part of a dynamical susceptibility χ″(ω, T), for instance, the dynamical spin susceptibility that can be derived from inelastic neutron scattering (INS) experiments. In this formulation, called the QFI density22, the susceptibility is an intensive quantity, that is, it is counted per site or moment. The benefit of this tool is that it extracts the entanglement content of the quantum correlations contained in χ″(ω, T) and, thus, provides complementary information to dynamical scaling analyses. A prediction of direct pertinence for the present work is that at strongly entangled quantum phase transitions, the QFI is expected to diverge in the T = 0 limit, whereas no signature should appear at a thermal phase transition21. By contrast, in a spin-chain material, enhanced values of the QFI were found to be tied to the Néel order parameter and to decrease as the order is suppressed22. Motivated by the finding that at the Kondo destruction quantum critical point (QCP) of a Kondo impurity model, the entanglement entropy becomes long ranged23, we set out to study a strange metal heavy-fermion compound by INS experiments. In what follows, we make the case that the QFI due to local fluctuations associated with the Kondo destruction process, observed at a strange metal QCP, increases strongly with decreasing temperature as the strange metal develops, providing evidence for a state with enhanced multipartite entanglement (Supplementary Section B).We chose the heavy-fermion metal Ce3Pd20Si6 for this study because quantum criticality of the Kondo destruction type, associated with strange metal behaviour, has been identified in previous experiments24 and because large single crystals suitable for INS experiments are available25. We focus on the material’s magnetic-field-induced strange metal QCP at 1.73 T (applied along the crystallographic [0 0 1] direction), where a phase with antiferroquadrupolar (AFQ) order is continuously suppressed (Fig. 1a)26 and where the quadrupole moments are expected to undergo quantum critical fluctuations of the Kondo destruction type24. This means that the fluctuations are between bare quadrupole moments and Kondo-screened ones, a process that is predominantly local in real space and, thus, broad in momentum space. Although neutrons do not directly couple to electric quadrupoles, small secondary magnetic dipole moments can be induced by a magnetic field on top of the primary quadrupole moments, which then act as their magnetic markers27 (Supplementary Section E). A hallmark of this effect is the initial increase in the AFQ ordering temperature with field (Fig. 1a), before the phase collapses in the Kondo destruction transition28.Fig. 1: Ce3Pd20Si6 with orbital moments undergoing Kondo destruction.The alternative text for this image may have been generated using AI.Full size imagea, Temperature–magnetic field phase diagram of Ce3Pd20Si6, for a magnetic field applied along the crystallographic [0 0 1] direction. At the QCP (red star; BQ ≈ 1.73 T) studied here, the AFQ order is continuously suppressed24. The antiferromagnetic (AFM) order is suppressed already at BM ≈ 0.7 T (ref. 33). Both QCPs feature signatures of Kondo destruction24,33. Inset: sketch of the crystal structure, displaying only the 4f orbitals of the magnetically active Ce atoms at the 8c position, which assume a Γ8 quartet ground state. b, Constant-energy map at 50 mK, for the same field configuration as in a, obtained by integrating time-of-flight data within the indicated energy range, and within ±0.08 reciprocal lattice units (r.l.u.), that is, wavevectors in units of \(\frac{2\pi }{a}\), where a is the length of the unit cell in the orthogonal momentum direction26. The red cross, grey square and grey circles indicate the position studied with the triple-axis spectrometer ThALES, the direction of the neutron beam and the nuclear Bragg peaks, respectively (note that nuclear Bragg peaks in this structure exist only at all-even and all-odd Miller indices30). The double arrow indicates where the magnetic Bragg peak of the AFQ order would appear (at (1 1 1)). The grey-shaded lines from the red cross to the grey crosses indicate trajectories remeasured on ThALES (Supplementary Section C and Supplementary Fig. 2). c, Electrical resistivity follows the Fermi liquid