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https://doi.org/10.1038/s42005-020-00448-5 OPEN Quantum-critical scale invariance in a transition metal alloy ✉ Yasuyuki Nakajima 1,2 , Tristin Metz2, Christopher Eckberg2, Kevin Kirshenbaum2, Alex Hughes2, Renxiong Wang2, Limin Wang2, Shanta R. Saha2, I-Lin Liu2,3,4, Nicholas P. Butch2,4, Daniel Campbell2, Yun Suk Eo2, David Graf 5, Zhonghao Liu6,7, Sergey V. Borisenko6, Peter Y. Zavalij8 & 1234567890():,; ✉ Johnpierre Paglione 2,9

Quantum-mechanical fluctuations between competing phases induce exotic collective exci- tations that exhibit anomalous behavior in transport and thermodynamic properties, and are often intimately linked to the appearance of unconventional Cooper pairing. High- temperature superconductivity, however, makes it difficult to assess the role of quantum- critical fluctuations in shaping anomalous finite-temperature physical properties. Here we report temperature-field scale invariance of non-Fermi liquid thermodynamic, transport, and

Hall quantities in a non-superconducting iron-pnictide, Ba(Fe1/3Co1/3Ni1/3)2As2, indicative of quantum criticality at zero temperature and applied magnetic field. Beyond a linear-in- temperature resistivity, the hallmark signature of strong quasiparticle scattering, we find a scattering rate that obeys a universal scaling relation between temperature and applied magnetic fields down to the lowest energy scales. Together with the dominance of hole-like carriers close to the zero-temperature and zero-field limits, the scale invariance, isotropic field response, and lack of applied pressure sensitivity suggests a unique quantum critical system unhindered by a pairing instability.

1 Department of Physics, University of Central Florida, Orlando, FL 32816, USA. 2 Maryland Quantum Materials Center, Department of Physics, University of Maryland, College Park, MD 20742, USA. 3 Chemical Physics Department, University of Maryland, College Park, MD 20742, USA. 4 NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA. 5 National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32310, USA. 6 IFW-Dresden, Helmholtzstraße 20, 01069 Dresden, Germany. 7 Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai 200050, China. 8 Department of Chemistry, University of Maryland, College Park, MD 20742, USA. ✉ 9 The Canadian Institute for Advanced Research, Toronto M5G 1Z8 ON, Canada. email: [email protected]; [email protected]

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on-Fermi liquid (NFL) behavior ubiquitously appears in ground state that persists to the T = 0 limit. Furthermore, this iron-based high-temperature superconductors with a behavior is strongly suppressed with magnetic field, which drives N ρ ∝ 2 novel type of superconducting pairing symmetry driven a recovery of Fermi liquid (FL) behavior (i.e., T ) at low by interband repulsion1, 2. The putative pairing mechanism is temperatures (Supplementary Note 1). thought to be associated with the temperature-doping phase Note that the unusual resistivity observed in Ba(Fe1/3Co1/3Ni1/ 24 diagram, bearing striking resemblance to cuprate and heavy- 3)2As2 cannot be ascribed to either Mooij correlations or fermion superconductors3, 4. In iron-based superconductors, the quantum interference25 due to randomness introduced by superconducting phase appears to be centered around the point counter-doping. Given that the Mooij correlations are dominant, of suppression of antiferromagnetic (AFM) and orthorhombic increasing randomness enhances the residual resistivity ρ0, structural order1. Close to the boundary between AFM order and accompanied by a gradual change in the slope of ρ(T) at high 25 superconductivity, the exotic metallic regime emerges in the temperatures, as observed in LuRh4B4 (ref. ). However, the “ ” normal state. The strange metallic behavior seems to be uni- overall slope of resistivity in Ba(Fe1/3Co1/3Ni1/3)2As2 is parallel versal in strongly correlated metals near a quantum