Pith. sign in

REVIEW 2 major objections 6 minor 1 cited by

Quantum Critical Eliashberg Theory

T0 review · 2 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read At low energies, quantum-critical Eliashberg theory and holographic superconductivity are the same theory, with the Cooper pair's relative time playing the role of an extra dimension.

desk verdict A useful review of the YSYK program, but the headline claim that Eliashberg theory and holographic superconductivity are identical at low energies is only demonstrated near Delta=1/4, not at the model's physical Delta≈0.42. read the letter →

arxiv 2506.11952 v1 pith:FGR2HKB5 submitted 2025-06-13 cond-mat.str-el

Ilya Esterlis , Joerg Schmalian This is my paper · ORCID
classification cond-mat.str-el
keywords QuantumcriticalityNon-FermiliquidEliashbergtheoryYukawa-SYKmodelHolographicsuperconductivityCooperpairingwithoutquasiparticlesStrangemetalLarge-Nlimit
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

This review argues that a family of exactly solvable large-N models, the Yukawa-SYK models, provides a controlled microscopic description of quantum-critical metals and their superconductivity. The paper's central claim is that the large-N saddle point of these models is Eliashberg theory, now with self-consistently dressed bosons and electrons, and that at low energies this Eliashberg theory is identical to holographic superconductivity. In that identity, the holographic scalar field is the Cooper pair, and the extra dimension of the gravitational description encodes the relative-time dynamics of the two electrons that form the pair. A sympathetic reader should care because the claim would unify three usually separate frameworks—strong-coupling superconductivity, SYK-style non-Fermi liquids, and gauge-gravity duality—into one theory of strange-metal superconductors, while giving explicit microscopic meaning to otherwise abstract ingredients of holographic models.

What carries the argument

The load-bearing object is the Yukawa-SYK model: fermions with $N$ flavor indices coupled to $M$ bosons through Gaussian-random Yukawa couplings, solved in the large-$N$ limit with $M/N$ fixed. The exact solution is organized by bilocal collective fields whose saddle point gives the Eliashberg equations (the paper's Eqs. 5-7): a normal self-energy $\Sigma$, an anomalous self-energy $\Phi$, and a boson self-energy $\Pi$ that dresses the boson propagator. Because the same singular boson self-energy produces both the non-Fermi-liquid damping and the pairing interaction, the equations describe Cooper pairing without quasiparticles. In the critical regime the gap equation reduces to the universal '$\gamma$-model' form with $\gamma = 4\Delta - 1$, and the holographic map is implemented by a Radon transform along geodesics of AdS2, $F((\tau_1+\tau_2)/2,\epsilon) = \int_\Gamma |\epsilon|^{(\gamma-1)/2}\,\psi(\tau,\zeta)\,dl$, which converts the Gaussian pairing action into the action of a holographic superconductor in Poincaré coordinates.

What would settle it

Perform numerically exact determinant quantum Monte Carlo on the YSYK quantum dot at strong coupling and compare the exact spectral function, pairing susceptibility, and ground-state order with the predictions of Eqs. 5-7: if the exact solution exhibits replica-symmetry-broken glassy order instead of the predicted superconducting state with g-independent Tc, the Eliashberg-large-N description—and hence the low-energy identity with holographic superconductivity—fails in precisely that regime.

Watch

Extended reading notes

Core claim

The paper's central discovery is that quantum-critical Eliashberg theory and holographic superconductivity are low-energy reformulations of the same theory. Starting from a zero-dimensional Yukawa-coupled SYK quantum dot with Gaussian-random couplings, the authors show that the replica trick and a saddle point over bilocal collective fields produce a closed set of Eliashberg equations for the normal and anomalous self-energies plus a self-consistent boson self-energy; the same structure re-emerges in higher dimensions. In the quantum-critical regime the linearized gap equation takes a scale-invariant power-law form, and a change of variables recasts it as a Klein-Gordon equation in two-dimensional anti-de Sitter space, with the onset of pairing occurring exactly at the Breitenlohner-Freedman bound. The paper further shows that finite temperature corresponds to an AdS2 black-hole metric with horizon set by temperature, and that a chemical potential maps to a boundary electric field experienced by a charge-2e scalar. The normal-state logic also yields a non-Fermi-liquid spectrum in the dot, a quantum-critical fan in two dimensions, and, with spatially random Yukawa couplings, the marginal-Fermi-liquid and T-linear resistivity phenomenology of strange metals.

