A Predictive Theory of Cuprate Superconductivity

Explain how superconductivity emerges from the cuprates’ unusual normal state, and predict their phase diagram from a microscopic theory.

Problem 01OpenUpdated Nov 2026

SuperconductivityCondensed matter physicsMany-body theory

Written with inputs from

Prof. Andrew MillisProfessor of Physics, Columbia University
Prof. Inna VishikAssociate Professor of Physics, UC Davis

…and condensed matter physicists who prefer to remain anonymous.

A Copper–Oxygen Plane

Superconductors are remarkable materials that conduct electricity without resistance and expel magnetic fields. Most conventional superconductors exhibit these properties only at extremely low temperatures, typically within a few tens of degrees of absolute zero. However, the discovery of copper-based superconductors, known as cuprates, four decades ago challenged this picture. These materials can remain superconducting at temperatures above 100 K, far higher than previously thought possible.

Developing superconductors that operate at even higher temperatures, ideally near room temperature and ambient pressure, could transform energy transmission, transportation, medical imaging, and computing. Yet despite decades of research, we still lack a widely accepted microscopic theory that quantitatively explains how superconductivity emerges in cuprates. This limits our ability to predict which materials will superconduct and at what temperatures, leaving the search for new superconductors heavily dependent on experimental trial and error.

Every cuprate superconducts in the same place: copper–oxygen planes, with holes (or, in a few families, electrons) doped onto them. The oxygen orbitals are where the holes live, bound to a copper spin as a Zhang–Rice singlet; the one-orbital Hubbard model throws the oxygens away and keeps an effective site with hopping t and repulsion U.

The problem

Develop a predictive theory of cuprate superconductivity that explains existing experimental observations and provides a foundation for discovering new high-temperature superconductors.

Benchmark

Schematic, not to scale. The arrangement of antiferromagnetism, d-wave superconductivity, pseudogap, strange-metal and Fermi-liquid behavior is what the theory must predict; comparing LSCO, YBCO and the Bi-, Hg- and Tl-based families establishes which features are shared.

We assess theories on their ability to explain, simultaneously and quantitatively, the following open results, among others:

#	Experimental target	What is experimentally observed	What the theory must predict
1	Superconducting phase diagram	The superconducting transition temperature T_c forms a dome as a function of doping, with its maximum and shape varying across cuprate families [1, 2].	T_c(p), including the onset of superconductivity, optimal doping, maximum T_c, and the disappearance of superconductivity upon overdoping.
2	Superconducting gap symmetry	Phase-sensitive Josephson experiments establish predominantly d-wave superconductivity, with nodes along the Brillouin-zone diagonals.	The symmetry of the superconducting order parameter, including its sign structure and nodal locations.
3	Superconducting gap magnitude	ARPES and STM measure the superconducting energy gap and its dependence on momentum, temperature, and doping.	The superconducting gap Δ(k, T, p), including its magnitude and momentum dependence.
4	Pseudogap in electronic spectra	ARPES and STM observe a suppression of low-energy electronic spectral weight above T_c in underdoped cuprates. The pseudogap evolves with temperature and doping [3, 4].	The single-particle spectral function A(k, ω), pseudogap energy scale, and their dependence on temperature and doping.
5	Strange-metal electrical transport	The in-plane resistivity is approximately linear in temperature over broad regimes in several cuprates, sometimes extending toward very low temperatures when superconductivity is suppressed [5].	The electrical resistivity ρ(T, p), including its magnitude, temperature dependence, and evolution with doping.
6	Electronic specific heat and entropy	Calorimetry reveals a suppression of electronic entropy in the pseudogap regime. A sharp specific-heat anomaly is generally absent at the operational pseudogap onset temperature T* [6, 7, 8].	The electronic specific heat C_el(T, p) and entropy S_el(T, p), including the magnitude of entropy suppression.
7	Superfluid density	Penetration-depth measurements establish the magnitude and temperature dependence of the superfluid density, with strong doping dependence.	The superfluid density ρ_s(T, p), or equivalently the magnetic penetration depth λ(T, p).
8	Antiferromagnetic order	Undoped and lightly doped cuprates exhibit antiferromagnetic order, which is suppressed with increasing hole doping [1].	The antiferromagnetic ordered moment and doping dependence of magnetic order, including the ordering temperature where applicable.
9	Magnetic excitation spectra	Neutron scattering and RIXS reveal strong magnetic excitations across a wide doping range, including dispersive paramagnons that persist into overdoped samples [9, 10, 13].	The dynamical spin structure factor S(q, ω), including excitation energies, linewidths, and spectral weights.
10	Charge-density-wave order	X-ray scattering detects charge-density-wave correlations in several cuprate families, with characteristic wavevectors, correlation lengths, and doping dependence.	The charge structure factor S_c(q, ω), including ordering wavevectors, correlation lengths, and their evolution with temperature and doping.
11	Symmetry-breaking signatures near the pseudogap	Resonant ultrasound measurements report anomalies near T*, including around 245 K in YBa₂Cu₃O_6.60. Their interpretation as evidence of a thermodynamic phase transition remains debated [8, 11].	The relevant elastic response, such as changes in elastic constants as a function of temperature. If the theory predicts broken symmetry, it must identify the order parameter and associated signatures.
12	Local electronic structure and pairing	STM measurements reveal spatial correlations between local charge-transfer energy and superconducting pair density [12].	The relationship between local electronic parameters, such as charge-transfer energy, and the resulting superconducting properties, including whether charge-transfer superexchange sets the pairing.

Submissions

We assess submissions on their ability to quantitatively explain multiple independent experimental observations within a consistent microscopic framework.

Submissions should include:

Method: A description of the proposed theory and its underlying scientific principles.
Implementation: Working code, or derivations sufficient for a third party to reproduce the calculations, together with the predictions for the experimental targets above.

Submissions must clearly distinguish experimental observations used to determine model parameters from those predicted independently.

Evaluation considers:

Predictive accuracy: Agreement between theoretical predictions and experimental measurements, including the magnitudes and dependence of observables on temperature and doping.
Breadth of explanation: The number and diversity of experimental phenomena explained by the theory, with particular emphasis on superconductivity, magnetic correlations, and the pseudogap.
Physical consistency: Whether the predictions emerge from a consistent microscopic Hamiltonian and a physically justified parameterization, without independently fitting each experimental observable.
Novelty and scientific significance: The extent to which the theory advances beyond existing microscopic descriptions of cuprate superconductivity.
Reproducibility: The ability to independently verify the reported calculations and predictions.

References

1C. Proust and L. Taillefer, The remarkable underlying ground states of cuprate superconductors, Annual Review of Condensed Matter Physics 10, 409 (2019) doi:10.1146/annurev-conmatphys-031218-013210
2D. Rybicki, M. Jurkutat, S. Reichardt, C. Kapusta and J. Haase, Perspective on the phase diagram of cuprate high-temperature superconductors, Nature Communications 7, 11413 (2016) doi:10.1038/ncomms11413
3A. Damascelli, Z. Hussain and Z.-X. Shen, Angle-resolved photoemission studies of the cuprate superconductors, Reviews of Modern Physics 75, 473 (2003) doi:10.1103/RevModPhys.75.473
4S.-D. Chen et al., Incoherent strange metal sharply bounded by a critical doping in Bi2212, Science 366, 1099 (2019) doi:10.1126/science.aaw8850
5A. Legros et al., Universal T-linear resistivity and Planckian dissipation in overdoped cuprates, Nature Physics 15, 142 (2019) doi:10.1038/s41567-018-0334-2
6J. L. Tallon and J. W. Loram, The doping dependence of T* – what is the real high-T_c phase diagram?, Physica C 349, 53 (2001) doi:10.1016/S0921-4534(00)01524-0
7J. W. Loram, J. Luo, J. R. Cooper, W. Y. Liang and J. L. Tallon, Evidence on the pseudogap and condensate from the electronic specific heat, Journal of Physics and Chemistry of Solids 62, 59 (2001) doi:10.1016/S0022-3697(00)00101-3
8J. R. Cooper, J. W. Loram, I. Kokanović, J. G. Storey and J. L. Tallon, Pseudogap in YBa₂Cu₃O_6+δ is not bounded by a line of phase transitions: Thermodynamic evidence, Physical Review B 89, 201104(R) (2014) doi:10.1103/PhysRevB.89.201104
9M. Le Tacon et al., Intense paramagnon excitations in a large family of high-temperature superconductors, Nature Physics 7, 725 (2011) doi:10.1038/nphys2041
10T. Dahm et al., Strength of the spin-fluctuation-mediated pairing interaction in a high-temperature superconductor, Nature Physics 5, 217 (2009) doi:10.1038/nphys1180
11A. Shekhter et al., Bounding the pseudogap with a line of phase transitions in YBa₂Cu₃O_6+δ, Nature 498, 75 (2013) doi:10.1038/nature12165
12S. M. O’Mahony et al., On the electron pairing mechanism of copper-oxide high temperature superconductivity, Proceedings of the National Academy of Sciences 119, e2207449119 (2022) doi:10.1073/pnas.2207449119
13M. P. M. Dean et al., Persistence of magnetic excitations in La_2−xSr_xCuO₄ from the undoped insulator to the heavily overdoped non-superconducting metal, Nature Materials 12, 1019 (2013) doi:10.1038/nmat3723

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