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

Separate material-specific pairing trends from transition-temperature predictions, and examine embedding, solver and phase-coherence limits.

## 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](#ref-source-51a15c56a9ac1472) [R5](#ref-source-12f2513b257d7194) [R7](#ref-source-a57ef549f45fe74c)

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](#ref-source-12f2513b257d7194) [R6](#ref-source-2b331df83384bc5c)

## 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](#ref-source-f9aa25c4ae4f8b5d)

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](#ref-source-f9aa25c4ae4f8b5d)

## 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](#ref-source-809aa49ee911889b)

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](#ref-source-0fac414ae312f585)

## 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](#ref-source-2b331df83384bc5c)

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](#ref-source-2f6a1c773323226a)

## 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](#ref-source-51a15c56a9ac1472) [R5](#ref-source-12f2513b257d7194) [R6](#ref-source-2b331df83384bc5c)

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](#ref-source-a57ef549f45fe74c)

## 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](#ref-source-51a15c56a9ac1472) [R5](#ref-source-12f2513b257d7194) [R6](#ref-source-2b331df83384bc5c)

## 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](#ref-source-51a15c56a9ac1472) [R7](#ref-source-a57ef549f45fe74c)

## Related topics

Continue with [theoretical tools](/scwiki/theoretical-tools), [cuprates](/scwiki/cuprates), [unifying pairing mechanisms](/scwiki/open-questions/n2), and [flat-band temperature bounds](/scwiki/open-questions/n10).

## Material cases

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

## Sources

### Ref source-f9aa25c4ae4f8b5d

[R1] Combining Eliashberg Theory with Density Functional Theory for the Accurate Prediction of Superconducting Transition Temperatures and Gap Functions (2020). [DOI 10.1103/PhysRevLett.125.057001](https://doi.org/10.1103/PhysRevLett.125.057001). Publication: journal-article. Check: selected-final-functional-and-benchmark-check. Checked: 2026-10-19. Published 27 July 2020. Selected final pp. 1–5; p. 4 equations and Figures 2–3 visually checked. Model calibration, static RPA and KS-gap limits retained. No supplement or full computational reproduction.
### Ref source-809aa49ee911889b

[R2] Ab initio quantum many-body description of superconducting trends in the cuprates (2025). [DOI 10.1038/s41467-025-56883-x](https://doi.org/10.1038/s41467-025-56883-x). Publication: journal-article. Check: selected-final-ground-state-methods-and-limits. Checked: 2026-10-19. Selected final pp. 1–5; p. 3 method and limitations visually checked. Auxiliary gap, finite-cell and charge-self-consistency limits retained. Full supplement and solver reproduction remain open.
### Ref source-0fac414ae312f585

[R3] Toward an Ab Initio Theory of High-Temperature Superconductors: A Study of Multilayer Cuprates (2025). [DOI 10.1103/PhysRevX.15.021071](https://doi.org/10.1103/PhysRevX.15.021071). Publication: journal-article. Check: selected-author-solver-postprocessing-and-temperature-check. Checked: 2026-10-19. Journal published 28 May 2025. Author v2 submitted 7 March 2025: selected pp. 9–12 and 28–29; p. 11 visually checked. Fictitious temperature and post-processing restrictions retained. Final-version comparison, complete appendices and reproduction remain open.
### Ref source-51a15c56a9ac1472

[R4] Solutions of the Two-Dimensional Hubbard Model: Benchmarks and Results from a Wide Range of Numerical Algorithms (2015). [DOI 10.1103/PhysRevX.5.041041](https://doi.org/10.1103/PhysRevX.5.041041). Publication: journal-article. Check: selected-published-qmc-dmrg-convergence-check. Checked: 2026-10-19. Published 14 December 2015. Selected pages 1–4, 7–8 and 10; the DMRG convergence discussion on page 8 visually checked. The review retains constrained-path and particle-hole conditions, finite-cylinder limits and method-specific errors. Full tables, datasets and all algorithms have not been reproduced.
### Ref source-12f2513b257d7194

[R5] Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions (1996). [DOI 10.1103/RevModPhys.68.13](https://doi.org/10.1103/RevModPhys.68.13). Publication: journal-article. Check: selected-published-self-consistency-and-limits-check. Checked: 2026-10-19. Selected printed pages 15–18 and 20; equations 6–13 on page 18 visually checked. Local temporal fluctuations, bath self-consistency, large-coordination scaling and atomic/noninteracting limits were reviewed. An independent four-state atomic Green function was checked; no interacting impurity solver or lattice DMFT calculation was reproduced.
### Ref source-2b331df83384bc5c