Load-bearing premise

The entire program rests on the assumption that the replica-diagonal, large-N saddle point gives the true low-energy ground state and pairing physics; exact simulations show signs of glassy behavior at strong coupling that could break the saddle point.

Editorial extensions

If this is right

  • Superconductivity can emerge from a normal state with no quasiparticles: at weak coupling the transition temperature is a power law in the coupling rather than exponentially small, and at strong coupling it saturates to a value of order $0.1\,\omega_0$, independent of coupling.
  • The superconducting state at strong coupling is a strongly interacting Cooper-pair fluid, characterized by gap-filling spectra, small Bogoliubov quasiparticle weight, and high sensitivity to pair-breaking disorder.
  • In two dimensions the same large-$N$ solution reproduces known quantum-critical results, for example $\Sigma \sim |\omega|^{2/3}$ at an Ising-nematic critical point, and with spatially random Yukawa couplings it produces a marginal Fermi liquid with $T$-linear resistivity and approximate $\omega/T$ scaling of the optical scattering rate.
  • The holographic dual is explicit: the extra dimension encodes the relative-time dynamics of the Cooper pair, the finite-temperature geometry is an AdS2 black hole with horizon $\zeta_T = 1/(2\pi T)$, and the effective scalar charge is $e^* = 2e$.
  • Because the linearized gap equation is shared across many quantum-critical systems, the YSYK formulation unifies those systems and makes the Eliashberg equations an exact large-$N$ statement rather than an approximation.

Reading between the lines

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

  • If the low-energy equivalence is exact, the holographic description inherits the restrictions of the large-$N$ saddle point: in strong-coupling regimes where exact simulations indicate glassy behavior, the dual geometry may describe an unstable or unphysical state rather than the true ground state.
  • The same Radon-transform derivation should apply to any quantum-critical superconductor whose gap equation is of the $\gamma$-model form, making the AdS2 description a universal statement about the pairing-fluctuation sector; deriving the dual geometry for a two-dimensional spin-density-wave critical point would test this directly.
  • One testable extension is to compute the quartic term in the dual scalar action directly from the YSYK bilocal action and compare it with the holographic superconductor action: agreement to that order would strengthen the identity beyond the Gaussian level, while a mismatch would show the equivalence is only asymptotic.
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.

Referee Report

2 major / 6 minor

Summary. This review article surveys quantum critical Eliashberg theory as realized in Yukawa-SYK (YSYK) models. It covers the (0+1)-dimensional quantum dot, its large-N saddle-point solution leading to Eliashberg equations, the normal non-Fermi liquid and superconducting phases, the extension to two-dimensional models with clean and spatially disordered Yukawa couplings, and a proposed explicit mapping between the linearized Eliashberg gap equation and holographic superconductivity in AdS2. The central claim is that at low energies quantum-critical Eliashberg theory and holographic superconductivity are identical, with the holographic scalar field representing the Cooper pair and the extra dimension encoding relative-time dynamics. The review is unusually candid about limitations, including the breakdown of the large-N replica-diagonal approach at strong coupling and the Gaussian-level character of the holographic mapping.