critical point shifted from that in BaCo2As2 with a sizable increase of residual (QCP), characterized by linear-in-T resistivity5–8. The universal resistivity by ~30 μΩ cm, indicative of the absence of Mooij transport behavior is known as Planckian dissipation, where the correlation (Supplementary Note 2). Also, the quasi-T-linear ℏ τ = transport scattering rate is constrained by thermal energy, / P dependence of the resistivity at low temperatures in Ba(Fe1/3Co1/ kBT, where ℏ is the reduced Planck constant and kB is the 3Ni1/3)2As2 cannot be reproduced by the quantum corrections in Boltzmann constant. Lacking an intrinsic energy scale, the scale- conductivity caused by interference that provide the power law σ invariant transport in strange metals is one of the unresolved ~ Tp/2 (or ρ ~ T−p/2), where p = 3/2 (dirty limit), 3 phenomena in condensed matter physics, but its microscopic (electron–phonon scattering), or 1 (enhanced electron–electron origin has yet to be fully understood. In iron-based super- interaction)25. The absence of Mooij correlations and quantum conductors, along with the AFM order, the presence of an elec- interference allows us to treat scattering sources in charge tronic nematic phase above the structural transition complicates transport independently. As demonstrated by a smooth change in the understanding of the superconductivity and NFL behavior9– the temperature slope of resistivity at ~30 K (Supplementary 12. Moreover, the robust superconducting phase prohibits inves- Fig. 2), the inelastic scattering dominates over the tigations of zero-temperature limit normal state physical prop- electron–phonon scattering in the charge transport at low erties associated with the quantum critical (QC) instability due to temperatures. the extremely high upper critical fields. Mimicking the quasi-linear behavior in the temperature While AFM spin fluctuations are widely believed to provide the dependence of Δρ(T) = ρ(T) − ρ(0) at 0 T (Fig. 1a inset), the pairing glue in the iron pnictides, other magnetic interactions are magnetoresistance (MR) at 1.31 K ΔR(B)/R(0) varies sublinearly prevalent in closely related materials, such as the cobalt-based with applied field up to 35 T (Fig. 1b). The quasi-linear T and B oxypnictides LaCoOX (X = P, As)13, which exhibit ferromagnetic dependence allow us to introduce a new energy scale involving (FM) orders, and Co-based intermetallic arsenides with coexisting the scattering rate, the quadrature sum of temperature and – qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi FM and AFM spin correlations14 16. For instance, a strongly 2 2 magnetic field ΓðT; BÞ ðk TÞ þðημ BÞ , where μ is the enhanced Wilson ratio R of ~7–10 at 2 K (ref. 17) and violation B B B – W η of the Koringa law14 16 suggest proximity to a FM instability in Bohr magneton and is a dimensionless parameter. Here, we treat η as a fitting parameter rather than a value extracted from BaCo2As2. BaNi2As2, on the other hand, seems to be devoid of magnetic order18 and rather hosts other ordering instabilities in other measurements or microscopic theoretical calculations. η = fi both structure and charge19. Confirmed by extensive study, Fe, Setting 0.67, we nd the unusual scaling in the inelastic ℏ τ = ℏ 2 ρ − ρ * Co, and Ni have the same 2+ oxidation state in the tetragonal scattering rate, / ne ( (T, B) (0, 0))/m , where n is the carrier density extracted from low-temperature Hall coefficient ThCr2Si2 structure, thus adding one d electron (hole) contribu- * tion by Ni (Fe) substitution for Co in BaCo As (refs. 20–23), and measured at 0.5 T and m is the effective mass obtained from low- 2 2 fi thereby modifying the electronic structure subtly, but significantly temperature-speci c heat measured at 10 T in the present work, Γ enough to tune in and out of different ground states and corre- as a function of (T, B), collapsing onto one universal curve as lation types. Utilizing this balance, counter-doping a system to shown