# Quantum-critical scale invariance in a transition metal alloy

## Abstract

Quantum-mechanical fluctuations between competing phases induce exotic collective excitations 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.

## Introduction

Non-Fermi liquid (NFL) behavior ubiquitously appears in iron-based high-temperature superconductors with a novel type of superconducting pairing symmetry driven by interband repulsion1, 2. The putative pairing mechanism is thought to be associated with the temperature-doping phase diagram, bearing striking resemblance to cuprate and heavy-fermion superconductors3, 4. In iron-based superconductors, the superconducting phase appears to be centered around the point of suppression of antiferromagnetic (AFM) and orthorhombic structural order1. Close to the boundary between AFM order and superconductivity, the exotic metallic regime emerges in the normal state. The “strange” metallic behavior seems to be universal in strongly correlated metals near a quantum critical point (QCP), characterized by linear-in-T resistivity5,6,7,8. The universal transport behavior is known as Planckian dissipation, where the transport scattering rate is constrained by thermal energy, /τP = kBT, where is the reduced Planck constant and kB is the Boltzmann constant. Lacking an intrinsic energy scale, the scale-invariant transport in strange metals is one of the unresolved phenomena in condensed matter physics, but its microscopic origin has yet to be fully understood. In iron-based superconductors, along with the AFM order, the presence of an electronic nematic phase above the structural transition complicates the understanding of the superconductivity and NFL behavior9,10,11,12. Moreover, the robust superconducting phase prohibits investigations of zero-temperature limit normal state physical properties associated with the quantum critical (QC) instability due to the extremely high upper critical fields.

While AFM spin fluctuations are widely believed to provide the pairing glue in the iron pnictides, other magnetic interactions are prevalent in closely related materials, such as the cobalt-based oxypnictides LaCoOX (X = P, As)13, which exhibit ferromagnetic (FM) orders, and Co-based intermetallic arsenides with coexisting FM and AFM spin correlations14,15,16. For instance, a strongly enhanced Wilson ratio RW of  ~7–10 at 2 K (ref. 17) and violation of the Koringa law14,15,16 suggest proximity to a FM instability in BaCo2As2. BaNi2As2, on the other hand, seems to be devoid of magnetic order18 and rather hosts other ordering instabilities in both structure and charge19. Confirmed by extensive study, Fe, Co, and Ni have the same 2+  oxidation state in the tetragonal ThCr2Si2 structure, thus adding one d electron (hole) contribution by Ni (Fe) substitution for Co in BaCo2As2 (refs. 20,21,22,23), and thereby modifying the electronic structure subtly, but significantly enough to tune in and out of different ground states and correlation types. Utilizing this balance, counter-doping a system to achieve the same nominal d electron count as BaCo2As2 can realize a unique route to the same nearly FM system, while disrupting any specific spin correlation in the system.

Here, we utilize this approach to stabilize a novel ground state in the counter-doped nonsuperconducting iron pnictide Ba(Fe1/3Co1/3Ni1/3)2As2, also nearly FM but with a unique type of spin fluctuation that leads to very strong quasiparticle scattering. We show that NFL behavior is prevalent in the very low-temperature charge transport and thermodynamic properties of Ba(Fe1/3Co1/3Ni1/3)2As2, with temperature and magnetic energy scale invariance arising from a QC ground state.

## Results

### Non-Fermi liquid magnetotransport

The hallmark of NFL behavior in Ba(Fe1/3Co1/3Ni1/3)2As2 is clearly observed in the resistivity (Fig. 1a), which exhibits a quasi-linear T dependence over three orders of magnitude variation, from 20 K down to at least 20 mK at B = 0 T. In this temperature range, we find no discernible anomaly associated with phase transitions down to 20 mK, suggestive of the realization of an anomalous metallic ground state that persists to the T = 0 limit. Furthermore, this behavior is strongly suppressed with magnetic field, which drives a recovery of Fermi liquid (FL) behavior (i.e., ρT2) at low temperatures (Supplementary Note 1).