Significance. If the holographic identification holds, the paper provides a concrete microscopic bridge between a controlled large-N quantum many-body model and AdS2 holographic superconductivity, giving physical meaning to the extra dimension and the scalar field. The review also usefully connects the YSYK approach to the older gamma-model literature, to DQMC simulations from other groups, and to strange-metal transport phenomenology, including T-linear resistivity and optical conductivities. The explicit statements of limitations, the acknowledgement of possible glassy behavior, and the falsifiable transport predictions are commendable strengths. The main significance is as a pedagogical and conceptual synthesis, but the strongest new claim, the exact low-energy equivalence with holographic superconductivity, is currently demonstrated only in a restricted parameter regime.

major comments (2)
  1. [4.2, Eq. (30)-(35); 4.4] The derivation of the holographic mapping is controlled only for gamma = 1 - 4Delta much less than 1, as stated around Eq. (30), and the subsequent Radon-transform step (Eq. (34)) is performed 'within a gradient expansion' without a stated control parameter. For the particle-hole symmetric YSYK model with M/N=1, the value Delta ≈ 0.420 (Eq. (9)) gives gamma ≈ -0.68, which is neither small nor positive. In this regime the replacement |omega - omega'|^gamma ≈ max(|omega|^gamma, |omega'|^gamma) is not accurate, and the nonlocal integral equation (13) cannot be recast as the local Klein-Gordon equation (31). Therefore the statement in Section 4.4 that 'at low energies, the two theories are identical' goes beyond what Eqs. (30)-(35) establish. The equivalence is demonstrated, at best, for Delta close to 1/4; for the model's physical parameters it remains an unproven conjecture. The authors should either extend the derivation to the relevant Delta or explicitly qualify the claim in Sections 4.2 and 4.4.
  2. [2.4, Sections 2.3 and 3.2] Section 2.4 explicitly concedes that exact DQMC simulations find a breakdown of the large-N replica-diagonal saddle point at sufficiently strong coupling, with signatures of glassy behavior that may be due to replica-symmetry breaking. However, the strong-coupling results in Section 2.3, notably the saturation of Tc, the gap-filling spectral function, and the 'impurity-like' normal state, and the strong-coupling transport results of Section 3.2.2, are all presented as reliable predictions of the YSYK framework. The manuscript should state more precisely which regions of the (g, alpha, M/N) phase diagram are protected by the DQMC comparisons and which strong-coupling conclusions could be altered by a glass or replica-symmetry-broken phase. This is load-bearing because the review's overall case for a controlled quantum-critical Eliashberg theory rests on the validity of the large-N solution in the very regimes where the most distinctive physical claims are made.
minor comments (6)
  1. [2.4] The text reads 'exact DMQC simulations'; the acronym should be DQMC.
  2. [2.3.2] There is a typo in 'Bogoliugbov quasi-particle peak'; it should be 'Bogoliubov'.
  3. [5] In the Conclusions, 'micropscopic' should be 'microscopic'.
  4. [2.4] Reference 172 is listed as 'Esterlis I. unpublished' for the large-N breakdown. For a review, it would be preferable to cite a published or arXiv-available source, or to mark the claim as private communication.
  5. [4.2] Eq. (30) is presented without derivation of the boundary conditions; a sentence indicating how the cutoff T and the upper cutoff Lambda enter the differential equation would improve readability.
  6. [3.2.2] The notation Delta m*/m in the caption of Figure 6 is defined only in the text; a brief definition in the caption would help the reader.

Circularity Check

0 steps flagged · score 1.0 of 10

No load-bearing circularity: the YSYK derivations and the explicit holographic map are carried out in the text, with independent DQMC and transport checks; remaining caveats are approximation validity, not circularity.