[R6] Maximally localized Wannier functions: Theory and applications (2012). [DOI 10.1103/RevModPhys.84.1419](https://doi.org/10.1103/RevModPhys.84.1419). Publication: journal-article. Check: selected-author-gauge-window-interpolation-check. Checked: 2026-10-19. Author version 2, dated 12 May 2012; journal publication 10 October 2012. Selected pages 4–5, 11–13 and 36–37; Figure 5 and the frozen-window explanation on page 13 visually checked. Gauge versus subspace selection, Chern obstruction and interpolation conditions were reviewed. The independent finite-range Fourier example is not a material wannierization. Full final-version comparison remains open.
### Ref source-a57ef549f45fe74c

[R7] Bounds on the Superconducting Transition Temperature: Applications to Twisted Bilayer Graphene and Cold Atoms (2019). [DOI 10.1103/PhysRevX.9.031049](https://doi.org/10.1103/PhysRevX.9.031049). Publication: journal-article. Check: selected-final-bounds-and-anisotropy. Checked: 2026-10-19. Final published 17 September 2019. Selected pp. 2–6 and 10–11; p. 3 equations and conditions visually checked. Peierls coupling, BKT thermodynamic limit, parabolic density convention, monotonicity assumption and input-dependent multiband estimates retained. Appendices A–G and original QMC data not fully reproduced.
### Ref source-2f6a1c773323226a

[R8] Doping a Mott insulator: Physics of high-temperature superconductivity (2006). [DOI 10.1103/RevModPhys.78.17](https://doi.org/10.1103/RevModPhys.78.17). Publication: journal-article. Check: earlier-author-manuscript-selected-passages-check. Checked: 2026-10-18. Selected manuscript pp.27–28 projected Hilbert space and constraint, p.33 Gaussian response and experimental mismatch, p.45 representation versus emergent gauge group. Page33 equations visually inspected. Full final-version comparison pending.

## Claim evidence and conditions

### scwiki-n3-claim-scope-1

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.

Evidence: hypothesis. Sources: source-51a15c56a9ac1472, source-12f2513b257d7194, source-a57ef549f45fe74c.

Conditions: Keep zero-temperature order, auxiliary gap, numerical mesh temperature and actual material transition distinct. Retain version, orbital space, solver and phase-fluctuation limits.

Verification scope: Selected primary passages/figures, named inherited reviews and a finite normal-state block-elimination check. No many-body/material calculation reproduced.

### scwiki-n3-claim-scope-2

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.

Evidence: hypothesis. Sources: source-12f2513b257d7194, source-2b331df83384bc5c.

Conditions: Keep zero-temperature order, auxiliary gap, numerical mesh temperature and actual material transition distinct. Retain version, orbital space, solver and phase-fluctuation limits.

Verification scope: Selected primary passages/figures, named inherited reviews and a finite normal-state block-elimination check. No many-body/material calculation reproduced.

### scwiki-n3-claim-what-conventional-calculations-demonstrate-1

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.

Evidence: theory. Sources: source-f9aa25c4ae4f8b5d.

Conditions: Keep zero-temperature order, auxiliary gap, numerical mesh temperature and actual material transition distinct. Retain version, orbital space, solver and phase-fluctuation limits.

Verification scope: Selected primary passages/figures, named inherited reviews and a finite normal-state block-elimination check. No many-body/material calculation reproduced.

### scwiki-n3-claim-what-conventional-calculations-demonstrate-2

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.

Evidence: theory. Sources: source-f9aa25c4ae4f8b5d.

Conditions: Keep zero-temperature order, auxiliary gap, numerical mesh temperature and actual material transition distinct. Retain version, orbital space, solver and phase-fluctuation limits.

Verification scope: Selected primary passages/figures, named inherited reviews and a finite normal-state block-elimination check. No many-body/material calculation reproduced.

### scwiki-n3-claim-two-cuprate-advances-and-their-boundaries-1

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.

Evidence: theory. Sources: source-809aa49ee911889b.

Conditions: Keep zero-temperature order, auxiliary gap, numerical mesh temperature and actual material transition distinct. Retain version, orbital space, solver and phase-fluctuation limits.

Verification scope: Selected primary passages/figures, named inherited reviews and a finite normal-state block-elimination check. No many-body/material calculation reproduced.

### scwiki-n3-claim-two-cuprate-advances-and-their-boundaries-2

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.

Evidence: theory. Sources: source-0fac414ae312f585.

Conditions: Keep zero-temperature order, auxiliary gap, numerical mesh temperature and actual material transition distinct. Retain version, orbital space, solver and phase-fluctuation limits.

