Quantum-critical scale invariance in a transition metal alloy

The study addresses how quantum-critical (QC) fluctuations influence anomalous metallic behavior in iron-based compounds when superconductivity does not obscure the zero-temperature normal state. In many iron-based superconductors, non-Fermi liquid (NFL) behavior with linear-in-T resistivity appears near the suppression of antiferromagnetic (AFM) and structural order and is thought to be connected to unconventional superconductivity and Planckian dissipation. However, superconductivity and additional phases (e.g., electronic nematicity) complicate access to the QC normal state. The authors employ a counter-doping strategy to realize a nonsuperconducting iron pnictide, Ba(Fe1/3Co1/3Ni1/3)2As2, to probe QC behavior down to millikelvin temperatures and in high magnetic fields without a superconducting dome. The central hypothesis is that this alloy stabilizes a QC ground state exhibiting scale invariance in transport, thermodynamics, and Hall effect, reflecting Planckian-limited scattering and a QCP at T=0, B=0.

Prior work shows that iron-based superconductors exhibit NFL behavior and unconventional pairing likely mediated by spin fluctuations, with superconductivity emerging near the suppression of AFM/orthorhombic order. Universal linear-in-T resistivity, or Planckian dissipation, has been reported in cuprates and pnictides, where ħ/τ ~ kB T reflects the absence of intrinsic energy scales. In iron pnictides, nematic phases above structural transitions further complicate the landscape. Magnetic interactions vary across related materials: LaCoOX can be ferromagnetic; BaCo2As2 shows proximity to FM instability (enhanced Wilson ratio and Korringa-law violations); BaNi2As2 lacks magnetic order but displays structural/charge ordering. Fe, Co, and Ni substitutions tune the d-electron count and electronic structure, enabling traversal across different ground states. Scaling between temperature and magnetic field in transport has been observed in BaFe2(As,P)2. Theories for quantum critical ferromagnets (Hertz-Moriya-Millis and beyond) predict certain critical exponents; quantum Griffiths phases can produce power-law singularities in disordered systems. The present work builds on these contexts to test for scale invariance and QC behavior in a tailored nonsuperconducting alloy.

- Sample growth: Single crystals of Ba(Fe1/3Co1/3Ni1/3)2As2 grown by transition-metal arsenide self-flux with molar ratio Ba:FeAs:CoAs:NiAs = 3:4:4:4; crystals cleaved from flux; typical size ~5×5×0.1 mm^3. - Magnetotransport: Measurements up to 15 T in a 3He-4He dilution refrigerator; high-field transport up to 35 T at the National High Magnetic Field Laboratory. Four-point contacts with silver epoxy; current in ab-plane; various field orientations (B||c, B||ab; transverse and longitudinal) studied; temperatures down to ~20 mK. - Heat capacity: Thermal relaxation method in a 3He-4He dilution refrigerator; RuO4 thermometer calibrated in fields to 15 T; phonon and nuclear Schottky contributions subtracted to obtain electronic C_el/T. - Magnetic susceptibility and magnetization: Vibrating sample magnetometer (14 T PPMS) and 17 T SQUID MPMS; fields primarily within ab-plane; temperature dependence and scaling analyses performed (e.g., -dM/dT vs T/B). - Hall effect: Hall resistivity ρ_xy(B) measured with B||c over 0.1–14 T and temperatures down to 100 mK; Hall coefficient R_H = ρ_xy/B analyzed versus T and B and scaled using combined energy variable. - Pressure-dependent transport: Nonmagnetic piston-cylinder cell up to 1.99 GPa; pressure medium n-pentane:1-methyl-3-butanol (1:1); Pb superconducting Tc as pressure gauge at base temperature; same crystal measured across pressures; resistance measured on warming in PPMS. - Angle-resolved photoemission spectroscopy (ARPES): Performed at 13-ARPES endstation, UE112-PGM2b beamline (BESSY II, HZB); Fermi surface mapping at 30 K; band dispersions including near 1 K for scattering rate extraction; identification of electron and hole pockets; momentum distribution curve widths used to estimate scattering rates. - Data analysis: Resistivity separated into residual and inelastic components; scaling variable defined Γ(T,B) = sqrt[(kB T)^2 + (η μB B)^2]; inelastic scattering rate estimated using ħ/τ = ħ n e^2 [ρ(T,B)-ρ(0,0)]/m* with n from low-T Hall and m* from low-T specific heat; scaling collapses tested for 1/τ, magnetization (-dM/dT), specific heat (ΔC/T), and Hall coefficient versus Γ or T/B; anisotropy assessed via MR ratios and angular dependence.

