Quantum-critical scale invariance in a transition metal alloy

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. Extensive theoretical and experimental efforts have been devoted to the effect of quantum criticality in our understanding of the physics of high-temperature superconductors and strongly correlated electron materials, yet it remains a puzzle in condensed matter physics. The authors report observations of a quantum criticality by investigating the non-Fermi liquid thermodynamics and transport behaviour in a non-superconducting iron pnictide.

N on-Fermi liquid (NFL) behavior ubiquitously appears in iron-based high-temperature superconductors with a novel type of superconducting pairing symmetry driven by interband repulsion 1,2 . The putative pairing mechanism is thought to be associated with the temperature-doping phase diagram, bearing striking resemblance to cuprate and heavyfermion superconductors 3,4 . In iron-based superconductors, the superconducting phase appears to be centered around the point of suppression of antiferromagnetic (AFM) and orthorhombic structural order 1 . 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 resistivity [5][6][7][8] . The universal transport behavior is known as Planckian dissipation, where the transport scattering rate is constrained by thermal energy, ℏ/τ P = k B T, where ℏ is the reduced Planck constant and k B is the Boltzmann constant. Lacking an intrinsic energy scale, the scaleinvariant 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 behavior [9][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 correlations [14][15][16] . For instance, a strongly enhanced Wilson ratio R W of~7-10 at 2 K (ref. 17 ) and violation of the Koringa law [14][15][16] suggest proximity to a FM instability in BaCo 2 As 2 . BaNi 2 As 2 , on the other hand, seems to be devoid of magnetic order 18 and rather hosts other ordering instabilities in both structure and charge 19 . Confirmed by extensive study, Fe, Co, and Ni have the same 2+ oxidation state in the tetragonal ThCr 2 Si 2 structure, thus adding one d electron (hole) contribution by Ni (Fe) substitution for Co in BaCo 2 As 2 (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 BaCo 2 As 2 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(Fe 1/ 3 Co 1/3 Ni 1/3 ) 2 As 2 , 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(Fe 1/3 Co 1/ 3 Ni 1/3 ) 2 As 2 , 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(Fe 1/3 Co 1/3 Ni 1/3 ) 2 As 2 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., ρ ∝ T 2 ) at low temperatures (Supplementary Note 1).
Note that the unusual resistivity observed in Ba(Fe 1/3 Co 1/3 Ni 1/ 3 ) 2 As 2 cannot be ascribed to either Mooij correlations 24 or quantum interference 25 due to randomness introduced by counter-doping. Given that the Mooij correlations are dominant, increasing randomness enhances the residual resistivity ρ 0 , accompanied by a gradual change in the slope of ρ(T) at high temperatures, as observed in LuRh 4 B 4 (ref. 25 ). However, the overall slope of resistivity in Ba(Fe 1/3 Co 1/3 Ni 1/3 ) 2 As 2 is parallel shifted from that in BaCo 2 As 2 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(Fe 1/3 Co 1/ 3 Ni 1/3 ) 2 As 2 cannot be reproduced by the quantum corrections in conductivity caused by interference that provide the power law σ T p/2 (or ρ~T −p/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 ΓðT; BÞ , where μ B 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, ℏ/τ = ℏne 2 (ρ(T, B) − 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 lowtemperature-specific heat measured at 10 T in the present work, as a function of Γ(T, B), collapsing onto one universal curve as shown in Fig. 1c. This scaling is reminiscent of the observation in QC iron pnictide BaFe 2 (As,P) 2 (ref. 5 ). Although Ba(Fe 1/3 Co 1/ 3 Ni 1/3 ) 2 As 2 and BaFe 2 (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 BaFe 2 (As, P) 2 , it holds in the low Γ region below~2 meV in Ba(Fe 1/3 Co 1/ 3 Ni 1/3 ) 2 As 2 , and that the extracted value of dimensionless parameter η (γ/α in ref. 5 ) is different from that reported in ref. 5 .
The Γ(T, B) scaling can closely be correlated to the Planckian bound of dissipation. Quantum mechanics allows the shortest dissipative time scale τ P = ℏ/k B T, constrained by the uncertainty principle between dissipative time scale τ and energy dissipation E~k B T, τ ⋅ k B T ≳ ℏ. Redefining Γ(T, B) as the dissipation energy scale in magnetic field, we can obtain the universal bound of dissipation, ℏ/τ P~Γ (T, B). Our experimental observation in Γ (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.
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(B ∥ c)/Δρ(B ∥ ab), as a function of temperature in Fig. 1d. The anisotropy between transverse MR in the out-of-plane field (B ∥ c, I ∥ ab) and transverse MR in the inplane field (B ∥ ab, B ⊥ I ∥ ab) 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(T, B) 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 (B ∥ c, I ∥ ab) and longitudinal (B ∥ I ∥ ab) configurations (Fig. 1d).