form ρ = ρ0 + AT2 at the lowest temperatures, in shrinking temperature ranges on approaching the critical fields, and with a strongly enhanced A coefficient on approaching those fields, consistent with divergences for fields near BQ (inset). At both critical fields and in quantum critical fans emerging from them, the resistivity assumes the strange metal form \(\rho ={\rho }_{0}^{{\prime} }+{A}^{{\prime} }T\) (ref. 24). d, Differential Hall resistance jumps in the extrapolated zero-temperature limit24, both at BQ as shown here and at BM (ref. 33). This is understood as resulting from the spin degree of freedom σ of the Γ8 quartet being incorporated into the Fermi surface at BM, and the orbital degree of freedom τ being incorporated at BQ (refs. 24,33), via the Kondo destruction (or construction) mechanism. Panels adapted with permission from: a, ref. 26, APS; b,c, ref. 24, National Academy of Sciences.As we are interested in multipartite entanglement associated with the critical Kondo destruction process, we have selected the wavevector \((0\,\bar{1}\,0)\) for our study, which is far away from the AFQ ordering wavevector (1 1 1) identified by neutron diffraction via the markers26,28. This minimizes contributions from magnetic order parameter fluctuations—fluctuations between mutually aligned and unaligned moments, which are generally not considered a source of strange metallicity29 (Supplementary Section A). Furthermore, this choice avoids the contamination of the expected quasielastic quantum critical signal with magnetic Bragg or quasi-Bragg intensity, as well as structural Bragg intensity, as there is also no structural Bragg peak at this position in momentum space30. This is vital for data analysis (Supplementary Section C). The very presence of an appreciable intensity detected at this wavevector is remarkable and, by itself, a confirmation of the local nature of the underlying quantum criticality. Note that at fields away from the quantum critical field, the intensity at \((0\,\bar{1}\,0)\) as well as the broad intensity distribution as such are suppressed26 (Supplementary Section C and Supplementary Fig. 1).As shown previously24, a magnetic field applied along [0 0 1] drives the material across a two-stage Kondo destruction transition31,32. At large fields, both spin (σ) and orbital (τ) degrees of freedom of the 4f1 Γ8 quartet ground state of the magnetically active Ce atoms situated at the 8c site (Fig. 1a, inset) are Kondo screened by the conduction electrons (c), and the Fermi surface is large (Fig. 1d, brown circle). With a decreasing magnetic field, at BQ ≈ 1.73 T, Kondo screening first breaks up for the quadrupole moments, leading to a jump in the Fermi volume to an intermediate size (Fig. 1d, green circle) and AFQ order with the ordering wavevector (1 1 1)28. With further decreasing field, at BM ≈ 0.7 T, Kondo screening breaks up for the spin degree of freedom33, leading to a small Fermi surface that contains only the conduction electrons, and antiferromagnetic (AFM) order with the incommensurate ordering wavevector (0 0 0.8)28. Near both critical fields, the effective mass as probed by the A coefficient of the Fermi liquid form Δρ = AT2 is strongly enhanced (Fig. 1c) before, at the two QCPs, the strange metal linear-in-temperature form prevails down to the lowest temperatures24.We now turn to the INS data of Ce3Pd20Si6 measured down to temperatures of 60 mK. The experiment was performed at the cold-neutron triple-axis spectrometer ThALES (Institut Laue-Langevin), which is the state of the art in terms of the combination of high neutron flux and excellent energy resolution, that is, 0.035 meV (half-width at half-maximum) for the chosen final neutron wavevector kf. Extreme care was taken to remove all background contributions and to bring the data into absolute units (Supplementary Sections C and D). The thus-obtained dynamical spin correlation function S(q, ω, T), measured at \({\bf{q}}=(0\,\bar{1}\,0)\), is shown in Fig. 2a. The extended dynamical mean-field theory of Kondo destruction quantum criticality7 predicts the dynamical spin susceptibility χ(q, ω, T) at the ordering wavevector q = Q to exhibit the scaling form$$\chi ({\bf{q}},\omega ,T)=\frac{1}{A{T}^{\alpha }W(\hslash \omega /{k}_{{\rm{B}}}T)}.