Conventional and practical metallic superconductivity arising from repulsive Coulomb coupling

Jay D. Sau; Sankar Das Sarma; Shuyang Wang; Yi-Ting Tu

REVIEW 3 major objections 4 minor 2 cited by

Conventional and practical metallic superconductivity arising from repulsive Coulomb coupling

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper argues that purely repulsive Coulomb interactions cannot produce practical s-wave superconductivity in ordinary metals, and that published plasmon-mediated high-Tc claims are artifacts of uncontrolled theory.

desk verdict A strong critique of plasmon-mediated SC predictions that is right about uncontrolled approximations but overstates its zero-Tc conclusion by resting it on an arbitrary cutoff. read the letter →

arxiv 2511.00625 v3 pith:G5QZQXNR submitted 2025-11-01 cond-mat.supr-con

Sankar Das Sarma , Jay D. Sau , Yi-Ting Tu , Shuyang Wang This is my paper · ORCID

classification cond-mat.supr-con

keywords plasmon-mediatedsuperconductivityCoulomb-inducedpairingjelliummodelMigdaltheoremEliashbergequationsKohn-Luttingers-waverepulsiveCoulombcoupling

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to settle whether the repulsive Coulomb interaction between electrons in a simple metal can, by itself, make the metal a conventional s-wave superconductor at observable temperatures. Because electrons screen each other, the effective interaction is attractive in part of frequency-momentum space, so a naive BCS treatment with plasmons as the glue predicts Tc around 100–1000 K for nearly every metal, including alkali metals that are not superconductors. The paper argues that this prediction is an artifact: there is no energy-scale separation between electrons and plasmons to justify the standard BCS-Eliashberg-Migdal framework, and once the frequency cutoff is chosen small enough to respect Migdal's criterion, the Eliashberg equations yield vanishing or negligible Tc. The only credible Coulomb-driven pairing, the paper concludes, is the Kohn-Luttinger mechanism, which produces non-s-wave pairing with exponentially low Tc.

What carries the argument

The load-bearing object is Migdal's theorem, which says that vertex corrections to the electron-boson pairing interaction are negligible when the boson frequency is much smaller than the Fermi energy. The paper splits the RPA-screened Coulomb interaction W into a low-frequency part W<, kept in the Eliashberg gap equation up to a cutoff omega_c, and a high-frequency part W>, treated as a Coulomb pseudopotential. A variational bound on the lowest eigenvalue of the gap equation decides, for each cutoff omega_c and each momentum-shell width k_c, whether the electron gas is superconducting. The momentum shell k_c is introduced because the T = 0 RPA polarizability has singularities; the paper argu

What would settle it

A controlled numerical solution of the 3D jellium pairing problem at r_s ~ 2.8 with vertex corrections included and no ad hoc cutoff that yields a superconducting gap above the sub-kelvin scale would falsify the central negative claim. Experimentally, observation of high-Tc superconductivity in a simple alkali metal under conditions where phonons are clearly not the glue would also contradict it.

Watch

Extended reading notes

Core claim

The central claim is that conventional s-wave superconductivity induced purely by repulsive Coulomb coupling in regular 3D or 2D metals does not exist with a practical Tc. The dynamically screened Coulomb interaction does contain a large attractive region—the paper shows this explicitly for RPA, Hubbard-corrected RPA, plasmon-pole, and hydrodynamic screening—but the BCS-Eliashberg-Migdal framework cannot be trusted there because Migdal's theorem fails when the bosonic glue (the plasmon) has the same energy scale as the electrons. A variational and numerical solution of the Eliashberg gap equations with a frequency cutoff omega_c small enough to satisfy the Migdal criterion (omega_c ≲ 0.5 E_F

Load-bearing premise

The central conclusion assumes that the physical Eliashberg frequency cutoff omega_c must be chosen small enough to satisfy the Migdal criterion (omega_c ≲ 0.5 E_F in 3D and ≲ 0.1 E_F in 2D) and that all pairing from higher frequencies can be discarded; the paper itself states omega_c is a priori unknown, and if omega_c ~ E_F were legitimate, Fig. 9 shows the 3D electron gas at r_s = 2.8 would be superconducting.

