What question did this study set out to answer?

Investigate the universality of the superconducting dome across different materials and establish a master curve.

May 8, 2026Open Access

The Universal 3/2 Master Curve of Unconventional Superconductivity: Cuprates, Iron-Pnictides, and Heavy Fermions Collapse onto a Single Quantum-Critical Phase Boundary

Key Points

Investigate the universality of the superconducting dome across different materials and establish a master curve.
Analyzed 84 data points from cuprates, iron-pnictides, and heavy fermions with independently measured tuning parameters.
Rescaled data using asymmetric dome half-width to collapse onto a single master curve.
Conducted bootstrap analysis to determine fitting parameters and validate findings.
Data points collapse onto a single master curve with a universal exponent β = 1.44±0.10, matching β = 3/2.
Confidence validation excludes β = 2 at 100%, confirming unique macroscopic phase boundary.
Establishes connection to a continuous transition in a d = 3, ν = 1/2 universality class, adjacent to a quantum critical point.

Abstract

For thirty years the Presland parabola Tc /Tc,max = 1 − 82.6 (p − 0.16)2 has been the unquestioned phenomeno-logical template of the cuprate superconducting dome, with analogous parabolic-in-tuning forms applied to iron-pnictides and heavy-fermion systems. We show three things. First, the apparent universality of the parabolicform is partly an artifact of circular calibration: in many cuprate families the hole-doping axis is itself derivedfrom Tc using the Presland relation, guaranteeing a perfect fit by construction. Second, restricting to mate-rials with independently measured tuning parameters across three microscopically distinct classes — cuprates(LSCO, Tl-2201, Bi-2212; doping p measured by stoichiometry/Hall/AMRO/ARPES), electron-doped iron-pnictides (BaFe2 As2 Co/Ni; x measured by WDS), and heavy fermions (CeCu2 Si2 , CeRhIn5 , CeCoIn5 , CeIrIn5 ,CeCoIn5 :Sn; tuned by hydrostatic pressure or chemical doping) — 84 data points collapse onto a single mastercurve when each system’s tuning axis is rescaled by its own asymmetric dome half-width. Third, the universalclosed formTc /Tc,max = max 0, 1 − 0.705 |u|3/2 , u ≡ (x − xopt )/Wfbeats the parabolic β = 2 form on the combined dataset in 82/100 random discovery/validation splits at ±0.10tolerance with sign-test p < 10−17 . Bootstrap analysis (N = 1000) yields β = 1.44±0.10 (1σ), with β = 3/2 insidethe band and β = 2 excluded at 100% confidence. The exponent β = 3/2 is the BCS–mean-field signature of acontinuous transition in a d = 3, ν = 1/2 universality class adjacent to a quantum critical point. Despite radicallydifferent microscopic origins — d-wave on doped Mott insulators (cuprates), multiband s± on Fermi surfaceswith hole and electron pockets (pnictides), and Cooper pairing from spin fluctuations near antiferromagneticquantum critical points (heavy fermions) — all three classes share a single macroscopic phase boundary. Wepre-register a falsifiable prediction for magic-angle twisted bilayer graphene: if its dome obeys β = 3/2, it fallswithin this universality class.

The Universal 3/2 Master Curve of Unconventional Superconductivity: Cuprates, Iron-Pnictides, and Heavy Fermions Collapse onto a Single Quantum-Critical Phase Boundary

Key Points

Abstract

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