in Fig. 1c. This scaling is reminiscent of the observation in QC iron pnictide BaFe (As,P) (ref. 5). Although Ba(Fe Co achieve the same nominal d electron count as BaCo2As2 can 2 2 1/3 1/ realize a unique route to the same nearly FM system, while dis- 3Ni1/3)2As2 and BaFe2(As,P)2 share the similar scaling relation in rupting any specific spin correlation in the system. magnetotransport with each other, we note that while the scaling Γ Here, we utilize this approach to stabilize a novel ground state relation holds in the high region above ~3 meV in BaFe2(As, P) , it holds in the low Γ region below ~2 meV in Ba(Fe Co in the counter-doped nonsuperconducting iron pnictide Ba(Fe1/ 2 1/3 1/ Co Ni ) As , also nearly FM but with a unique type of spin 3Ni1/3)2As2, and that the extracted value of dimensionless 3 1/3 1/3 2 2 η γ α 5 5 fluctuation that leads to very strong quasiparticle scattering. We parameter ( / in ref. ) is different from that reported in ref. . Γ show that NFL behavior is prevalent in the very low-temperature The (T, B) scaling can closely be correlated to the Planckian charge transport and thermodynamic properties of Ba(Fe Co bound of dissipation. Quantum mechanics allows the shortest 1/3 1/ τ = ℏ Ni ) As , with temperature and magnetic energy scale invar- dissipative time scale P /kBT, constrained by the uncertainty 3 1/3 2 2 τ iance arising from a QC ground state. principle between dissipative time scale and energy dissipation τ ⋅ ≳ ℏ fi Γ E ~ kBT, kBT . Rede ning (T, B) as the dissipation energy scale in magnetic field, we can obtain the universal bound of ℏ τ Γ Γ Results dissipation, / P ~ (T, B). Our experimental observation in Non-Fermi liquid magnetotransport. The hallmark of NFL (T, B) scaling for the inelastic scattering gives a linear relation, ℏ/τ = AΓ(T, B) with A = 1.80, in good agreement with expected behavior in Ba(Fe1/3Co1/3Ni1/3)2As2 is clearly observed in the resistivity (Fig. 1a), which exhibits a quasi-linear T dependence behavior. over three orders of magnitude variation, from 20 K down to at Notwithstanding the quasi-two-dimensional layered structure, fi least 20 mK at B = 0 T. In this temperature range, we find no the NFL magnetotransport is independent of applied eld discernible anomaly associated with phase transitions down to 20 orientations with respect to the FeAs layers. We plot the Δρ ∥ Δρ ∥ mK, suggestive of the realization of an anomalous metallic anisotropy of the MR, (B c)/ (B ab), as a function of

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Fig. 1 Scale invariance in the resistivity of Ba(Fe1/3Co1/3Ni1/3)2As2.aTemperture dependence of resistivity for Ba(Fe1/3Co1/3Ni1/3)2As2 in the 46 configuration of B ∥ ab, B ⊥ I ∥ ab. Upper inset: crystal structure for Ba(Fe1/3Co1/3Ni1/3)2As2 (ref. ). Lower inset: Δρ(T) = ρ(T) − ρ(0)) as a function of T at B = 0 T for Ba(Fe1/3Co1/3Ni1/3)2As2. A dashed line is a guide to the eye to highlight quasi-linear-in-T dependence of resistivity. b Magnetic field dependence of ΔR(B)/R(0) ≡ (R(1.31 K, B) − R(1.31 K, 0))/R(1.31 K, 0) at T = 1.31 K. A red dashed line is a guide to the eye to highlight sublinear-in-B behavior of = ∥ ⊥ = ℏ τ magnetoresistance.qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Inset: angular dependence of magnetoresistance at T 550 mK and B( c I) 35 T. c Inelastic scattering rate / as a function of Γ ¼ ð Þ2 þðημ Þ2 η = kBT BB , with 0.67, suggestive of a universal scale invariance in the scattering mechanism in Ba(Fe1/3Co1/3Ni1/3)2As2. A blue sold line is a linear fit to data using ℏ/τ = AΓ with A = 1.8. d Temperature dependence of anisotropy of magnetoresistance between Δρ(B ∥ c) and Δρ(B ∥ ab ⊥ I; sample #1) and between Δρ(B ∥ c) and Δρ(B ∥ ab ∥ I; sample #2) at B = 14 T, showing lack of anisotropy in the scattering rate. temperature in Fig. 1d. The anisotropy between transverse MR in than logarithmic in the temperature dependence down to ~150 the out-of-plane field (B ∥ c, I ∥ ab) and transverse MR in the in- mK (Fig. 