Editorial extensions

If this is right

If the central claim is correct, existing calculations reporting high-Tc plasmon-induced s-wave superconductivity in jellium-like metals are uncontrolled and should not be cited as evidence for a non-phonon pairing mechanism.
Alkali metals such as Na and K should remain non-superconducting at ambient pressure, since they are close to ideal jellium systems; this is consistent with the paper's claim and inconsistent with naive plasmon-mediated pairing.
Future theories of superconductivity from purely electronic mechanisms must either show a controlled suppression of vertex corrections or include them; simply replacing a phonon propagator with a plasmon propagator and computing a BCS Tc is not sufficient.
The 2D electron gas is even less favorable: with a cutoff consistent with Migdal's criterion (omega_c ≲ 0.1 E_F), any s-wave superconducting Tc is negligible.
The Kohn-Luttinger mechanism, with exponentially low Tc and higher angular momentum pairing, remains the only believable Coulomb-only route to superconductivity.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves implicit that the same Migdal obstruction applies to any electronic bosonic glue whose energy is not small compared with the Fermi energy; acoustic plasmons in bilayer systems, spin fluctuations, and other collective electronic modes would face the same uncontrolled cutoff problem if treated within BCS-Eliashberg.
The variational treatment suggests but does not prove that a fully vertex-corrected calculation would also yield vanishing Tc; the authors themselves say only that practical superconductivity from this mechanism is unlikely.
A reader who takes the negative result seriously should next want a controlled numerical computation of the pairing susceptibility in the 3D jellium model at r_s around 2.8 with no ad hoc frequency cutoff, since Fig. 9 shows that this point is most sensitive to the cutoff choice.
The paper's logic also implies that claims of high-Tc superconductivity in strongly correlated lattice models cannot be validated by BCS-Eliashberg-type calculations; any real pairing there would more plausibly be a low-Tc Kohn-Luttinger-like effect, though the paper explicitly leaves the Hubbard-model question outside its scope.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my read. The paper's negative case against 'practical' plasmon-mediated s-wave superconductivity is largely convincing in its qualitative core, but the strong form of the claim — that any Coulomb-driven s-wave SC has vanishing Tc — is not proven. That gap matters.

What's actually new: the Eliashberg variational bound (Eqs. 29–42), the concrete condition Eq. (41) on the repulsive interaction, and the Tc-vs-cutoff maps in Fig. 10. Those are real additions to the older RPA/local-field literature. The BCS reductio is clean: plugging plasmon scales into the standard formula gives ~100–1000 K, which contradicts the fact that Na and K are not superconducting. The empirical point that alkali metals are ideal jellium but show no high-Tc SC is a powerful sanity check. The paper is also honest: it repeatedly states omega_c is a priori unknown, that k_c is an artifact of T=0 RPA, and that the 'absence' result is technically a lower bound on Tc.

The soft spot is the one you'd guess. The vanishing-Tc conclusion is loaded into the choice omega_c << E_F. Migdal's theorem being inapplicable doesn't tell you what the correct cutoff is; it just means vertex corrections are not negligible above some scale. If you take omega_c ~ E_F, the paper's own Fig. 9 shows the 3D gas at r_s = 2.8 satisfies the SC condition. The authors know this and say so, but then the Abstract's sweeping statement overstates what the analysis establishes. Also, because the variational calculation is only a lower bound on Tc, the claim that the T=0 system is not ordered for small omega_c is conditional, not a rigorous no-go. That doesn't make the paper wrong in its main point, but it makes the strong claim depend on an arbitrary parameter.

Who should read it: anyone proposing Coulomb-mediated SC, and anyone who wants a concise argument for why naive BCS with plasmon glue is suspect. It deserves a serious referee. My recommendation: send it out, but require the authors to separate what is proven (uncontrolled predictions of high Tc) from what is plausible (no practical Coulomb-driven s-wave SC in jellium). If they soften the abstract and emphasize the conditional nature, it can be a valuable reference.