Note that the unusual resistivity observed in Ba(Fe1/3Co1/3Ni1/3)2As2 cannot be ascribed to either Mooij correlations24 or quantum interference25 due to randomness introduced by counter-doping. Given that the Mooij correlations are dominant, increasing randomness enhances the residual resistivity ρ0, accompanied by a gradual change in the slope of ρ(T) at high temperatures, as observed in LuRh4B4 (ref. 25). However, the overall slope of resistivity in Ba(Fe1/3Co1/3Ni1/3)2As2 is parallel shifted from that in BaCo2As2 with a sizable increase of residual resistivity by  ~30 μΩ cm, indicative of the absence of Mooij correlation (Supplementary Note 2). Also, the quasi-T-linear dependence of the resistivity at low temperatures in Ba(Fe1/3Co1/3Ni1/3)2As2 cannot be reproduced by the quantum corrections in conductivity caused by interference that provide the power law σ ~ Tp/2 (or ρ ~ Tp/2), where p = 3/2 (dirty limit), 3 (electron–phonon scattering), or 1 (enhanced electron–electron interaction)25. The absence of Mooij correlations and quantum interference allows us to treat scattering sources in charge transport independently. As demonstrated by a smooth change in the temperature slope of resistivity at  ~30 K (Supplementary Fig. 2), the inelastic scattering dominates over the electron–phonon scattering in the charge transport at low temperatures.

Mimicking the quasi-linear behavior in the temperature dependence of Δρ(T) = ρ(T) − ρ(0) at 0 T (Fig. 1a inset), the magnetoresistance (MR) at 1.31 K ΔR(B)/R(0) varies sublinearly with applied field up to 35 T (Fig. 1b). The quasi-linear T and B dependence allow us to introduce a new energy scale involving the scattering rate, the quadrature sum of temperature and magnetic field $$\Gamma (T,B)\equiv \sqrt{{({k}_{{\rm{B}}}T)}^{2}+{(\eta {\mu }_{{\rm{B}}}B)}^{2}}$$, where μB is the Bohr magneton and η is a dimensionless parameter. Here, we treat η as a fitting parameter rather than a value extracted from other measurements or microscopic theoretical calculations. Setting η = 0.67, we find the unusual scaling in the inelastic scattering rate, /τ = ne2(ρ(TB) − ρ(0, 0))/m*, where n is the carrier density extracted from low-temperature Hall coefficient measured at 0.5 T and m* is the effective mass obtained from low-temperature-specific heat measured at 10 T in the present work, as a function of Γ(TB), collapsing onto one universal curve as shown in Fig. 1c. This scaling is reminiscent of the observation in QC iron pnictide BaFe2(As,P)2 (ref. 5). Although Ba(Fe1/3Co1/3Ni1/3)2As2 and BaFe2(As,P)2 share the similar scaling relation in magnetotransport with each other, we note that while the scaling relation holds in the high Γ region above  ~3 meV in BaFe2(As,P)2, it holds in the low Γ region below  ~2 meV in Ba(Fe1/3Co1/3Ni1/3)2As2, and that the extracted value of dimensionless parameter η (γ/α in ref. 5) is different from that reported in ref. 5.

The Γ(TB) scaling can closely be correlated to the Planckian bound of dissipation. Quantum mechanics allows the shortest dissipative time scale τP = /kBT, constrained by the uncertainty principle between dissipative time scale τ and energy dissipation E ~ kBT, τ kBT. Redefining Γ(TB) as the dissipation energy scale in magnetic field, we can obtain the universal bound of dissipation, /τP ~ Γ(TB). Our experimental observation in Γ(TB) scaling for the inelastic scattering gives a linear relation, /τ = AΓ(TB) with A = 1.80, in good agreement with expected behavior.