full rationale

No load-bearing circular steps were identified. The YSYK saddle-point equations (Eqs. 5-7) are derived in the text from the model action via the replica trick and large-N evaluation, and their validity is checked against independent DQMC studies cited in Section 2.4. The holographic mapping in Section 4.2 is performed explicitly in the paper: the linearized gap equation Eq. 13 is transformed to Eq. 30 under stated approximations, the change of variables zeta = 1/epsilon yields the AdS2 Klein-Gordon equation Eq. 31, and the Radon transform Eq. 34 converts the bilocal pairing action into the holographic matter action Eq. 35. The derivation does not reduce to a fitted parameter or to a self-citation chain; Ref. 137, which has overlapping authorship, is followed closely but the relevant equations are reproduced in this review rather than imported as an unverified premise. The finite-temperature and chemical-potential maps in Section 4.3 rely on standard SYK reparametrization invariance (Ref. 129) and external AdS results (Refs. 204, 206, 207, 210). The paper does lean on the authors' prior works for context and for some technical details, but those references are not the load-bearing argument for the central claims. Two caveats are worth stating as correctness risks, not as circularity. First, Section 2.4 concedes that the replica-diagonal large-N solution is not reliable at sufficiently strong coupling, where DQMC shows possible glassy behavior. Second, the reduction of the nonlocal gap equation to a local differential equation in Section 4.2 assumes gamma = 1 - 4Delta much less than 1 and a gradient expansion, while the model's particle-hole-symmetric exponent is Delta approximately 0.420, so the strong claim in Section 4.4 that the two theories are identical goes beyond what the presented derivation establishes for the physical parameter point. These are approximation-validity concerns, not identity-by-construction or fitted-input circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 1 invented entities

No experimental data are fitted; the free parameters listed are model inputs, not fit constants. The central claims rest on the large-N random-matrix saddle point, the replica trick, emergent conformal symmetry, and the Radon-transform dictionary used in the holographic mapping. The review discloses that these assumptions break down in some strong-coupling variants.

free parameters (3)
  • boson-to-fermion flavor ratio M/N = M/N = 1 for most of the review; general ratio in Ref. 105
    Controls the SYK-NFL exponent Delta in (1/4, 1/2) and the phase diagram. It is a model parameter, not fitted to data.
  • pair-breaking parameter alpha = 0 to 1; critical alpha_c ~ 0.62
    Models time-reversal symmetry-breaking disorder. It determines the superconducting Tc and the BKT vanishing. It is a model knob, not fitted to experiment.
  • dimensionless Yukawa coupling g^2 = g^2/omega_0^3 = weak and strong regimes, e.g. g = 0.5 and g = 4 in figures
    Sets the interaction strength and crossover scales. It is varied by hand in the review to explore different regimes.
assumptions (5)
  • domain assumption Random all-to-all Gaussian-distributed Yukawa couplings with large-N limit and fixed M/N produce the Eliashberg saddle point.
    Defines the YSYK model in Eq. 1. All results are statements about this solvable limit rather than about a specific material.
  • domain assumption The replica trick with only replica-diagonal solutions is valid.
    Section 2.2 and 2.4; the paper acknowledges possible replica-symmetry breaking and glassy behavior at strong coupling.
  • domain assumption The low-energy SYK-NFL has emergent conformal reparametrization invariance, used for finite-T and finite-mu propagators.
    Borrowed from the SYK literature, Ref. 129. It underpins the finite-T black-hole metric in Section 4.3.
  • standard math Bilocal pairing fields map to a scalar in AdS2 via a Radon transform, with a gradient expansion yielding the holographic action.
    Mathematical tool from Ref. 206. The validity of the gradient expansion and boundary terms is assumed.
  • domain assumption In two dimensions, the model uses a quadratic band with constant density of states, c ~ v_F, long-wavelength bosons, and no form factors; g-prime disorder is delta-correlated.
    Section 3.1. These modeling choices determine the Landau-damped or marginal-Fermi-liquid results.
invented entities (1)
  • Extra radial coordinate zeta in AdS2 (holographic dimension)
    purpose: Parameterizes the internal relative-time and frequency dynamics of the Cooper pair and maps the quantum-critical Eliashberg gap equation to a Klein-Gordon equation in AdS2.
    Not a physical spatial dimension but an emergent mathematical coordinate obtained via a Radon transform of bilocal pairing fields. Its observational handle is indirect, through predictions of the boundary Eliashberg theory, not an independent external signal.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Quantum Critical Eliashberg Theory." pith.science (2026). https://pith.science/paper/FGR2HKB5

@misc{pith2026250611952,
  author       = {Pith},
  title        = {Pith review of: Quantum Critical Eliashberg Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FGR2HKB5}},
  note         = {Machine review of arXiv:2506.11952}
}
read the original abstract