- Transport: Resistivity exhibits quasi-linear temperature dependence from 20 K down to at least 20 mK at B=0 T, without phase-transition anomalies, indicating an anomalous metallic ground state. Magnetic field suppresses NFL behavior and restores Fermi-liquid (ρ∝T^2) at low T. - Magnetoresistance: At 1.31 K, ΔR(B)/R(0) is sublinear in B up to 35 T. MR is isotropic with respect to field orientation; anisotropy ratio Δρ(B||c)/Δρ(B||ab) approaches unity at low T. - Universal scaling of scattering: Defining Γ(T,B)=sqrt[(kB T)^2 + (η μB B)^2] with η=0.67, the inelastic scattering rate collapses onto a single curve: ħ/τ = A Γ with A≈1.80, consistent with Planckian dissipation and a universal temperature–field scaling down to low energies (Γ ≲ 2 meV). - Thermodynamics: Magnetic susceptibility χ shows χ ∝ T^(-1/3) below ~8 K at low fields; with increasing field up to 7 T, behavior crosses over toward FL-like. Electronic specific heat coefficient C_el/T diverges as ~T^(-0.25) down to ~150 mK at B=0 and is suppressed by field, with FL recovered by ~10 T. Wilson ratio R_W ≈ 3.2 at 1.8 K (B=0), indicating magnetic instability. - T/B scaling: -dM/dT and ΔC/T exhibit scale invariance with single-parameter scaling in T/B. The scaling implies a free energy of the form F(T,B) with critical exponents satisfying z/γ=1 and δ/γ−1=−1/3, leading (with d=3) to d/z=2/3 and z=γ≈4.5. Specific heat scaling: ΔC/T ∝ B^(−2/3) f_C(T/B), equivalently ΔC ∝ B^(1/3) g_c(T/B), with data collapsing over nearly three decades in T/B. - Hall effect: ρ_xy is negative and linear in B at 20 K (electron-like dominant). Upon cooling, ρ_xy becomes nonlinear and changes sign at low fields (<2 T) below ~1 K, indicating hole-like carriers dominate near the QCP. R_H(T,B) exhibits scale invariance and collapses when plotted versus I(T,B)=(kB T)^2+(η μB B)^2 with η=1. - ARPES: Electronic structure comprises a large hole-like pocket and cross-shaped electron-like Fermi surface at Γ, with shallow elongated electron pockets at M. The large hole pocket dominates low-T, low-B transport and is implicated in QC behavior. The scattering rate of hole bands (from MDC widths) increases linearly with kinetic energy up to ~100 meV, consistent with Planckian dissipation. - Robustness: Linear-in-T resistivity below ~20 K is robust under hydrostatic pressure up to 1.99 GPa and persists upon Ba→Sr substitution, suggesting insensitivity to lattice compression and pointing to a role of substitutional alloying/site dilution. - QCP location: Thermodynamic T/B scaling and transport/Hall scaling indicate a QCP at T=0, B=0 in Ba(Fe1/3Co1/3Ni1/3)2As2, without interference from superconductivity.

The comprehensive observation of temperature–field scale invariance across resistivity, magnetization, specific heat, and Hall effect strongly evidences quantum criticality centered at T=0 and B=0 in Ba(Fe1/3Co1/3Ni1/3)2As2. The linear-in-T resistivity and the linear relation ħ/τ ≈ AΓ(T,B) indicate Planckian-limited scattering without intrinsic energy scales. Isotropic magnetotransport and magnetization imply effective three-dimensionality despite the layered structure. The dominance of hole-like carriers near the QCP, revealed by sign-changing Hall response and ARPES, connects the QC fluctuations to the large hole pocket. The extracted critical exponents (d/z=2/3 with d=3, yielding z≈4.5) and scaling forms are inconsistent with standard Hertz-Moriya-Millis predictions for clean or dirty ferromagnetic QCPs, and also do not match preasymptotic asymptotics or known quantum Griffiths scenarios. Discrepancies between exponents derived from χ and C/T further argue against a Griffiths phase. The results suggest an unconventional QC regime stabilized by counter-doping and substantial substitutional alloying, wherein site dilution and multiband electronic structure may play essential roles. The robustness of the NFL scattering to pressure and to A-site substitution (Ba→Sr) underscores that tuning beyond simple d-electron count or lattice parameters is at play. The absence of superconductivity enables direct access to the QC normal state and highlights that QC fluctuations need not necessarily precipitate pairing. Overall, the findings broaden the phenomenology of metallic quantum criticality, revealing a planckian, scale-invariant state in a multiband transition metal alloy without a proximate superconducting instability.

This work demonstrates a quantum-critical, scale-invariant metallic state in Ba(Fe1/3Co1/3Ni1/3)2As2, a nonsuperconducting iron pnictide. Key contributions include: (i) linear-in-T resistivity from 20 K to 20 mK at zero field and its suppression by magnetic field toward Fermi-liquid behavior; (ii) universal temperature–field scaling of the inelastic scattering rate (ħ/τ = AΓ with A≈1.8, η≈0.67), magnetization (−dM/dT vs T/B), specific heat (ΔC/T vs T/B), and Hall coefficient (vs (kB T)^2+(η μB B)^2); (iii) identification of hole-like carriers dominating near the QCP and ARPES evidence for a large hole pocket with planckian energy-linear scattering up to 100 meV; (iv) isotropic field response indicating 3D behavior; and (v) robustness of NFL behavior under pressure and A-site substitution, implying a key role of substitutional alloying. The QCP is located at T=0, B=0, with unusual critical exponents not captured by existing ferromagnetic QCP theories or quantum Griffiths descriptions, motivating the development of new theoretical frameworks that incorporate strong disorder/site dilution and multiband effects. Future studies could probe the nature of spin fluctuations (e.g., neutron scattering), disorder landscapes, and compare with related alloys to further elucidate the mechanism behind the unconventional QC state and its relation to the absence of superconductivity.

- The extracted critical exponents (e.g., z≈4.5) do not conform to established theoretical models for ferromagnetic quantum criticality, leaving the microscopic origin unresolved. - While disorder and substitutional alloying are implicated, the precise role of randomness/site dilution versus band-structure effects is not disentangled. - No direct microscopic probe of magnetic fluctuations (e.g., inelastic neutron scattering) is reported to identify the order parameter or fluctuation spectrum. - Multiband transport analysis relies on scaling collapses and effective parameters (η, A, n, m*), introducing uncertainties tied to their determination from separate measurements. - ARPES measurements are performed at finite temperatures and selected k-space cuts; full three-dimensional Fermi surface and possible kz dispersion effects are not exhaustively mapped.