Notwithstanding the quasi-two-dimensional layered structure, the NFL magnetotransport is independent of applied field orientations with respect to the FeAs layers. We plot the anisotropy of the MR, Δρ(Bc)/Δρ(Bab), as a function of temperature in Fig. 1d. The anisotropy between transverse MR in the out-of-plane field (BcIab) and transverse MR in the in-plane field (BabBIab) decreases down to unity with decreasing temperatures, suggesting the spatial dimension of the system is three. The isotropy in MR remains even at 35 T, as shown in the angular dependence of MR (Fig. 1 inset). Due to the three dimensionality, we observe similar Γ(TB) scaling in the resistivity regardless of applied field orientations (Supplementary Note 3). Moreover, the observed positive MR appears not to be driven by the orbital effect due to the Lorentz force, but rather associated with Zeeman energy-tuned scattering, evidenced by the isotropy in the MR between in-plane transverse (BcIab) and longitudinal (BIab) configurations (Fig. 1d).

### Thermodynamic properties

In addition to resistivity, magnetic susceptibility χ  =  M/B and electronic heat capacity Ce/T also exhibit canonical NFL behavior, i.e., diverging temperature dependence associated with QC instabilities26. The magnetic susceptibility varies as χT−1/3 at low temperatures <8 K (inset of Fig. 2a), in contrast to the T-independent Pauli paramagnetic susceptibility $${\chi }_{{\rm{P}}}=2g{\mu }_{{\rm{B}}}^{2}D({E}_{{\rm{F}}})$$ (with electron g-factor and density of states at the Fermi energy D(EF)) observed in FL metals, and observed upon increasing magnetic field to 7 T (Fig. 2a). A similar crossover is also observed in the heat capacity. Obtained form the subtraction of phonon (Cph) and nuclear Schottky contributions (CSch) from the total heat capacity (Ctot), the electronic specific heat coefficient Ce/T  =  (Ctot  −  Cph −  CSch)/T exhibits power law divergence, Ce/T  ~  T−0.25 (Supplementary Note 4), stronger than logarithmic in the temperature dependence down to  ~150 mK (Fig. 2b). Diminished with applying field, the NFL behavior observed in zero field completely disappears at applied field of 10 T, indicative of the recovery of FL (Supplementary Note 5). We note that the obtained specific heat coefficient γ = Ce/T at B = 0 T, combined with the magnetic susceptibility χ, provides large Wilson ratio $${R}_{{\rm{W}}}={\pi }^{2}{k}_{{\rm{B}}}^{2}\chi /3{\mu }_{{\rm{B}}}^{2}\gamma =3.2$$ at T = 1.8 K, suggestive of the presence of magnetic instabilities similar to BaCo2As2.

The observation of FL recovery with magnetic field corroborates the presence of a new energy scale kBT*, distinctive of crossover between the QC (kBTgμBB) and FL (kBTgμBB) regimes. Intriguingly, this new energy scale allows a single scaling function of T/B in the magnetization, written by,

$$-\frac{dM}{dT}={B}^{-\frac{1}{3}}{f}_{M}\left(\frac{T}{B}\right),$$
(1)

as shown in Fig. 2c. This scaling relation indeed reveals the underlying free energy given by a universal function of T/B,

$$F(T,B)={B}^{(d+z)/{y}_{b}}{f}_{F}\left(\frac{T}{{B}^{z/{y}_{b}}}\right),$$
(2)

where d is the spatial dimensionality, z is the dynamic exponent, and yb is the scaling exponent related to the tuning parameter B (refs. 27,28,29,30). Here, fF(x) is a universal function of x. Hence, the magnetization can be derived from the derivative of the free energy,

$$-\frac{dM}{dT}=-\frac{d}{dT}\left(\frac{dF}{dB}\right)={B}^{d/{y}_{b}-1}{f}_{M}\left(\frac{T}{{B}^{z/{y}_{b}}}\right).$$
(3)

Directly comparing this with the observed QC scaling relation in Fig. 2c, we can extract the critical exponents in the free energy, namely, z/yb = 1 and d/yb − 1 = −1/3, yielding z = yb and d/z = 2/3. These values of the critical exponents describe the specific heat by using the same free energy,

$$\frac{{C}_{{\rm{e}}}(B,T)}{T}=-\frac{{\partial }^{2}F}{\partial {T}^{2}}={B}^{\frac{d-z}{{y}_{b}}}{f}_{C}\left(\frac{T}{{B}^{z/{y}_{b}}}\right).$$
(4)