$$ (2) Here we measure the imaginary part χ″(q, ω, T) at \({\bf{q}}=(0\,\bar{1}\,0)\) which, as discussed above, is far away from any magnetic (and structural) Bragg peak. As χ″(q, ω, T) is expected to decrease (smoothly) away from Q (ref. 7), our data represent a lower bound of the quantum critical fluctuation strength (Supplementary Section A). Here χ″(q, ω, T) is related to S(q, ω, T) via the fluctuation–dissipation theorem34,35 as$${\chi }^{{\prime\prime} }({\bf{q}},\omega ,T)=\pi (1-{{\rm{e}}}^{-\hslash \omega /{k}_{{\rm{B}}}T})S({\bf{q}},\omega ,T).$$ (3) A minimization procedure of our S(q, ω, T) data for energies below 0.58 meV and temperatures below 5 K produces the best data collapse for the exponent α = 0.88 ± 0.02 (Fig. 2b). The excellent quality of the scaling, together with the fractional exponent α, provides strong evidence for the beyond-order-parameter nature of quantum criticality. The fact that the scaling form of equation (2) describes our data so well far away from the ordering wavevector of the AFQ phase indicates that contributions from order parameter fluctuations are small in Ce3Pd20Si6, at least on the scale of the energy resolution of the experiment. The broad intensity distribution in the reciprocal-space map (Fig. 1b) supports this assessment (Supplementary Section A).Fig. 2: Dynamical spin correlation function and dynamical scaling analysis of Ce3Pd20Si6.The alternative text for this image may have been generated using AI.Full size imagea, Selected isotherms of the dynamical spin correlation function S(q, ω) versus energy ℏω, measured at \({\bf{q}}=(0\,\bar{1}\,0)\) and in a magnetic field of 1.73 T applied along [0 0 1]. b, S(q, ω) from a, multiplied with \({({k}_{{\rm{B}}}T)}^{\alpha }\) and plotted versus ℏω/kBT. Data in the temperature range of 0.06–5 K and for energy transfers in the range of 0.025–0.58 meV show the best overlap for the exponent α = 0.88 ± 0.02, as seen from the minimum in the quality factor \({\chi }_{{\rm{reduced}}}^{2}\) of the minimization procedure (inset), a scaling that is compatible with Kondo destruction quantum criticality as described in ref. 7. The error bars result from statistical errors and uncertainties in other quantities via error propagation (Supplementary Section D).Source dataNext, we determine the temperature-dependent QFI density as$${f}_{{\rm{Q}}}(T)=4{\int }_{0}^{\infty }\tanh \left(\frac{\hslash \omega }{2{k}_{{\rm{B}}}T}\right)(1-{{\rm{e}}}^{-\hslash \omega /{k}_{{\rm{B}}}T})S(\omega ,T){\rm{d}}(\hslash \omega )$$ (4) from the different S(q, ω, T) isotherms at \({\bf{q}}=(0\,\bar{1}\,0)\). As our data extend only up to 1.5 meV, we terminate the integral at this energy (Fig. 3 and Supplementary Section I discuss integration range effects). The resulting fQ shows a pronounced increase with decreasing temperature, by almost a factor of 40 when cooling from 10 K to 60 mK (Fig. 3), indicating that entanglement is building up as the Kondo destruction QCP is approached. The temperature dependence is smooth, with no sign of a characteristic energy scale or saturation trend. Note that fluctuations from a classical phase transition freeze out below the ordering temperature as spectral weight accumulates near the ordering wavevector and ultimately becomes Bragg intensity. As the \(\tanh\) filter function in equation (1) suppresses contributions with ℏω m{({h}_{\max }-{h}_{\min })}^{2}\), where m is an integer21,36. Thus, the normalized QFI35$${\rm{nQFI}}=\frac{{f}_{{\rm{Q}}}}{{({h}_{\max }-{h}_{\min })}^{2}}$$ (5) witnesses at least (m + 1)-partite entanglement if nQFI > m. Previous work has focused on the case of localized spin-1/2 systems22,35, where \({({h}_{\max }-{h}_{\min })}^{2}=c{g}^{2}{[(+1/2)-(-1/2)]}^{2}=c{g}^{2}\). c counts the spin directions that are probed by S(q, ω, T) (refs. 22,37) and g is the Landé factor.As described above, at the QCP of Ce3Pd20Si6 we study here, local quantum critical fluctuations derive from the destruction of the Kondo screening of electric quadrupoles (Fig. 1)24, made visible to the neutrons through the secondary magnetic dipole moments \({\mu }_{\sec }\) induced by the applied magnetic field (along its direction) on top of the primary electric quadrupole moments. For Ce3Pd20Si6, with a Landé factor g = 1 (ref. 38), c = 1 because the secondary moments