Thermodynamic properties. In addition to resistivity, magnetic susceptibility χ = M/B and electronic heat capacity C e /T also exhibit canonical NFL behavior, i.e., diverging temperature dependence associated with QC instabilities 26 . The magnetic susceptibility varies as χ ∝ T −1/3 at low temperatures <8 K (inset of Fig. 2a), in contrast to the T-independent Pauli paramagnetic susceptibility χ P ¼ 2gμ 2 B DðE F Þ (with electron g-factor and density of states at the Fermi energy D(E F )) 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 (C ph ) and nuclear Schottky contributions (C Sch ) from the total heat capacity (C tot ), the electronic specific 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 = C e /T at B = 0 T, combined with the magnetic susceptibility χ, provides large Wilson ratio R W ¼ π 2 k 2 B χ=3μ 2 B γ ¼ 3:2 at T = 1.8 K, suggestive of the presence of magnetic instabilities similar to BaCo 2 As 2 .
The observation of FL recovery with magnetic field corroborates the presence of a new energy scale k B T * , distinctive of crossover between the QC (k B T ≫ gμ B B) and FL (k B T ≪ gμ B B) regimes. Intriguingly, this new energy scale allows a single scaling function of T/B in the magnetization, written by, as shown in Fig. 2c. This scaling relation indeed reveals the underlying free energy given by a universal function of T/B, where d is the spatial dimensionality, z is the dynamic exponent, and y b is the scaling exponent related to the tuning parameter B (refs. [27][28][29][30]. Here, f F (x) is a universal function of x. Hence, the magnetization can be derived from the derivative of the free energy, Directly comparing this with the observed QC scaling relation in Fig. 2c, we can extract the critical exponents in the free energy, namely, z/y b = 1 and d/y b − 1 = −1/3, yielding z = y b and d/z = 2/3. These values of the critical exponents describe the specific heat by using the same free energy, Rewriting the free energy, where g C (x) is field-dependent part of f C (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 YbAlB 4 (ref. 31 ) and YFe 2 Al 10 (ref. 32 ). More notably, the multiband nature in iron pnictides affixes the uniqueness of quantum criticality for Ba(Fe 1/3 Co 1/3 Ni 1/3 ) 2 As 2 . Dominated by electron-like carriers, the Hall resistivity ρ yx is negative and perfectly linear in field at high temperatures (T = 20 K), as shown in Fig. 3a. Upon cooling, ρ yx 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 R H defined by ρ yx /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 R H (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, R H obeys Γ(T, B) 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 BaCo 2 As 2 , the electronic structure for Ba(Fe 1/3 Co 1/3 Ni 1/3 ) 2 As 2 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 k x = 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(Fe 1/3 Co 1/3 Ni 1/3 ) 2 As 2 . While sharing an enhancement of the Wilson ratio with BaCo 2 As 2 indicative of a FM instability, the critical behavior in Ba(Fe 1/3 Co 1/ 3 Ni 1/3 ) 2 As 2 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 = y b yield z = y b = 4.5.
The high residual resistivity observed in Ba(Fe 1/3 Co 1/3 Ni 1/3 ) 2 As 2 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~T λ−1 ), specific heat (C/T~T λ−1 ), and magnetization (M~B λ ), determined by the nonuniversal Griffiths exponent λ that takes 0 at the QCP, and increases with distance from criticality 42 . 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(Fe 1/3 Co 1/3 Ni 1/3 ) 2 As 2 do not agree with those obtained experimentally in other quantum Griffith systems 37  with η = 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.
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 counterdoping. Indeed, the anomalous behavior observed in Ba(Fe 1/3 Co 1/ 3 Ni 1/3 ) 2 As 2 is more prominent than that observed in both of the end members of the 3d 7 configuration line, namely, BaCo 2 As 2 and Ba(Fe,Ni) 2 As 2 (Supplementary Notes 8 and 9), signifying that the specific 1/3 equal ratios of Fe:Co:Ni in BaCo 2 As 2 are indeed important to stabilizing a unique QC ground state. In fact, as shown in Fig. 5, the observed NFL scattering behavior in Ba(Fe 1/3 Co 1/3 Ni 1/ 3 ) 2 As 2 is completely robust against pressure and even replacement of Ba for Sr (i.e., in Sr(Fe 1/3 Co 1/3 Ni 1/3 ) 2 As 2 ), 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 alloys 38,45 , in which similar NFL behavior has been realized 38,45 . Together with the pressure insensitivity of the T-linear scattering in Ba(Fe 1/3 Co 1/3 Ni 1/3 ) 2 As 2 , 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.
Magnetotransport measurements. Magnetotransport measurements up to 15 T were conducted in a 3 He-4 He 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 3 He-4 He dilution refrigerator. A RuO 2 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 npentane 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(Fe 1/3 Co 1/3 Ni 1/3 ) 2 As 2 crystal with 200 Fig. 4 Fermi surfaces and anomalous scattering rates in Ba(Fe 1/3 Co 1/3 Ni 1/3 ) 2 As 2 . a Angle-resolved photoemission study of Fermi surface map for Ba (Fe 1/3 Co 1/3 Ni 1/3 ) 2 As 2 , 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 k x = 0 direction, highlighting the elongated shallow electron-like pockets observed near the BZ corners. c Energy dispersion of holelike 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.
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.

Data availability
All data presented in this manuscript are available from the corresponding author upon reasonable request. Fig. 5 Robust non-Fermi liquid behavior against pressure. a Overall temperature dependence of resistivity for Ba(Fe 1/3 Co 1/3 Ni 1/3 ) 2 As 2 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(Fe 1/3 Co 1/3 Ni 1/3 ) 2 As 2 increases, approaching that for Sr(Fe 1/3 Co 1/3 Ni 1/3 ) 2 As 2 with smaller lattice constants than Ba(Fe 1/