Rewriting the free energy, $$F(T,B)={B}^{\frac{d+z}{{y}_{b}}}f(T/{B}^{z/{y}_{b}})={T}^{\frac{d+z}{z}}\tilde{f}(B/{T}^{{y}_{b}/z})$$, we find

$$\frac{\Delta {C}_{{\rm{e}}}(T,B)}{T}=\frac{{C}_{{\rm{e}}}(T,B)}{T}-\frac{{C}_{{\rm{e}}}(T,0)}{T}={B}^{-\frac{1}{3}}{g}_{C}\left(\frac{T}{B}\right),$$
(5)

where gC(x) is field-dependent part of fC(x) (Supplementary Notes 6 and 7). As demonstrated in Fig. 2d, this expression illustrates scale invariance in the specific heat that persists over nearly three orders of magnitude in the scaling variable T/B.

### Hall resistivity and electronic structure

The T/B scaling in thermodynamics clearly discloses the presence of the QCP located exactly at zero field and absolute zero, similar to the layered QC metals YbAlB4 (ref. 31) and YFe2Al10 (ref. 32). More notably, the multiband nature in iron pnictides affixes the uniqueness of quantum criticality for Ba(Fe1/3Co1/3Ni1/3)2As2. Dominated by electron-like carriers, the Hall resistivity ρyx is negative and perfectly linear in field at high temperatures (T = 20 K), as shown in Fig. 3a. Upon cooling, ρyx develops a nonlinearity with negative curvature. More prominent <1 K, the nonlinear Hall resistivity switches its sign at low fields <2 T. The sign change is more readily observed in the temperature dependence of Hall coefficient RH defined by ρyx/B at low-T and low-H region (Fig. 3b), implying that hole-like carriers dominate the transport in the vicinity of the QCP. The radial shape of the dominant carrier crossover in the field-temperature phase diagram confirms the absence of an intrinsic energy scale in RH (Fig. 3c), or in other words, the presence of scale invariance in the Hall effect tuned by temperature and magnetic field. Similar to the resistivity, RH obeys Γ(TB) scaling (Fig. 3d), consolidating the existence of scale invariance near the QCP in this system beyond any doubt.

Angle-resolved photoemission measurements identify a unique electronic structure and confirm the anomalous scattering rate correlated with Planckian dissipation. Unlike heavily electron-doped BaCo2As2, the electronic structure for Ba(Fe1/3Co1/3Ni1/3)2As2 consists of a large hole-like pocket and a cross-shaped electron-like Fermi surface around the Γ point, together with oval and elongated electron pockets around the M points, exhibited by the Fermi surface map (Fig. 4a), the band dispersion along kx = 0 direction (Fig. 4b) at 30 K, and a schematic illustration (Fig. 4a, inset). The elongated electron pockets are very shallow, and the chemical potential is located close to the bottom of the shallow bands. Dominating transport at low temperatures and fields, the large hole-like pocket is identified as the one responsible for QC behavior. Amazingly, the scattering rate (obtained from the dispersion of the hole-like bands at 1 K) varies linearly with the kinetic energy up to 100 meV, consistent with Planckian dissipation as observed in the resistivity (Fig. 4c, d).

## Discussion

While our primary observations of the scale invariance in the thermodynamics are consistent with quantum criticality overall, they indicate a highly unusual critical behavior in Ba(Fe1/3Co1/3Ni1/3)2As2. While sharing an enhancement of the Wilson ratio with BaCo2As2 indicative of a FM instability, the critical behavior in Ba(Fe1/3Co1/3Ni1/3)2As2 is not described by any known theoretical predictions. Assuming spacial dimensionality of three (d = 3) based on the observed isotropic response in MR and magnetization (Supplementary Note 3), the observed critical exponents of d/z = 2/3 and z = yb yield z = yb = 4.5.