Verification scope: Selected primary passages/figures, named inherited reviews and a finite normal-state block-elimination check. No many-body/material calculation reproduced.

### scwiki-n3-claim-mathematical-assumptions-3

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.

Evidence: theory. Sources: source-2b331df83384bc5c.

Conditions: Keep zero-temperature order, auxiliary gap, numerical mesh temperature and actual material transition distinct. Retain version, orbital space, solver and phase-fluctuation limits.

Verification scope: Selected primary passages/figures, named inherited reviews and a finite normal-state block-elimination check. No many-body/material calculation reproduced.

### scwiki-n3-claim-mathematical-assumptions-7

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.

Evidence: theory. Sources: source-2f6a1c773323226a.

Conditions: Keep zero-temperature order, auxiliary gap, numerical mesh temperature and actual material transition distinct. Retain version, orbital space, solver and phase-fluctuation limits.

Verification scope: Selected primary passages/figures, named inherited reviews and a finite normal-state block-elimination check. No many-body/material calculation reproduced.

### scwiki-n3-claim-experimental-criteria-1

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.

Evidence: theory. Sources: source-51a15c56a9ac1472, source-12f2513b257d7194, source-2b331df83384bc5c.

Conditions: Keep zero-temperature order, auxiliary gap, numerical mesh temperature and actual material transition distinct. Retain version, orbital space, solver and phase-fluctuation limits.

Verification scope: Selected primary passages/figures, named inherited reviews and a finite normal-state block-elimination check. No many-body/material calculation reproduced.

### scwiki-n3-claim-experimental-criteria-2

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 $T_c$.

Evidence: theory. Sources: source-a57ef549f45fe74c.

Conditions: Keep zero-temperature order, auxiliary gap, numerical mesh temperature and actual material transition distinct. Retain version, orbital space, solver and phase-fluctuation limits.

Verification scope: Selected primary passages/figures, named inherited reviews and a finite normal-state block-elimination check. No many-body/material calculation reproduced.

### scwiki-n3-claim-limits-and-disputes-1

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.

Evidence: hypothesis. Sources: source-51a15c56a9ac1472, source-12f2513b257d7194, source-2b331df83384bc5c.

Conditions: Keep zero-temperature order, auxiliary gap, numerical mesh temperature and actual material transition distinct. Retain version, orbital space, solver and phase-fluctuation limits.

Verification scope: Selected primary passages/figures, named inherited reviews and a finite normal-state block-elimination check. No many-body/material calculation reproduced.

### scwiki-n3-claim-discovery-implications-1

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.

Evidence: hypothesis. Sources: source-51a15c56a9ac1472, source-a57ef549f45fe74c.

Conditions: Keep zero-temperature order, auxiliary gap, numerical mesh temperature and actual material transition distinct. Retain version, orbital space, solver and phase-fluctuation limits.

Verification scope: Selected primary passages/figures, named inherited reviews and a finite normal-state block-elimination check. No many-body/material calculation reproduced.

## Approved scope items

N3. Checked: 2026-10-19. Selected SCDFT final methods/benchmark, two cuprate embedding studies with explicit temperature limits, 60 independent block-elimination examples and previously checked finite-dimer arithmetic.

- Complete scientific acceptance remains open.
- The cluster-DMFT author manuscript still needs a full final-version comparison; complete supplements and raw computations were not reproduced.
- This is a selected methodological review, not an exhaustive benchmark of all strongly correlated superconductors.
- Finite normal-state matrices and dimer splittings do not predict material Tc.

## Revision and review

Article ID: scwiki-n3. Revision: 6332bc1ae0a7931b9111. Updated: 2026-10-18T20:28:26.066288+00:00.

Review scope: Selected SCDFT final methods/benchmark, two cuprate embedding studies with explicit temperature limits, 60 independent block-elimination examples and previously checked finite-dimer arithmetic.

- Complete scientific acceptance remains open.
- The cluster-DMFT author manuscript still needs a full final-version comparison; complete supplements and raw computations were not reproduced.
- This is a selected methodological review, not an exhaustive benchmark of all strongly correlated superconductors.
- Finite normal-state matrices and dimer splittings do not predict material Tc.

Source: [02_BATCH_2_IN_PLAN.md](https://github.com/JackZH26/jzis-superconductivity/blob/0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf/02_BATCH_2_IN_PLAN.md), commit 0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf, SHA-256 502293e5ed2fe8463b61a6c5b78a2c33bc33f3265bd36fe534fa2c469e42ff48.