Quantum criticality plays a central role in understanding non-Fermi liquid behavior and unconventional superconductivity in strongly correlated systems. In this review, we explore the quantum critical Eliashberg theory, which extends conventional Eliashberg approaches to non-Fermi liquid regimes governed by critical fluctuations. We discuss the theoretical foundations and recent developments in the field, focusing on the interplay between electronic interactions and bosonic modes near quantum phase transitions as described in the Yukawa-coupled version of the Sachdev-Ye-Kitaev model. Special emphasis is placed on the breakdown of quasiparticle coherence, anomalous scaling behaviour, Cooper pairing without quasiparticles, and emergent universality in different physical settings. Starting from a zero-dimensional "quantum-dot" model, we discuss the generalization to higher spatial dimensions and demonstrate the connection between quantum-critical Eliashberg theory and holographic superconductivity. Our analysis provides a perspective on how quantum criticality shapes the dynamics of strongly correlated metals and superconductors.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Intertwined Orders and the Physics of High Temperature Superconductors

    cond-mat.supr-con 2025-06 unverdicted novelty 1.0 of 10

    A review lecture arguing that complex cuprate phase diagrams are best understood through intertwined orders, with the pair-density wave state as the central example.

Reference graph

Works this paper leans on

214 extracted references · 78 canonical work pages · cited by 1 Pith paper

  1. [1]

    Landau LD. 1957. Soviet Physics Jetp-Ussr3(6):920–925

    1957

  2. [2]

    Abrikosov AA, Gorkov LP, Dzyaloshinski IE. 2012. Methods of quantum field theory in sta- tistical physics. Courier Corporation

    2012

  3. [3]

    Baym G, Pethick C. 2008. Landau fermi-liquid theory: concepts and applications. John Wiley & Sons

    2008

  4. [4]

    Polchinski J. 1992. arXiv preprint hep-th/9210046

    arXiv 1992

  5. [5]

    Shankar R. 1994. Rev. Mod. Phys.66(1):129–192

    1994

  6. [6]

    Bardeen J, Cooper LN, Schrieffer JR. 1957. Phys. Rev. 106(1):162–164

    1957

  7. [7]

    Bardeen J, Cooper LN, Schrieffer JR. 1957. Phys. Rev. 108(5):1175–1204

    1957

  8. [8]

    Kohn W, Luttinger JM. 1965. Phys. Rev. Lett.15(12):524–526

    1965

Show all 214 references
  1. [9]

    Cooper LN. 1956. Physical Review 104(4):1189

    1956

  2. [10]

    Legros A, Benhabib S, Tabis W, Lalibert´ e F, Dion M, et al. 2019. Nature Physics 15(2):142– 147

    2019

  3. [11]

    Hayes IM, McDonald RD, Breznay NP, Helm T, Moll PJW, et al. 2016. Nature Physics 12(10):916–919

    2016

  4. [12]

    Jiang X, Qin M, Wei X, Xu L, Ke J, et al. 2023. Nature Physics 19(3):365–371

    2023

  5. [13]

    Nguyen DH, Sidorenko A, Taupin M, Knebel G, Lapertot G, et al. 2021. Nature Communi- cations 12(1):4341

    2021

  6. [14]

    Lee K, Wang BY, Osada M, Goodge BH, Wang TC, et al. 2023. Nature 619(7969):288–292

    2023

  7. [15]

    2022.Nature Physics18(6):633–638

    Jaoui A, Das I, Di Battista G, D ´ ıez-M´ erida J, Lu X, et al. 2022.Nature Physics18(6):633–638

    2022

  8. [16]

    Lee SS. 2018. Annual Review of Condensed Matter Physics9(1):227–244

    2018

  9. [17]

    Millis AJ. 1993. Phys. Rev. B48(10):7183–7196

    1993

  10. [18]

    Altshuler BL, Ioffe LB, Millis AJ. 1995. Phys. Rev. B52(8):5563–5572

    1995

  11. [19]