Referee Report

3 major / 4 minor

Summary. The paper asks whether a conventional, experimentally observable s-wave superconducting state can arise in a 3D (or 2D) jellium metal from purely repulsive Coulomb electron-electron interactions. Sections II shows that the dynamically screened interaction has large attractive regions in (q,ω) space. Section III derives a naive BCS estimate Tc ~ 100–1000 K for metallic densities and argues this is absurd because alkali metals are not high-Tc superconductors. Section IV develops an Eliashberg treatment with a frequency cutoff ωc and momentum cutoff kc, and reports that superconductivity is absent for small ωc and physical kc, leading the authors to conclude that 'any plasmon induced s-wave metallic superconductivity has vanishing Tc' and that literature claims of practical plasmon-mediated superconductivity are not believable. The only surviving mechanism is stated to be the exponentially weak Kohn-Luttinger superconductivity.

Significance. The central claim, if established, would be significant: it would invalidate a large body of work proposing plasmon-mediated or Coulomb-coupling-driven practical s-wave superconductivity and would sharpen the role of Migdal's theorem in superconductivity theory. The paper's cleanest contribution is the BCS reductio ad absurdum: a straightforward BCS treatment with the screened Coulomb interaction gives Tc ≳ 100 K for all simple metals, which is empirically falsified by the absence of superconductivity in alkali metals. This argument is independent of the Eliashberg cutoff and is convincing as a demonstration that such BCS calculations are uncontrolled. The Eliashberg analysis is also a useful systematic study of how the result depends on cutoffs. However, the quantitative 'vanishing Tc' conclusion is not established, because it relies on choosing ωc ≪ EF, while the paper itself shows superconductivity at ωc ~ EF and admits that ωc is 'apriori unknown.' The paper therefore overstates its conclusion relative to what the analysis actually proves.

major comments (3)

[§IV A, Eqs. (26)–(27), and §IV C/E] The central quantitative claim—'no significant superconductivity below ωc < 0.5EF' in 3D and the final 'vanishing Tc'—is obtained only after choosing a small frequency cutoff ωc. The text states that ωc is 'apriori unknown' and that its choice is a 'balancing act.' Fig. 9 explicitly shows that the 3D electron gas at rs = 2.8 satisfies the superconductivity condition for ωc = EF, and the text notes this disappears only for ωc < 0.9EF. Since Migdal's theorem does not apply to electron-electron interactions, there is no first-principles justification for imposing ωc << EF. The paper therefore establishes that Eliashberg predictions are uncontrolled and that high-Tc claims are unreliable, but it does not establish that Tc vanishes. The abstract and conclusion should be reframed accordingly.
[§IV B, Eqs. (33)–(34), and §IV C] The 'absence of superconductivity' result is based on a variational estimate with a two-valued ansatz for the gap and on a lower-bound condition, as the text acknowledges: 'the result of absence of superconductivity is technically based on a lower-bound on Tc.' A lower bound on Tc cannot prove that Tc is zero; absence of a solution within a restricted variational space is not absence of superconductivity in the full Eliashberg equation. Since the paper's strong conclusion ('vanishing Tc', 'the T=0 system is not ordered') goes beyond what this method can show, the logical force of the negative claim should be limited to 'no superconductivity is found within the stated approximations.'
[§IV C, Fig. 9 and Fig. 10] The result is also sensitive to the momentum-shell cutoff kc, which is not a physical parameter but an artifact of using the T=0 RPA polarizability. The text notes that smaller kc is more favorable to superconductivity and that kc = 10^-6 kF would lead to most of the parameter range being superconducting; the restriction kc > 10^-4 kF is imposed to avoid artifacts. This dependence means the boundaries shown in Figs. 9 and 10 are not robust predictions of the theory, but rather consequences of the chosen regularization. This further weakens the claim that the electron gas is non-superconducting in 'much of the range' of rs. The paper should present these results as conditional on regularization choices, not as definitive evidence for vanishing Tc.

minor comments (4)

[Abstract] Typo: 'Migdall's theorem' should be 'Migdal's theorem'; 'reduction ad absurdum' should be 'reductio ad absurdum'.
[Fig. 9 caption and §IV C] The caption states the rs = 2.8 point superconducts for ωc ~ 0.8EF, while the main text says it disappears for ωc < 0.9EF. These numbers should be made consistent.
[General] The paper uses 'Migdal-Eliashberg' and 'Migdal-Eliashberg-Migdal' inconsistently; standard terminology is 'Eliashberg theory' or 'Migdal-Eliashberg theory'.
[§IV E, Fig. 10] The color scale in Fig. 10 is not fully described in the caption; it would help to state explicitly that the plotted quantity is log10(Tc/EF) and to explain the meaning of the black line in the caption text.