The extracted dynamical exponents from our measurements do not match the predictions for either mean-field Hertz–Moriya–Millis theory for d = 3 (which predict z = 3 for clean FM and z = 4 for dirty FM quantum criticality with v = 1/2)27,28,29, or predictions for clean FM beyond mean field, which predict the appearance of a weak first-order transition, with z = 3 and ν = 1/4 for d = 3 and quantum wing critical points with the same critical exponents, as the mean-field theory33,34,35,36,37. QC behavior in disordered 3d FM has been well explained by the Belitz–Kirkpatrick–Vojta theory, predicting critical exponents ν = 1 and z = 3 for the asymptotic limit, and ν = 0.25 and z = 6 for the preasymptotic limit37, 38, neither of which is in agreement with our observation. Experimentally, previously measured exponents in QC materials, such as YbNi4(P1−xAsx)2 (FM QCP, νz ~ 5)39, CeCu6−xAux (AFM QCP, d/z = 1/4, νz = 1)40, β-YbAlB4 (mixed-valence meal, d/z = 1/2, νz = 1)31, YFe2Al10 (layered QC metal, d/z = 1, νz = 0.59)32, and Sr0.3Ca0.7RuO3 (disordered FM QCP, z = 1.76)41 are also incompatible with the measured dynamical exponent.

The high residual resistivity observed in Ba(Fe1/3Co1/3Ni1/3)2As2 evokes the possible realization of quantum Griffiths phase, where the QC behavior is dominated by FM rare regions. The quantum Griffiths model predicts power law singularities in the magnetic susceptibility (χ ~ Tλ−1), specific heat (C/T ~ Tλ−1), and magnetization (M ~ Bλ), determined by the nonuniversal Griffiths exponent λ that takes 0 at the QCP, and increases with distance from criticality42. In the present system, however, λ = 2/3 extracted from the magnetic susceptibility (χ ~ T−1/3; Fig. 1a inset) disagrees with λ = 0.75 obtained from the specific heat (C/T ~ T−0.25; Fig. 2b inset), irreconcilable with the quantum Griffiths model. Besides, the critical exponents in Ba(Fe1/3Co1/3Ni1/3)2As2 do not agree with those obtained experimentally in other quantum Griffith systems37. For instance, disordered weak ferromagnet Ni1−xVx show critical behavior dominated by quantum Griffiths singularities, χ ~ Tλ−1 and M ~ Bλ, over a wide range of vanadium concentration43, 44. On the other hand, in Ba(Fe1/3Co1/3Ni1/3)2As2, λ = 2/3 derived from the magnetic susceptibility contradicts λ obtained from magnetization M ~ B0.75 (Supplementary Fig. 5), in conflict with the quantum Griffiths phase.

Highly unusual dynamical critical behavior in this material cannot be simply explained by existing FM QCP theories, but instead, it can be attributed to substitutional alloying by counter-doping. Indeed, the anomalous behavior observed in Ba(Fe1/3Co1/3Ni1/3)2As2 is more prominent than that observed in both of the end members of the 3d7 configuration line, namely, BaCo2As2 and Ba(Fe,Ni)2As2 (Supplementary Notes 8 and 9), signifying that the specific 1/3 equal ratios of Fe:Co:Ni in BaCo2As2 are indeed important to stabilizing a unique QC ground state. In fact, as shown in Fig. 5, the observed NFL scattering behavior in Ba(Fe1/3Co1/3Ni1/3)2As2 is completely robust against pressure and even replacement of Ba for Sr (i.e., in Sr(Fe1/3Co1/3Ni1/3)2As2), implying either an electronic structure modification beyond d electron tuning, or a significant role for transition metal site dilution. In fact, while generally obscuring the critical behavior, high randomness due to substitution indeed plays an important role in some QC materials, such as medium entropy alloys38, 45, in which similar NFL behavior has been realized38, 45. Together with the pressure insensitivity of the T-linear scattering in Ba(Fe1/3Co1/3Ni1/3)2As2, our experimental observations of scale invariance in this system indicates that substitutional alloying is a key ingredient to tune the quantum criticality that may provide the key to understanding the lack of superconductivity driven by QC fluctuations.