2b). Diminished with applying field, the NFL behavior plane field (B ∥ ab, B ⊥ I ∥ ab) decreases down to unity with observed in zero field completely disappears at applied field of 10 decreasing temperatures, suggesting the spatial dimension of the T, indicative of the recovery of FL (Supplementary Note 5). We system is three. The isotropy in MR remains even at 35 T, as note that the obtained specific heat coefficient γ = Ce/T at B = 0 shown in the angular dependence of MR (Fig. 1 inset). Due to the T, combined with the magnetic susceptibility χ, provides large three dimensionality, we observe similar Γ(T, B) scaling in the Wilson ratio R ¼ π2k2 χ=3μ2 γ ¼ 3:2atT = 1.8 K, suggestive of fi W B B resistivity regardless of applied eld orientations (Supplementary the presence of magnetic instabilities similar to BaCo2As2. Note 3). Moreover, the observed positive MR appears not to be The observation of FL recovery with magnetic field corrobo- * driven by the orbital effect due to the Lorentz force, but rather rates the presence of a new energy scale kBT , distinctive of ≫ μ ≪ μ associated with Zeeman energy-tuned scattering, evidenced by the crossover between the QC (kBT g BB) and FL (kBT g BB) isotropy in the MR between in-plane transverse (B ∥ c, I ∥ ab) and regimes. Intriguingly, this new energy scale allows a single scaling longitudinal (B ∥ I ∥ ab) configurations (Fig. 1d). function of T/B in the magnetization, written by,  Thermodynamic properties. In addition to resistivity, magnetic dM À1 T χ = À ¼ B 3f ; ð1Þ susceptibility M/B and electronic heat capacity Ce/T also dT M B exhibit canonical NFL behavior, i.e., diverging temperature dependence associated with QC instabilities26. The magnetic as shown in Fig. 2c. This scaling relation indeed reveals the susceptibility varies as χ ∝ T−1/3 at low temperatures <8 K (inset underlying free energy given by a universal function of T/B, of Fig. 2a), in contrast to the T-independent Pauli paramagnetic χ ¼ μ2 ð Þ  susceptibility P 2g BD EF (with electron g-factor and density ðdþzÞ=y T of states at the Fermi energy D(EF)) observed in FL metals, and ð ; Þ¼ b ; ð Þ F T B B f F z=y 2 observed upon increasing magnetic field to 7 T (Fig. 2a). A similar B b crossover is also observed in the heat capacity. Obtained form the subtraction of phonon (Cph) and nuclear Schottky contributions where d is the spatial dimensionality, z is the dynamic exponent, fi (CSch) from the total heat capacity (Ctot), the electronic speci c and yb is the scaling exponent related to the tuning parameter B fi = − − 27–30 heat coef cient Ce/T (Ctot Cph CSch)/T exhibits power (refs. ). Here, fF(x) is a universal function of x. Hence, the −0.25 law divergence, Ce/T ~ T (Supplementary Note 4), stronger magnetization can be derived from the derivative of the free

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Fig. 2 Non-Fermi liquid to Fermi liquid crossover and scale invariance in thermodynamic quantities. a Temperature dependence of magnetic 1/3 susceptibility for Ba(Fe1/3Co1/3Ni1/3)2As2 (B ∥ ab). Inset: 1/χ as a function of T at low temperatures. b Electronic specific heat Ce/T = (Ctot − Csch − Cph)/ −0.25 T for Ba(Fe1/3Co1/3Ni1/3)2As2 under several fields parallel to ab-plane. Inset: log–log plot for Ce/T vs T for B = 0 T. The dashed line emphasizes the T - power law behavior observed <1 K. c Temperature-magnetic field scale invariance in magnetization and d specific heat. The measured magnetization and −1/3 −1/3 specific heat collapse onto universal scaling curves in the forms of −dM/dT = B fM(T/B) and ΔCe/T = B gC(T/B), respectively, indicating the presence of the underlying free energy given by a universal function of T/B and the existence of the quantum critical point located at T = 0 and B = 0. Black lines represent scaling functions fM(T/B) for magnetization and gC(T/B) for specific heat derived from the underlying free energy, described in the main text. energy, Hall resistivity and electronic structure. The T/B scaling in   thermodynamics clearly discloses the presence of the QCP located dM d dF = À T fi À ¼À ¼ Bd yb 1f : ð3Þ exactly at zero eld and absolute zero, similar to the layered QC M z=y 31 32 dT dT dB B b metals YbAlB4 (ref. ) and YFe2Al10 (ref. ). More notably, the multiband nature in iron pnictides affixes the uniqueness of Directly comparing this with the observed QC scaling relation in quantum criticality for Ba(Fe1/3Co1/3Ni1/3)2As2. Dominated by Fig. 2c, we can extract the critical exponents in the free energy, ρ = − = − = = electron-like carriers, the Hall resistivity yx is negative and per- namely, z/yb 1 and d/yb 1 1/3, yielding z yb and d/z fectly linear in field at high temperatures (T = 20 K), as shown in 2/3. These values of the critical exponents describe the specific ρ Fig. 3a. Upon cooling, yx develops a nonlinearity with negative heat by using the same free energy, curvature. More prominent <1 K, the nonlinear Hall resistivity  fi ð ; Þ ∂2 À switches its sign at low elds <2 T. The sign change is more Ce B T F d z T ¼À ¼ yb : ð4Þ fi 2 B f C z=y readily observed in the temperature dependence of Hall coef - T ∂T B b fi ρ cient RH de ned by yx/B at low-T and low-H region (Fig. 3b), Rewriting the free energy, implying that hole-like carriers dominate the transport in the dþz þ z=y d z~ y =z vicinity of the QCP. The radial shape of the dominant carrier FðT; BÞ¼B yb f ðT=B b Þ¼T z f ðB=T b Þ,wefind fi fi  crossover in the eld-temperature phase diagram con rms the absence of an intrinsic energy scale in R (Fig. 3c), or in other ΔC ðT; BÞ C ðT; BÞ C ðT; 0Þ À1 T H e ¼ e À e ¼ 3 ; ð Þ B gC 5 words, the presence of scale invariance in the Hall effect tuned by T T T B fi temperature and magnetic eld. Similar to the resistivity, RH fi obeys Γ(T, B) scaling (Fig. 3d), consolidating the existence of scale where gC(x)is eld-dependent part of fC(x) (Supplementary Notes 6 and 7). As demonstrated in Fig. 2d, this expression invariance near the QCP in this system beyond any doubt. illustrates scale invariance in the specific heat that persists over Angle-resolved photoemission measurements identify a unique fi nearly three orders of magnitude in the scaling variable T/B. electronic structure and con rm the anomalous scattering rate

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Fig. 3 Sign change due to dominant hole-like carriers near the quantum critical point and scale invariance in Hall effect. a Hall resistivity ρyx as a function of B.Athightemperatures,ρyx is negative and linear in field. Upon cooling temperatures, ρyx becomes nonlinear and its sign switches to positive at low fields <2 * T. b Temperature dependence of Hall coefficient RH defined by ρyx/B. c T − B phase diagram with color plot of RH. Crossover temperatures T obtained from the 2 * quantum scaling in dM/dT and ΔCe/T and TFL from the T -fit are also plotted in the phase diagram.qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The error bars of T represent the standard deviation, and fi Γð ; Þ ð Þ2 þðημ Þ2 η = the error bars of TFL are estimated by changing the tting range. d RH as a function of T B kBT BB with 1. All the data collapse onto one universal curve, suggesting unusual scaling between temperature and applied field similar to that found in the longitudinal resistivity. correlated with Planckian dissipation. Unlike heavily electron-doped dirty FM quantum criticality with v = 1/2)27–29,orpredictionsfor fi BaCo2As2, the electronic structure for Ba(Fe1/3Co1/3Ni1/3)2As2 clean FM beyond mean eld, which predict the appearance of a weak consists of a large hole-like pocket and a cross-shaped electron-like first-order transition, with z = 3andν = 1/4 for d = 3andquantum Fermi surface around the Γ point, together with oval and elongated wing critical points with the same critical exponents, as the mean- electron pockets around the M points, exhibited by the Fermi surface field theory33–37. QC behavior in disordered 3d FM has been well = – – map (Fig. 4a), the