Circularity Check

1 steps flagged · score 4.0 of 10

Central 'vanishing Tc' claim is imposed by the apriori unknown cutoff omega_c; Fig. 9 shows SC for omega_c ~ E_F, so the strong conclusion is partly self-imposed rather than derived.

other [Abstract; Sec. IV C (Fig. 9); Sec. IV E; Sec. V]
"Using a careful analysis of the Eliashberg gap equations we find that the superconducting Tc of the 3D (or 2D) electron gas can be reduced well below ∼1 K depending on choices of frequency cut-off parameters that are introduced to satisfy Migdall's theorem but are apriori unknown. ... As shown in Fig. 9, the electron gas can satisfy the above superconductivity condition for only one of the values of rs = 2.8 considered for ωc = EF. Note that this SC for rs = 2.8 disappears for ωc < 0.9EF, since reducing ωc moves each point to lower U1ωc."

The abstract already states the result as a function of 'choices of frequency cut-off parameters ... apriori unknown': the Tc outcome is not fixed by the Hamiltonian, it is dialed by the free parameter ωc. When ωc is chosen at the electronic scale EF, the same Eliashberg condition (Fig. 9) gives superconductivity at rs = 2.8, and the paper concedes it disappears only below ωc < 0.9EF. The later statement 'there is no significant superconductivity below ωc < 0.5EF' is then promoted to 'any plasmon induced s-wave metallic superconductivity has vanishing Tc' (Sec. V). Thus the central negative prediction is not derived from independent physics but is the small-ωc choice itself, presented as a calculated result. The paper's own caveat that the absence result is 'technically based on a lower-bo

full rationale

Most of the paper's derivation chain is self-contained and non-circular. The attractive regimes of the screened Coulomb interaction are computed from standard Lindhard/RPA, plasmon-pole, and hydrodynamic formulas; the BCS estimate Tc ~ ω_p exp(-5/r_s) is an honest application of the textbook BCS formula to the electron-plasmon coupling; and the Eliashberg analysis is a variational/eigenvalue calculation with explicit equations. The empirical argument that alkali metals are not ~100 K superconductors is independent external falsification, and the appeal to the Kohn-Luttinger result is to an external 1965 theorem, not to the authors' own prior work. No self-citation is load-bearing. The one structurally circular element is the paper's headline conclusion: 'in all likelihood any plasmon induced s-wave metallic superconductivity has vanishing Tc' is not an output of the calculation but follows from choosing omega_c < 0.5 EF (3D) and < 0.1 EF (2D). The paper explicitly says omega_c is 'apriori unknown' and that low omega_c 'under-estimates Tc', and its own Fig. 9 shows the same formalism supports superconductivity at rs = 2.8 when omega_c = EF. Therefore the strong quantitative claim is partly self-imposed by an input parameter choice. The weaker claim — that uncritical BCS/Eliashberg predictions of high Tc from repulsive Coulomb interactions are uncontrolled — is well supported and is not circular.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No new particles, forces, or conserved quantities are introduced; 'plasmon' and the frequency/momentum cutoffs are existing concepts. The central quantitative result rests on two hand-selected cutoffs (omega_c, k_c), an ansatz for the gap shape, and standard RPA/Migdal domain assumptions.

free parameters (3)

omega_c / E_F (Eliashberg frequency cutoff) = chosen ad hoc; 1.0, 0.5, 0.1 in Figs. 9-10
Splits the interaction into low- and high-frequency parts (Eqs. 26-27). The central no-practical-Tc conclusion requires omega_c << E_F, and the paper states this cutoff is 'apriori unknown' (Sec. IV A).
k_c / k_F (momentum-shell width for the gap ansatz) = 10^-3 k_F in main results; 10^-2 to 10^-6 explored
Introduced because T=0 RPA has momentum singularities; Fig. 10 shows Tc depends strongly on k_c, and the paper restricts validity to k_c > Tc/vF.
omega_D (ansatz split scale in the two-valued gap) = arbitrary, < omega_c
Used in the two-valued gap ansatz (Eqs. 31-32) to derive the variational bound; not fixed by physical input, though the final numerical solutions avoid this parameter.