## Methods

### Sample preparation

The single crystals of Ba(Fe1/3Co1/3Ni1/3)2As2 were grown by TMAs (TM = Fe, Co, and Ni) self-flux method with the molar ratios of 3:4:4:4 = Ba:FeAs:CoAs:NiAs. Resulting crystals were cleaved out of the flux. The typical crystal size is 5 × 5 × 0.1 mm3.

### Magnetotransport measurements

Magnetotransport measurements up to 15 T were conducted in a 3He–4He dilution refrigerator, and high magnetic field transport measurements up to 35 T were performed at the National High Magnetic Field Laboratory in Tallahassee.

### Heat capacity measurements

Heat capacity was measured using the thermal relaxation method in a 3He–4He dilution refrigerator. A RuO2 thermometer on the calorimeter was calibrated in magnetic fields up to 15 T.

### Magnetic susceptibility measurements

Magnetic susceptibility was measured using the vibrating sample magnetometer option in a 14 T Quantum Design DynaCool Physical Properties Measurement System and a 7 T SQUID Magnetic Properties Measurement System.

### Pressure measurements

A nonmagnetic piston–cylinder pressure cell was used 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 temperature of lead as pressure gauge at base temperature. All transport measurements were performed on the same Ba(Fe1/3Co1/3Ni1/3)2As2 crystal with 200 μm thickness, using four point contacts made with silver epoxy. The pressure and temperature dependence of the resistivity were measured during warming process in a Quantum Design Physical Properties Measurement System. Mention of commercial equipment does not imply endorsement by NIST.

### Angle-resolved photoemission spectroscopy

Angle-resolved photoemission spectroscopy for Ba(Fe1/3Co1/3Ni1/3)2As2 was performed using the 13-ARPES end station of the UE112-PGM2b beam-line at BESSY II (Helmholtz Zentrum Berlin) synchrotron radiation center.

## Data availability

All data presented in this manuscript are available from the corresponding author upon reasonable request.

## References

1. 1.

Paglione, J. & Greene, R. L. High-temperature superconductivity in iron-based materials. Nat. Phys. 6, 645–658 (2010).

2. 2.

Shibauchi, T., Carrington, A. & Matsuda, Y. A quantum critical point lying beneath the superconducting dome in iron pnictides. Annu. Rev. Condens. Matter Phys. 5, 113–135 (2014).

3. 3.

Mazin, I. I., Singh, D. J., Johannes, M. D. & Du, M. H. Unconventional superconductivity with a sign reversal in the order parameter of LaFeAsO1−xFx. Phys. Rev. Lett. 101, 057003 (2008).

4. 4.

Kuroki, K., Usui, H., Onari, S., Arita, R. & Aoki, H. Pnictogen height as a possible switch between high-Tc nodeless and low-Tc nodal pairings in the iron-based superconductors. Phys. Rev. B 79, 224511 (2009).

5. 5.

Hayes, I. M. et al. Scaling between magnetic field and temperature in the high-temperature superconductor BaFe2(As1−xPx)2. Nat. Phys. 12, 916–919 (2016).

6. 6.

Giraldo-Gallo, P. et al. Scale-invariant magnetoresistance in a cuprate superconductor. Science 361, 479–481 (2018).

7. 7.

Legros, A. et al. Universal T-linear resistivity and planckian dissipation in overdoped cuprates. Nat. Phys. 15, 142–147 (2019).

8. 8.

Guo, H., Gu, Y. & Sachdev, S. Linear in temperature resistivity in the limit of zero temperature from the time reparameterization soft mode. Ann. Phys. 418, 168202 (2020).

9. 9.

Chu, J.-H. et al. In-plane resistivity anisotropy in an underdoped iron arsenide superconductor. Science 329, 824–826 (2010).

10. 10.

Yi, M. et al. Symmetry-breaking orbital anisotropy observed for detwinned Ba(Fe1−xCox)2As2 above the spin density wave transition. Proc. Natl Acad. Sci. USA 108, 6878–6883 (2011).