are B induced, and assuming that the size of the field-induced moment corresponds to a full Bohr magneton μB, we obtain nQFI = 8.2 ± 0.9. Any ratio \(r={\mu }_{\sec }/{\mu }_{{\rm{B}}} < 1\) will boost nQFI as nQFI/r2 (Supplementary Section E). As nQFI provides—by its very nature21—a lower bound for multipartite entanglement, the present results witness a state with at least 9-partite entanglement. Note that a reduction of r from 1 is only one of several reasons why this lower bound of multipartite entanglement is conservative (Supplementary Section I).We now describe auxiliary-field quantum Monte Carlo simulations (Supplementary Section F) of a Kondo destruction transition and compare them with our experimental results. As a (sign-problem-free) model, we use a spin-1/2 Heisenberg chain on a two-dimensional Dirac semimetal akin to graphene10. The exchange interaction among the local moments of the spin chain competes with the Kondo coupling JK of the local moments to the conduction electrons, which possess a pseudogap39. In the Kondo-screened phase at large JK, a new particle described by the composite fermion operator \({\widehat{\Psi }}_{{\bf{i}}}^{\dagger }\) (ref. 40) emerges. It carries the quantum numbers of the electron and participates in the Luttinger volume such that this state can be identified as the heavy-fermion phase with a large Fermi surface. In the Kondo destruction phase at low JK, the composite fermion spectral function is purely incoherent (Supplementary Section F and Supplementary Fig. 7). The transition between these two regimes is driven by charge degrees of freedom and a sudden change in the Luttinger volume count at zero temperature, which are clear signs of Kondo destruction physics.The differences between our model and material are obvious (Supplementary Section F), but what they share is the presence of a Kondo destruction QCP. This, together with the unbiased nature of the simulations, allows us to assess whether the enhanced QFI found in the experiment is a generic feature of Kondo destruction transitions.We define two different types of QFI: one for the (bosonic) spin fluctuations denoted as fQ, defined as in equation (1), and one for the composite fermions \({\widehat{\Psi }}_{{\bf{i}}}^{\dagger }\), denoted as \({f}_{{\rm{Q}}}^{\,\Psi }\). The latter is related to the single-particle spectral function A(q, ω) of the composite fermion via$${f}_{{\rm{Q}}}^{\Psi }(T)=2{\int }_{-\infty }^{\infty }{\tanh }^{2}\left(\frac{\hslash \omega }{2{k}_{{\rm{B}}}T}\right)A(q,\omega ){\rm{d}}(\hslash \omega ).$$ (6) Even though this quantity is not ideal for detecting multipartite entanglement—it is bounded for all wavevectors by a sum rule for ∫A(q, ω)dω at T = 0 (Supplementary Section F and ref. 40)—its temperature dependence provides valuable information. This is because the temperature corrections to the sum rule at the Fermi wavevector qF = π/2 depend on the nature of the spectral function.In the Kondo destruction phase (small JK), the spins are decoupled from the conduction electrons. In our specific model, fQ matches that of an isolated spin-1/2 chain, with critical AFM spin fluctuations at the wavevector q = π (ref. 34). The composite fermions are purely incoherent, leading to a small and only weakly temperature-dependent \({f}_{{\rm{Q}}}^{\,\Psi }\). In the Kondo-screened phase (large JK), fQ is suppressed and saturates at low temperatures. \({f}_{\rm{Q}}^{\,\Psi }\) near the Fermi wavevector qF shows T2 corrections to the sum rule as expected for a heavy Fermi liquid, and a T3 law away from it (Supplementary Section F and Supplementary Figs. 8 and 9 show details on both these phases).We now describe the characteristics of fQ and \({f}_{{\rm{Q}}}^{\,\Psi }\) at the critical value of JK. The value of fQ at the wavevector q = π increases strongly with decreasing temperature, without a characteristic scale (Fig. 4a), reaching values much enhanced compared with the Kondo-screened phase (though still limited by the finite system size of our simulations; Fig. 4a, shaded area). This reflects the destruction of the composite fermion operator and the concomitant liberated critical spin degrees of freedom. The scale-free increase in fQ and the large values reached at the lowest temperatures are common characteristics of