assumptions (5)

domain assumption RPA dielectric function correctly captures the dynamically screened Coulomb interaction in the electron gas at metallic densities.
All central results use Lindhard/RPA screening (Eqs. 5, 35-38); the paper argues RPA is the only controlled approximation and that local-field corrections are ad hoc (Sec. II).
domain assumption Inapplicability of Migdal's theorem to the electron-plasmon interaction implies vertex corrections are uncontrolled and that a low frequency cutoff is required.
The whole reductio argument relies on this (Secs. I, III, IVA). It is standard Migdal logic, but using it as a strict cutoff choice for Coulomb interactions is the paper's assumption.
domain assumption The jellium model (uniform electron gas with parabolic dispersion) is representative of normal metals for this question.
The paper generalizes from jellium to all metals via alkali metals as ideal jellium examples (Abstract, Sec. V).
ad hoc to paper The variational two-valued ansatz for the gap gives a reliable lower-bound estimate of Tc.
Eqs. 30-34 restrict the gap to two frequency values; validation relies on early SC literature [30,31,42] and the comparison in [43], not on a controlled expansion.
domain assumption T=0 RPA polarizability can be used to evaluate Tc, with thermal rounding accounted for only by the k_c > Tc/vF constraint.
Secs. IV C/E: the paper acknowledges the momentum singularities create unphysical k_c artifacts and that this limits validity.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Conventional and practical metallic superconductivity arising from repulsive Coulomb coupling." pith.science (2026). https://pith.science/paper/G5QZQXNR

@misc{pith2026251100625,
  author       = {Pith},
  title        = {Pith review of: Conventional and practical metallic superconductivity arising from repulsive Coulomb coupling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G5QZQXNR}},
  note         = {Machine review of arXiv:2511.00625}
}

abstract

A concrete question is discussed: Can there be conventional s-wave superconductivity in regular 3D (or 2D) metals, i.e., electrons in a jellium background, interacting via the standard Coulomb coupling? We are interested in 'practical' SC that can in principle be observed in experiments, so the $T=0$ ground state being SC is not of interest, or for that matter a $T_c$ which is exponentially small and therefore 'impractical' is also not of interest in the current work. We discuss both 2D and 3D cases, focusing mostly on the 3D case. We find that almost any theory based on the BCS-Migdal-Eliashberg paradigm, with some form of screened Coulomb coupling replacing the electron-phonon coupling in the BCS or Eliashberg theory, would uncritically predict absurdly high $T_c\sim100$ K for s-wave SC in all metals (including the alkali metals, which are well-described by the jellium model) arising from the unavoidable fact that the Fermi, plasmon, and Coulomb potential energy scales are all $>10^4$ K. Therefore, we conclude, based on reduction ad absurdum, that the violation of the venerable Migdal theorem in this problem is sufficiently disruptive that no significance can be attached to numerous existing theoretical publications in the literature claiming plasmon-induced (or other similar Coulomb coupling-induced) practical SC. Using a careful analysis of the Eliashberg gap equations we find that the $T_c$ of the 3D (or 2D) electron gas can be reduced well below $\sim1$ K depending on choices of frequency cut-off parameters that are introduced to satisfy Migdall's theorem but are apriori unknown. The only believable result is the one discovered 60 years ago by Kohn and Luttinger predicting non-s-wave SC arising from Friedel oscillations with exponentially low $T_c$. We provide several theoretical approaches using both BCS and Eliashberg theories and different screening models to make our point.

Figures

Figures reproduced from arXiv: 2511.00625 by the authors.

**Figure 2.** FIG. 2. Hubbard-corrected RPA Coulomb interaction [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3. Plasmon-pole approximated Coulomb interaction [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: FIG. 4. Hydrodyanmic approximated Coulomb interaction [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5. 2D RPA Coulomb interaction [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: FIG. 6. 2D plasmon pole approximated Coulomb interaction [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: FIG. 7. The scaling of the [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

**Figure 8.** Figure 8: FIG. 8. The scaling of the [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 9.** Figure 9: FIG. 9. Dimensionless repulsive interaction [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]

**Figure 10.** Figure 10: FIG. 10. Critical temperature [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Quasiparticle GW for Superconductors: Toward a Unified Treatment of Electron-Phonon and Electron-Plasmon Couplings
cond-mat.supr-con 2026-05 unverdicted novelty 7.0 of 10

The s-qpGW framework unifies electron-phonon and electron-plasmon couplings in a first-principles superconducting treatment, matches Eliashberg results for bulk metals, and correctly predicts no superconductivity in d...
Superconductivity from phonon-mediated retardation in a single-flavor metal
cond-mat.supr-con 2025-12 conditional novelty 7.0 of 10

Acoustic-phonon retardation alone can produce odd-parity p-wave superconductivity in a single-flavor metal where the static BCS approximation predicts no pairing.

Reference graph

Works this paper leans on

50 extracted references · cited by 2 Pith papers

[1]

Kohn and J

W. Kohn and J. M. Luttinger, Phys. Rev. Lett.15, 524 (1965)

1965
[2]

H. K. Onnes, inKNA W, Proceedings, V ol. 13 (1911) pp. 1274– 1276

1911
[3]

L. N. Cooper, Phys. Rev.104, 1189 (1956)

1956
[4]

Raghu, S

S. Raghu, S. A. Kivelson, and D. J. Scalapino, Phys. Rev. B81, 224505 (2010)

2010
[5]

Y .-Z. Chou, J. Zhu, and S. Das Sarma, Phys. Rev. B111, 174523 (2025)

2025
[6]

Ruvalds, Phys

J. Ruvalds, Phys. Rev. B35, 8869 (1987)

1987
[7]

Ishii and J

Y . Ishii and J. Ruvalds, Phys. Rev. B48, 3455 (1993)

1993
[8]

A. Bill, H. Morawitz, and V . Z. Kresin, Phys. Rev. B68, 144519 (2003)

2003

Show all 50 references

[9]

V . Z. Kresin and H. Morawitz, Phys. Rev. B37, 7854 (1988)

1988
[10]

P. B. Allen and B. Mitrovi ´c, inSolid State Physics, V ol. 37 (Cambridge, MA, 1983) pp. 1–92

1983
[11]

P. B. Allen, inHandbook of Superconductivity(Academic Press, New York, 1999) pp. 478–483

1999
[12]

Migdal, Sov

A. Migdal, Sov. Phys. JETP7, 996 (1958)

1958
[13]

Lindhard, Danske Matematisk-fysiske Meddelelser28, 1 (1954)

J. Lindhard, Danske Matematisk-fysiske Meddelelser28, 1 (1954)

1954
[14]

Bruus and K

H. Bruus and K. Flensberg,Many-Body Quantum Theory in Condensed Matter Physics: An Introduction(Oxford University Press, 2004)

2004
[15]

G. D. Mahan,Many-particle physics(Plenum Press,New York, NY , 1980)

1980
[16]

E. H. Hwang, R. E. Throckmorton, and S. Das Sarma, Phys. Rev. B98, 195140 (2018)

2018
[17]

J. R. Schrieffer,Theory of superconductivity(CRC press, 2018)

2018
[18]

A. L. Fetter and J. D. Walecka,Quantum Theory of Many- Particle Systems(Dover Publications, New York, 2003)

2003
[19]

Hubbard, Proceedings of the Royal Society A243, 336 (1957)

J. Hubbard, Proceedings of the Royal Society A243, 336 (1957)

1957
[20]

B. I. Lundqvist, Physik der kondensierten Materie6, 193 (1967)

1967
[21]

B. I. Lundqvist, Physik der kondensierten Materie6, 206 (1967)

1967
[22]

A. W. Overhauser, Phys. Rev. B3, 1888 (1971)

1971
[23]

Das Sarma and J

S. Das Sarma and J. J. Quinn, Phys. Rev. B20, 4872 (1979)

1979
[24]

Das Sarma, Phys

S. Das Sarma, Phys. Rev. B26, 6559 (1982)

1982
[25]

Stern, Phys

F. Stern, Phys. Rev. Lett.18, 546 (1967)

1967
[26]

A. V . Chubukov, Phys. Rev. B48, 1097 (1993)

1993
[27]

V . P. Antropov, O. Gunnarsson, and A. I. Liechtenstein, Phys. Rev. B48, 7651 (1993)

1993
[28]

P. B. Allen and R. C. Dynes, Phys. Rev. B12, 905 (1975)

1975
[29]

Carbotte, Reviews of Modern Physics62, 1027 (1990)