11. 11.

Nakajima, M. et al. Unprecedented anisotropic metallic state in undoped iron arsenide BaFe2As2 revealed by optical spectroscopy. Proc. Natl Acad. Sci. USA 108, 12238–12242 (2011).

12. 12.

Chu, J.-H., Kuo, H.-H., Analytis, J. G. & Fisher, I. R. Divergent nematic susceptibility in an iron arsenide superconductor. Science 337, 710–712 (2012).

13. 13.

Yanagi, H. et al. Itinerant ferromagnetism in the layered crystals LaCoOX (X=P,As). Phys. Rev. B 77, 224431 (2008).

14. 14.

Anand, V. K. et al. Crystallography and physical properties of BaCo2As2, Ba0.94K0.06Co2As2, and Ba0.78K0.22Co2As2. Phys. Rev. B 90, 064517 (2014).

15. 15.

Wiecki, P. et al. Competing magnetic fluctuations in iron pnictide superconductors: role of ferromagnetic spin correlations revealed by NMR. Phys. Rev. Lett. 115, 137001 (2015).

16. 16.

Wiecki, P., Ogloblichev, V., Pandey, A., Johnston, D. C. & Furukawa, Y. Coexistence of antiferromagnetic and ferromagnetic spin correlations in SrCo2As2 revealed by 59Co and 75As NMR. Phys. Rev. B 91, 220406 (2015).

17. 17.

Sefat, A. S. et al. Renormalized behavior and proximity of BaCo2As2 to a magnetic quantum critical point. Phys. Rev. B 79, 024512 (2009).

18. 18.

Sefat, A. S. et al. Structure and anisotropic properties of BaFe2xNixAs2 (x = 0, 1, and 2) single crystals. Phys. Rev. B 79, 094508 (2009).

19. 19.

Eckberg, C. et al. Evolution of structure and superconductivity in Ba(Ni1−xCox)2As2. Phys. Rev. B 97, 224505 (2018).

20. 20.

Ni, N. et al. Phase diagrams of Ba(Fe1−xMx)2As2 single crystals (M=Rh and Pd). Phys. Rev. B 80, 024511 (2009).

21. 21.

Canfield, P. C., Bud’ko, S. L., Ni, N., Yan, J. Q. & Kracher, A. Decoupling of the superconducting and magnetic/structural phase transitions in electron-doped BaFe2As2. Phys. Rev. B 80, 060501 (2009).

22. 22.

Liu, C. et al. Evidence for a lifshitz transition in electron-doped iron arsenic superconductors at the onset of superconductivity. Nat. Phys. 6, 419–423 (2010).

23. 23.

Neupane, M. et al. Electron-hole asymmetry in the superconductivity of doped BaFe2As2 seen via the rigid chemical-potential shift in photoemission. Phys. Rev. B 83, 094522 (2011).

24. 24.

Ciuchi, S., Di Sante, D., Dobrosavljević, V. & Fratini, S. The origin of mooij correlations in disordered metals. npj Quantum Mater. 3, 44 (2018).

25. 25.

Lee, P. A. & Ramakrishnan, T. V. Disordered electronic systems. Rev. Mod. Phys. 57, 287–337 (1985).

26. 26.

Stewart, G. Non-fermi-liquid behavior in d- and f-electron metals. Rev. Mod. Phys. 73, 797 (2001).

27. 27.

Hertz, J. A. Quantum critical phenomena. Phys. Rev. B 14, 1165 (1976).

28. 28.

Moriya, T. Spin Fluctuations in Itinerant Electron Magnetism (Springer, Berlin, Heidelberg, Germany, 1985).

29. 29.

Millis, A. J. Effect of a nonzero temperature on quantum critical points in itinerant fermion systems. Phys. Rev. B 48, 7183–7196 (1993).

30. 30.

Sachdev, S. Quantum Phase Transitions (Cambridge University Press, Cambridge, UK, 1999).

31. 31.

Matsumoto, Y. et al. Quantum criticality without tuning in the mixed valence compound β-YbAlB4. Science 331, 316–319 (2011).