our simulations and experiments, and we, thus, identify them as signatures of Kondo destruction quantum criticality. The saturation of fQ seen for wavevectors away from q = π is due to the fact that the Kondo destruction quantum criticality is not local in our model; this discrepancy with the experiment conforms to our expectation and further underpins the above result. Regarding the composite fermions, \({f}_{{\rm{Q}}}^{\,\Psi }\) at the Fermi wavevector qF increases strongly with decreasing temperature (Fig. 4b), and only saturates at the lowest temperature due to the femionic sum rule (Fig. 4b, inset). Outside the saturation region (Fig. 4b, shaded area), we find an approximately linear temperature dependence. This agrees with the theoretically expected linear-in-temperature correction from the smooth incoherent part of the Green’s function at the Kondo destruction transition, which dominates the spectral weight at qF (Supplementary Section F), and reflects the loss of quasiparticles evidenced by shot noise experiments6.Fig. 4: Quantum Monte Carlo simulations of the QFI at the Kondo destruction transition.The alternative text for this image may have been generated using AI.Full size imagea, QFI density fQ for the spin degree of freedom at the wavevector q = π of AFM fluctuations and at several wavevectors away from it. At q = π, fQ grows substantially in the low-temperature limit, but at other vectors, it converges to a finite value. The shaded region indicates where effects due to finite system size set in. The dashed curve is for a larger system (L = 22) and tends to saturate only at lower temperatures. As our fQ takes into account all three components of spin–spin correlations, we have \({({h}_{\max }-{h}_{\min })}^{2}=3{g}^{2}=12\) in equation (5), and hence, nQFI = fQ/12. b, QFI density \({f}_{{\rm{Q}}}^{\,\Psi }(T)\) for the composite fermion. Wavevectors closer to the Fermi surface qF = π/2 show the most pronounced temperature dependence, reflecting the spectral weight corresponding to the heavy-fermion quasiparticle. The shaded area has the same meaning as in a. Inset: wavevector-independent and weakly temperature-dependent contribution to \({f}_{{\rm{Q}}}^{\,\Psi }(T)\) from the sum rule (Supplementary Section F). The error bars (smaller than the symbol sizes) result from statistical errors of the Algorithms for Lattice Fermions library52 implementation of stochastic maximum entropy calculations53,54,55 used to obtain the QFI from imaginary-time data obtained in quantum Monte Carlo simulations.Source dataA few comments are due on the Kondo destruction side of the QCP. Here the spin degrees of freedom can form various phases of matter, both with and without long-range order2,41, but this does not change the behaviour at the QCP. For instance, changing the symmetry from SU(2) to SU(4) (antisymmetric self-adjoint representation) in our model calculations will frustrate the AFM interactions, but should not alter the overall behaviour of the QFI at criticality in both spin and single-particle channels. Within the extended dynamical mean-field theory description of Kondo destruction quantum criticality7, the QFI peaks at the QCP42. The rise of fQ(T) without a scale42 is similar to what we find here.In summary, the dynamical scaling of the INS data found here (Fig. 2b), together with the previously observed momentum space structure of the critical fluctuations (Fig. 1b)26 and transport (Fig. 1c,d) and thermodynamic characteristics24 provide evidence that Ce3Pd20Si6 exhibits a Kondo destruction QCP near 1.73 T. In the Kondo destruction scenario2,3,4,5,6,7,8,9 (Fig. 5), the strange metal state emerges from such a Kondo destruction QCP, which separates a Kondo destruction phase to its left (at small Kondo coupling) from a Kondo-screened phase to its right (at large Kondo coupling). Our quantum Monte Carlo simulations reveal that in the former (where the spins are decoupled from the conduction electrons), the QFI is model dependent. In the latter, for a Fermi liquid with quasiparticles composed of localized spins and conduction electrons, the entanglement depth measured by fQ is small. The question we have addressed is whether the destruction of the composite fermion at the QCP is accompanied by an enhancement in the entanglement depth. Our results, both experimental and numerical—for widely different realizations of the Kondo destruction transition (Supplementary Section F)—show that this is indeed the case. The coherent-to-incoherent transition of the composite fermion that underlies this transition, in which the spectral weight is pushed away from the Fermi energy, is picked up by the QFI. The enhanced multipartite entanglement it witnesses points to the microscopic basis of scale invariance5 and the loss of quasiparticles6 and, therefore, of linear-in-temperature strange metallicity: a collective action of multiple parties resulting from the quantum superposition of their wavefunctions appears to create this behaviour.Fig. 5: Visualization of enhanced multipartite entanglement in the Kondo destruction scenario.The alternative text for this image may have been generated using AI.Full size imageSchematic of the phase diagram of the temperature-tuning parameter, exhibiting a Kondo destruction QCP, which separates a Kondo destruction phase to its left from a Kondo-screened phase to its right, as addressed previously2,3,4,5,6,7,8,9. The red arrows represent spins (or pseudo-spins in the case of the orbital Kondo effect) of localized electrons, and blue shadings indicate the conduction electrons. The Kondo destruction phase to the left of the QCP has no static Kondo screening, and the spins typically (but not necessarily) are in order. Violet shadings indicate Kondo screening. In the Kondo-screened phase to the right of the QCP, the individual Kondo clouds (which may have more complex internal character than sketched) form the heavy quasiparticles. The present work provides evidence that in the strange metal realized in the quantum critical fan emanating from the QCP (delimited by the blue lines), a distinct quantum state with high multipartite entanglement forms.To the best of our knowledge, the pronounced scale-free increase in the QFI with decreasing temperature, as observed in our INS investigation of a stoichiometric heavy-fermion compound, points to the largest entanglement depth reported so far in any quantum material, including the proximate quantum spin liquid material KYbSe2 above its Néel temperature43. This is particularly noticeable because, unlike in KYbSe2, we probed the QFI away from any magnetic ordering wavevector, clearly evidencing the beyond-order-parameter nature of the underlying quantum criticality. Disorder effects can be ruled out as away from the QCP, the system behaves as a normal Fermi liquid. Whether high multipartite entanglement is a universal property of strange metals is a question of central importance, and we hope that our work will motivate QFI experiments across the different strange metal platforms1. If so, this would open a constructive avenue for scrutinizing the strange metal problem.Because the strength of quantum critical fluctuations increases steeply with decreasing temperature and energy, such experiments should be performed on spectrometers with the highest possible energy resolution at the lowest possible temperatures and be combined with careful background measurements (Supplementary Section G and Supplementary Fig. 10 show the case of CeCu5.9Au0.1 (ref. 4)). Methods based on inelastic X-ray scattering44, angle-resolved photoemission spectroscopy45 and electron energy-loss spectroscopy46 have recently been proposed as alternatives to probe the QFI. Apart from challenges to quantify their responses and, thus, determine the QFI in absolute units, their energy resolution and the lowest accessible temperatures are still orders of magnitude away from what can be achieved with state-of-the-art INS experiments; as such, these techniques are limited to systems with large intrinsic energy scales.We anticipate that our work will boost the dialogue with the quantum information science community. On one hand, to fully understand strange metals and related phenomena in other settings such as topological semimetals47, further experimentally accessible entanglement witnesses, for example, to detect the entanglement structure48,49, should be developed. On the other hand, genuine multipartite entanglement is necessary to reach the maximum sensitivity in certain metrological tasks36,50, making strange metals interesting material candidates.