band dispersion along kx 0direction(Fig.4b) at explained by the Belitz Kirkpatrick Vojta theory, predicting critical 30 K, and a schematic illustration (Fig. 4a, inset). The elongated exponents ν = 1andz = 3 for the asymptotic limit, and ν = 0.25 and electron pockets are very shallow, and the chemical potential is z = 6 for the preasymptotic limit37, 38, neither of which is in agree- located close to the bottom of the shallow bands. Dominating ment with our observation. Experimentally, previously measured fi ν transport at low temperatures and elds, the large hole-like pocket is exponentsinQCmaterials,suchasYbNi4(P1−xAsx)2 (FM QCP, z ~ fi 39 = ν = 40 β identi ed as the one responsible for QC behavior. Amazingly, the 5) ,CeCu6−xAux (AFM QCP, d/z 1/4, z 1) , -YbAlB4 = ν = 31 scattering rate (obtained from the dispersion of the hole-like bands at (mixed-valence meal, d/z 1/2, z 1) ,YFe2Al10 (layered QC = ν = 32 1 K) varies linearly with the kinetic energy up to 100 meV, consistent metal, d/z 1, z 0.59) ,andSr0.3Ca0.7RuO3 (disordered FM with Planckian dissipation as observed in the resistivity (Fig. 4c, d). QCP, z = 1.76)41 are also incompatible with the measured dynamical exponent. Discussion The high residual resistivity observed in Ba(Fe1/3Co1/3Ni1/3)2As2 While our primary observations of the scale invariance in the ther- evokes the possible realization of quantum Griffiths phase, where the modynamics are consistent with quantum criticality overall, they QC behavior is dominated by FM rare regions. The quantum Grif- fiths model predicts power law singularities in the magnetic sus- indicate a highly unusual critical behavior in Ba(Fe1/3Co1/3Ni1/3)2As2. λ− λ− While sharing an enhancement of the Wilson ratio with BaCo As ceptibility (χ ~ T 1), specificheat(C/T ~ T 1), and magnetization 2 2 λ fi λ indicative of a FM instability, the critical behavior in Ba(Fe1/3Co1/ (M ~ B ), determined by the nonuniversal Grif ths exponent that 42 3Ni1/3)2As2 is not described by any known theoretical predictions. takes 0 at the QCP, and increases with distance from criticality .In = the present system, however, λ = 2/3 extracted from the magnetic Assuming spacial dimensionality of three (d 3) based on the − observed isotropic response in MR and magnetization (Supplemen- susceptibility (χ ~ T 1/3;Fig.1a inset) disagrees with λ = 0.75 = = fi −0.25 tary Note 3), the observed critical exponents of d/z 2/3 and z yb obtained from the speci cheat(C/T ~ T ;Fig.2b inset), irre- = = fi yield z yb 4.5. concilable with the quantum Grif ths model. Besides, the critical The extracted dynamical exponents from our measurements do exponents in Ba(Fe1/3Co1/3Ni1/3)2As2 do not agree with those not match the predictions for either mean-field Hertz–Moriya–Millis obtained experimentally in other quantum Griffith systems37.For theory for d = 3(whichpredictz = 3forcleanFMandz = 4for instance, disordered weak ferromagnet Ni1−xVx show critical

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Fig. 4 Fermi surfaces and anomalous scattering rates in Ba(Fe1/3Co1/3Ni1/3)2As2.aAngle-resolved photoemission study of Fermi surface map for Ba (Fe1/3Co1/3Ni1/3)2As2, measured at 30 K. White lines denote the Brillouin zone (BZ) boundary, and white arrow corresponds to the cut shown in c. The inset depicts a schematic Fermi surface corresponding to the experimentally observed data, with shallow elongated electron pockets shown in gray. b