32. 32.

Wu, L. S., Kim, M. S., Park, K., Tsvelik, A. M. & Aronson, M. C. Quantum critical fluctuations in layered YFe2Al10. Proc. Natl Acad. Sci. USA 111, 14088–14093 (2014).

33. 33.

Belitz, D., Kirkpatrick, T. R. & Vojta, T. Nonanalytic behavior of the spin susceptibility in clean fermi systems. Phys. Rev. B 55, 9452–9462 (1997).

34. 34.

Rech, J., Pépin, C. & Chubukov, A. V. Quantum critical behavior in itinerant electron systems: Eliashberg theory and instability of a ferromagnetic quantum critical point. Phys. Rev. B 74, 195126 (2006).

35. 35.

Conduit, G. J., Green, A. G. & Simons, B. D. Inhomogeneous phase formation on the border of itinerant ferromagnetism. Phys. Rev. Lett. 103, 207201 (2009).

36. 36.

Kirkpatrick, T. R. & Belitz, D. Exponent relations at quantum phase transitions with applications to metallic quantum ferromagnets. Phys. Rev. B 91, 214407 (2015).

37. 37.

Brando, M., Belitz, D., Grosche, F. M. & Kirkpatrick, T. R. Metallic quantum ferromagnets. Rev. Mod. Phys. 88, 025006 (2016).

38. 38.

Sales, B. C. et al. Quantum critical behavior in the asymptotic limit of high disorder in the medium entropy alloy NiCoCr0.8. npj Quantum Mater. 2, 33 (2017).

39. 39.

Steppke, A. et al. Ferromagnetic quantum critical point in the heavy-fermion metal YbNi4(P1−xAsx)2. Science 339, 933–936 (2013).

40. 40.

Schroeder, A. et al. Onset of antiferromagnetism in heavy-fermion metals. Nature 407, 351 (2000).

41. 41.

Huang, C. L. et al. Anomalous quantum criticality in an itinerant ferromagnet. Nat. Commun. 6, 8188 (2015).

42. 42.

Vojta, T. Quantum griffiths effects and smeared phase transitions in metals: theory and experiment. J. Low. Temp. Phys. 161, 299–323 (2010).

43. 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).

44. 44.

Wang, R. et al. Quantum griffiths phase inside the ferromagnetic phase of Ni1−xVx. Phys. Rev. Lett. 118, 267202 (2017).

45. 45.

Sales, B. C. et al. Quantum critical behavior in a concentrated ternary solid solution. Sci. Rep. 6, 26179 (2016).

46. 46.

Momma, K. & Izumi, F. VESTA3 for three-dimensional visualization of crystal, volumetric and morphology data. J. Appl. Crystallogr. 44, 1272–1276 (2011).

## Acknowledgements

Experimental research was supported by the National Science Foundation Division of Materials Research Award No. DMR-1610349, and materials development supported by the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant no. GBMF9071. A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement No. DMR-1644779 and the State of Florida. Pressure measurements were supported by NIST.

## Author information

Authors

### Contributions

Y.N., T.M., K.K., A.H., and R.W. performed the low-temperature transport and heat capacity measurements, and analyzed the data. C.E. and S.R.S. grew and characterized single crystals. I.L. and N.B. conducted the pressure measurements, and D.C., Y.S.E., and D.G. performed transport measurements at high magnetic fields. Z.L. and S.V.B. measured the angle-resolved photoemission spectroscopy, and L.W. performed the theoretical support. P.Y.Z. performed the single-crystal x-ray diffraction. Y.N. and J.P. conceived and designed the experiments, and all authors contributed to the editing of the manuscript.

### Corresponding authors

Correspondence to Yasuyuki Nakajima or Johnpierre Paglione.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Nakajima, Y., Metz, T., Eckberg, C. et al. Quantum-critical scale invariance in a transition metal alloy. Commun Phys 3, 181 (2020). https://doi.org/10.1038/s42005-020-00448-5

• Accepted:

• Published: