Can first-principles calculations predict Tc in strongly correlated materials?

Scope

Material-specific first-principles calculations have made substantial progress, but the reviewed studies do not establish a generally reliable, uncertainty-calibrated prediction of $T_c$ across strongly correlated superconductors. A useful answer must identify the predicted observable: a pairing tendency, zero-temperature order, excitation gap, mean-field instability, or thermodynamic transition. These are different targets. Predicting a trend can be valuable without already predicting an absolute transition temperature. R4 R5 R7

Here, ab initio concerns how the Hamiltonian and interactions are constructed; it does not remove approximations from their solution. A calculation should disclose fitted inputs, the correlated orbital space, screened interactions, treatment of double counting, solver limitations and convergence. A declared absence of material-by-material fitting is distinct from a proof that the chosen functional or embedding is exact. R5 R6

What conventional calculations demonstrate

Sanna, Pellegrini and Gross developed a superconducting density-functional approximation informed by an Einstein-phonon Eliashberg reference system. Their final paper compares transition temperatures and gaps with measurements for selected conventional superconductors. The calculation uses phonons from density-functional perturbation theory and static-RPA Coulomb screening. Its constants are calibrated to the reference model, rather than fitted separately to each compound. This is a concrete computational achievement within stated approximations, not a universal benchmark for strongly correlated pairing. R1

The same paper distinguishes the auxiliary Kohn–Sham pairing potential from a physical excitation gap. Agreement of an approximate gap equation with a reference calculation must therefore be tied to the quantity actually compared. A plotted auxiliary gap should not silently become an experimental gap or a measured $T_c$ in a database. R1

Two cuprate advances and their boundaries

Cui and colleagues calculate ground-state pairing using ab initio density-matrix embedding, with a finite $2\times2$ impurity and primarily a coupled-cluster solver. They recover selected pressure and layer trends. Phonons and temperature effects are omitted; the reported auxiliary bulk gap is not the interacting bulk excitation gap. Their conversion to a temperature invokes a weak-coupling relationship whose applicability is explicitly qualified. Incomplete charge self-consistency, implicit doping and omitted long-wavelength orders remain limitations. This work supports progress in material-specific pairing calculations, rather than an independently established finite-temperature prediction protocol. R2

Bacq-Labreuil and colleagues combine charge-self-consistent DFT with cluster DMFT. Their selected author manuscript uses zero-temperature exact diagonalization with a discrete bath; a fictitious temperature sets a Matsubara mesh. Superconductivity is then evaluated in a post-processing step with other orbitals frozen in their normal state. The authors explicitly distinguish zero-temperature order from $T_c$ and leave calculation of the actual transition, including phase fluctuations, for further work. Thus the mesh temperature is not a computed transition. The local solver's name also does not make the finite-bath embedding exact for the entire material. R3

Mathematical assumptions

For a noninteracting block Hamiltonian, eliminating auxiliary orbitals gives the exact retained-orbital Green's function

$$G_{dd}(z)=\left[z-\epsilon_d-V^\dagger(z-H_p)^{-1}V\right]^{-1}.$$

Here $z$ is complex energy, $\epsilon_d$ a scalar energy, $H_p$ a Hermitian auxiliary block and $V$ a coupling column. The term $V^\dagger(z-H_p)^{-1}V$ retains frequency dependence. This algebraic identity explains one reason a static reduced Hamiltonian needs justification; it is not a derivation of an interacting pairing vertex. Choices of orbital space and retained energy window matter when constructing practical effective descriptions. R6

An original three-orbital illustration compares direct matrix inversion with this elimination for 60 parameter/frequency choices. They agree to numerical precision. Replacing the eliminated term by its zero-frequency value changes the selected Green's functions, with a largest relative difference of about 46% in this deliberately chosen set. That percentage is an illustrative error, not an error bar for either cuprate paper. All energies use arbitrary common units; no material, interaction or superconducting transition is simulated.

Even a correctly computed magnetic scale is not itself $T_c$. In a half-filled, one-orbital, two-site repulsive Hubbard example, the singlet–triplet separation is

$$J_{\rm dimer}=\frac{\sqrt{U^2+16t^2}-U}{2} =\frac{4t^2}{U}+O(t^4/U^3),\qquad U\gg |t|.$$

The previously checked finite-dimer calculation gives about 21% and 6% overestimation by the leading term at $U/t=4$ and $8$, respectively. Neither the exact finite-system splitting nor its asymptotic approximation establishes superconductivity. The example illustrates why a descriptor, its approximation and a bulk ordering temperature require separate validation. R8

Experimental criteria

A useful validation chain connects specified crystal structure and composition to electronic inputs, an adequately controlled many-body calculation and the intended experimental observable. Report convergence with orbital basis, cluster size, bath discretization or sampling where applicable; separate numerical convergence from errors in the physical model. Cross-method agreement on a matched Hamiltonian is evidence about that Hamiltonian, not automatic validation of its correspondence to a compound. R4 R5 R6

The temperature question adds another layer. A nonzero pairing amplitude or a mean-field instability can occur without the phase coherence required for a thermodynamic superconducting transition. Dimensionality, superfluid stiffness and long-distance fluctuations must be addressed in the applicable regime. A two-dimensional phase-stiffness bound is neither a three-dimensional universal formula nor a procedure that turns any calculated gap into $T_c$. R7

Limits and disputes

This selected review does not rank every available method, exclude successful calculations in particular correlated families, or independently reproduce the quoted ab initio studies. Final-version comparison of the cluster-DMFT author manuscript, complete supplements and broad material benchmarks remain open. The underlying methodological question is quantitative: how much do plausible changes in input structure, screening, correlated subspace and solver treatment move the predicted observable? A single successful comparison cannot answer this for an entire material class. R4 R5 R6

Discovery implications

Use calculations to generate testable candidates with an explicit evidence level. Store the structure, conditions, method and predicted quantity together; compare against measurements on the corresponding sample and distinguish onset, bulk magnetic and zero-resistance criteria. Favor prospective checks on withheld materials and preserve failed predictions. This is a proposed validation standard: the aim is increasingly reliable material prediction, while retaining the difference between a promising pairing trend and a demonstrated transition temperature. R4 R7

Related topics

Continue with theoretical tools, cuprates, unifying pairing mechanisms, and flat-band temperature bounds.

Material cases

No sample-specific material records are included in this article. Model results are not experimental transition temperatures.