Energy cut along the kx = 0 direction, highlighting the elongated shallow electron-like pockets observed near the BZ corners. c Energy dispersion of hole- like pocket measured at 1 K near the BZ boundary along the cut indicated by white arrow in a, where sharp crossings of the Fermi level are found. The momentum axis originates at the crossing point. The white spectrum is the momentum distribution curve (MDC) at the Fermi level, with indicated width a representative measure of the scattering rate. d Scattering rate energy dependence obtained from the half width at half maximum (HWHM) of energy dispersion in panel c, with linear fit up to 100 meV. behavior dominated by quantum Griffiths singularities, χ ~ Tλ−1 and substitutional alloying is a key ingredient to tune the quantum M ~ Bλ,overawiderangeofvanadium concentration43, 44.Onthe criticalitythatmayprovidethekeytounderstandingthelackof λ = fl other hand, in Ba(Fe1/3Co1/3Ni1/3)2As2, 2/3 derived from the superconductivity driven by QC uctuations. magnetic susceptibility contradicts λ obtained from magnetization M ~ B0.75 (Supplementary Fig. 5), in conflict with the quantum Methods fi Grif ths phase. Sample preparation. The single crystals of Ba(Fe1/3Co1/3Ni1/3)2As2 were grown by Highly unusual dynamical critical behavior in this material TMAs (TM = Fe, Co, and Ni) self-flux method with the molar ratios of 3:4:4:4 = fl cannot be simply explained by existing FM QCP theories, but Ba:FeAs:CoAs:NiAs. Resulting crystals were cleaved out of the ux. The typical crystal size is 5 × 5 × 0.1 mm3. instead, it can be attributed to substitutional alloying by counter- doping. Indeed, the anomalous behavior observed in Ba(Fe Co 1/3 1/ Magnetotransport measurements. Magnetotransport measurements up to 15 T 3Ni1/3)2As2 is more prominent than that observed in both of the were conducted in a 3He–4He dilution refrigerator, and high magnetic field 7 fi end members of the 3d con guration line, namely, BaCo2As2 and transport measurements up to 35 T were performed at the National High Magnetic Ba(Fe,Ni)2As2 (Supplementary Notes 8 and 9), signifying that the Field Laboratory in Tallahassee. fi speci c 1/3 equal ratios of Fe:Co:Ni in BaCo2As2 are indeed important to stabilizing a unique QC ground state. In fact, as shown Heat capacity measurements. Heat capacity was measured using the thermal 3 –4 in Fig. 5, the observed NFL scattering behavior in Ba(Fe Co Ni relaxation method in a He He dilution refrigerator. A RuO2 thermometer on the 1/3 1/3 1/ calorimeter was calibrated in magnetic fields up to 15 T. 3)2As2 is completely robust against pressure and even replacement ofBaforSr(i.e.,inSr(Fe1/3Co1/3Ni1/3)2As2), implying either an fi Magnetic susceptibility measurements. Magnetic susceptibility was measured electronic structure modi cation beyond d electron tuning, or a using the vibrating sample magnetometer option in a 14 T Quantum Design significant role for transition metal site dilution. In fact, while DynaCool Physical Properties Measurement System and a 7 T SQUID Magnetic generally obscuring the critical behavior, high randomness due to Properties Measurement System. substitution indeed plays an important role in some QC materials, such as medium entropy alloys38, 45, in which similar NFL behavior Pressure measurements. A nonmagnetic piston–cylinder pressure cell was used has been realized38, 45. Together with the pressure insensitivity of for transport measurements under pressure up to 1.99 GPa, using a 1:1 ratio of n- pentane to 1-methyl-3-butanol as the pressure medium, and superconducting the T-linear scattering in Ba(Fe1/3Co1/3Ni1/3)2As2, our experimental temperature of lead as pressure gauge at base temperature. All transport mea- observations of scale invariance in this system indicates that surements were performed on the same Ba(Fe1/3Co1/3Ni1/3)2As2 crystal with 200

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Fig. 5 Robust non-Fermi liquid behavior against pressure. a Overall temperature dependence of resistivity for Ba(Fe1/3Co1/3Ni1/3)2As2 under applied pressure. The linear-T resistivity <20 K is robust against applying pressure up to 1.99 GPa as shown in the inset. b Normalized resistance R/R(300 K) vs T under pressure.

On applying pressure, R/R(300 K) for Ba(Fe1/3Co1/3Ni1/3)2As2 increases, approaching that for Sr(Fe1/3Co1/3Ni1/3)2As2 with smaller lattice constants than Ba(Fe1/ 3Co1/3Ni1/3)2As2 (Supplementary Notes 10 and 11, and Supplementary Table 1), indicative of robustness of linear-T behavior in the resistivity against pressure.