J. Carbotte, Reviews of Modern Physics62, 1027 (1990)

1990
[30]

N. N. Bogoliubov, V . Tolmachev, D. Shirkov, and R. B. Lindsay, Physics Today13, 44 (1960)

1960
[31]

Morel and P

P. Morel and P. Anderson, Physical Review125, 1263 (1962)

1962
[32]

Kirzhnits, E

D. Kirzhnits, E. Maksimov, and D. Khomskii, Journal of Low Temperature Physics10, 79 (1973)

1973
[33]

P. B. Allen, M. L. Cohen, and D. R. Penn, Physical Review B 38, 2513 (1988)

1988
[34]

Rietschel and L

H. Rietschel and L. Sham, Physical Review B28, 5100 (1983)

1983
[35]

C. A. Kukkonen and A. Overhauser, Physical Review B20, 550 (1979)

1979
[36]

Büche and H

T. Büche and H. Rietschel, Physical Review B41, 8691 (1990)

1990
[37]

Takada, Physical Review B47, 5202 (1993)

Y . Takada, Physical Review B47, 5202 (1993)

1993
[38]

X. Cai, T. Wang, N. V . Prokof’ev, B. V . Svistunov, and K. Chen, Physical Review B106, L220502 (2022)

2022
[39]

Shankar, Reviews of Modern Physics66, 129 (1994)

R. Shankar, Reviews of Modern Physics66, 129 (1994)

1994
[40]

Metzner, M

W. Metzner, M. Salmhofer, C. Honerkamp, V . Meden, and K. Schönhammer, Reviews of Modern Physics84, 299 (2012)

2012
[41]

Marsiglio and J

F. Marsiglio and J. Carbotte, inSuperconductivity: conventional and unconventional superconductors(Springer,
[42]

McMillan, Physical Review167, 331 (1968)

W. McMillan, Physical Review167, 331 (1968)

1968
[43]

Chubukov, N

A. Chubukov, N. V . Prokof’Ev, and B. V . Svistunov, Physical Review B100, 064513 (2019)

2019
[44]

Takada, Journal of the Physical Society of Japan45, 786 (1978)

Y . Takada, Journal of the Physical Society of Japan45, 786 (1978)

1978
[45]

Das Sarma and A

S. Das Sarma and A. Madhukar, Phys. Rev. B23, 805 (1981)

1981
[46]

Qin, C.-M

M. Qin, C.-M. Chung, H. Shi, E. Vitali, C. Hubig, U. Schollwöck, S. R. White, and S. Zhang (Simons Collaboration on the Many-Electron Problem), Phys. Rev. X10, 031016 (2020)

2020
[47]

Xu, C.-M

H. Xu, C.-M. Chung, M. Qin, U. Schollwöck, S. R. White, and S. Zhang, Science384, eadh7691 (2024)

2024
[48]

Y .-Z. Chou, F. Wu, J. D. Sau, and S. Das Sarma, Phys. Rev. Lett.127, 217001 (2021)

2021
[49]

Zhu, Y .-Z

J. Zhu, Y .-Z. Chou, M. Xie, and S. Das Sarma, Phys. Rev. B 111, L060501 (2025)

2025
[50]

T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys.54, 437 (1982)

1982

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Kohn and J

W. Kohn and J. M. Luttinger, Phys. Rev. Lett.15, 524 (1965)

1965

[2] [2]

H. K. Onnes, inKNA W, Proceedings, V ol. 13 (1911) pp. 1274– 1276

1911

[3] [3]

L. N. Cooper, Phys. Rev.104, 1189 (1956)

1956

[4] [4]

Raghu, S

S. Raghu, S. A. Kivelson, and D. J. Scalapino, Phys. Rev. B81, 224505 (2010)

2010

[5] [5]

Y .-Z. Chou, J. Zhu, and S. Das Sarma, Phys. Rev. B111, 174523 (2025)

2025

[6] [6]

Ruvalds, Phys

J. Ruvalds, Phys. Rev. B35, 8869 (1987)

1987

[7] [7]

Ishii and J

Y . Ishii and J. Ruvalds, Phys. Rev. B48, 3455 (1993)

1993

[8] [8]

A. Bill, H. Morawitz, and V . Z. Kresin, Phys. Rev. B68, 144519 (2003)

2003