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34. Rech, J., Pépin, C. & Chubukov, A. V. Quantum critical behavior in itinerant Author contributions electron systems: Eliashberg theory and instability of a ferromagnetic Y.N., T.M., K.K., A.H., and R.W. performed the low-temperature transport and heat quantum critical point. Phys. Rev. B 74, 195126 (2006). capacity measurements, and analyzed the data. C.E. and S.R.S. grew and characterized 35. Conduit, G. J., Green, A. G. & Simons, B. D. Inhomogeneous phase formation single crystals. I.L. and N.B. conducted the pressure measurements, and D.C., Y.S.E., and on the border of itinerant ferromagnetism. Phys. Rev. Lett. 103, 207201 (2009). D.G. performed transport measurements at high magnetic fields. Z.L. and S.V.B. mea- 36. Kirkpatrick, T. R. & Belitz, D. Exponent relations at quantum phase sured the angle-resolved photoemission spectroscopy, and L.W. performed the theore- transitions with applications to metallic quantum ferromagnets. Phys. Rev. B tical support. P.Y.Z. performed the single-crystal x-ray diffraction. Y.N. and J.P. 91, 214407 (2015). conceived and designed the experiments, and all authors contributed to the editing of the 37. Brando, M., Belitz, D., Grosche, F. M. & Kirkpatrick, T. R. Metallic quantum manuscript. ferromagnets. Rev. Mod. Phys. 88, 025006 (2016). 38. Sales, B. C. et al. Quantum critical behavior in the asymptotic limit of high Competing interests disorder in the medium entropy alloy NiCoCr0.8. npj Quantum Mater. 2,33 (2017). The authors declare no competing interests. 39. Steppke, A. et al. Ferromagnetic quantum critical point in the heavy-fermion – metal YbNi4(P1−xAsx)2. Science 339, 933 936 (2013). Additional information 40. Schroeder, A. et al. Onset of antiferromagnetism in heavy-fermion metals. Supplementary information is available for this paper at https://doi.org/10.1038/s42005- Nature 407, 351 (2000). 020-00448-5. 41. Huang, C. L. et al. Anomalous quantum criticality in an itinerant ferromagnet. Nat. Commun. 6, 8188 (2015). Correspondence and requests for materials should be addressed to Y.N. or J.P. 42. Vojta, T. Quantum griffiths effects and smeared phase transitions in metals: theory and experiment. J. Low. Temp. Phys. 161, 299–323 (2010). Reprints and permission information is available at http://www.nature.com/reprints 43. Ubaid-Kassis, S., Vojta, T. & Schroeder, A. Quantum griffiths phase in the weak itinerant ferromagnetic alloy Ni1−xVx. Phys. Rev. Lett. 104, 066402 (2010). Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in fi 44. Wang, R. et al. Quantum grif ths phase inside the ferromagnetic phase of Ni1 published maps and institutional affiliations. −xVx. Phys. Rev. Lett. 118, 267202 (2017). 45. Sales, B. C. et al. Quantum critical behavior in a concentrated ternary solid solution. Sci. Rep. 6, 26179 (2016). Open Access This article is licensed under a Creative Commons 46. Momma, K. & Izumi, F. VESTA3 for three-dimensional visualization of Attribution 4.0 International License, which permits use, sharing, crystal, volumetric and morphology data. J. Appl. Crystallogr. 44, 1272–1276 (2011). adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless Acknowledgements indicated otherwise in a credit line to the material. If material is not included in the ’ Experimental research was supported by the National Science Foundation Division of article s Creative Commons license and your intended use is not permitted by statutory Materials Research Award No. DMR-1610349, and materials development supported by regulation or exceeds the permitted use, you will need to obtain permission directly from the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant no. the copyright holder. To view a copy of this license, visit http://creativecommons.org/ GBMF9071. A portion of this work was performed at the National High Magnetic Field licenses/by/4.0/. Laboratory, which is supported by National Science Foundation Cooperative Agreement No. DMR-1644779 and the State of Florida. Pressure measurements were supported © The Author(s) 2020 by NIST.

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