    Castellani C, Di Castro C, Grilli M. 1995. Phys. Rev. Lett.75(25):4650–4653

    1995

  12. [20]

    Abanov A, Chubukov A V. 2000. Phys. Rev. Lett.84(24):5608–5611

    2000

  13. [21]

    Abanov A VC, Schmalian J

    Ar. Abanov A VC, Schmalian J. 2003. Advances in Physics52(3):119–218

    2003

  14. [22]

    Pankov S, Florens S, Georges A, Kotliar G, Sachdev S. 2004. Phys. Rev. B69(5):054426

    2004

  15. [23]

    Chubukov A V, Schmalian J. 2005. Phys. Rev. B72(17):174520

    2005

  16. [24]

    2007.Rev

    L¨ ohneysen Hv, Rosch A, Vojta M, W¨ olfle P. 2007.Rev. Mod. Phys.79(3):1015–1075

    2007

  17. [25]

    Metlitski MA, Sachdev S. 2010. Phys. Rev. B82(7):075128

    2010

  18. [26]

    Efetov KB, Meier H, P´ epin C. 2013. Nature Physics 9(7):442–446

    2013

  19. [27]

    2014.Phys

    Abrahams E, Schmalian J, W¨ olfle P. 2014.Phys. Rev. B90(4):045105

    2014

  20. [28]

    Meier H, P´ epin C, Einenkel M, Efetov KB. 2014. Phys. Rev. B89(19):195115 www.annualreviews.org • QC Eliashberg Theory 27

    2014

  21. [29]

    Varma CM. 2015. Phys. Rev. Lett.115(18):186405

    2015

  22. [30]

    Schlief A, Lunts P, Lee SS. 2017. Phys. Rev. X7(2):021010

    2017

  23. [31]

    Lunts P, Schlief A, Lee SS. 2017. Phys. Rev. B95(24):245109

    2017

  24. [32]

    Oganesyan V, Kivelson SA, Fradkin E. 2001. Phys. Rev. B64(19):195109

    2001

  25. [33]

    Metzner W, Rohe D, Andergassen S. 2003. Phys. Rev. Lett.91(6):066402

    2003

  26. [34]

    Lawler MJ, Barci DG, Fern´ andez V, Fradkin E, Oxman L. 2006. Phys. Rev. B73(8):085101

    2006

  27. [35]

    2006.Phys

    Rech J, P´ epin C, Chubukov A V. 2006.Phys. Rev. B74(19):195126

    2006

  28. [36]

    Aji V, Varma CM. 2007. Phys. Rev. Lett.99(6):067003

    2007

  29. [37]

    2009.Phys

    Zacharias M, W¨ olfle P, Garst M. 2009.Phys. Rev. B80(16):165116

    2009

  30. [38]

    Metlitski MA, Sachdev S. 2010. Phys. Rev. B82(7):075127

    2010

  31. [39]

    Maslov DL, Chubukov A V. 2010. Phys. Rev. B81(4):045110

    2010

  32. [40]

    Dalidovich D, Lee SS. 2013. Phys. Rev. B88(24):245106

    2013

  33. [41]

    Fitzpatrick AL, Kachru S, Kaplan J, Raghu S. 2014. Phys. Rev. B89(16):165114

    2014

  34. [42]

    Lee PA. 1989. Physical review letters63(6):680

    1989

  35. [43]

    Halperin BI, Lee PA, Read N. 1993. Phys. Rev. B47(12):7312–7343

    1993

  36. [44]

    Polchinski J. 1994. Nuclear Physics B422(3):617–633

    1994

  37. [45]

    Nayak C, Wilczek F. 1994. Nuclear Physics B430(3):534–562

    1994

  38. [46]

    1995.Phys

    Chakravarty S, Norton RE, Sylju ˚ asen OF. 1995.Phys. Rev. Lett.74(8):1423–1426

    1995

  39. [47]

    Bonesteel NE, McDonald IA, Nayak C. 1996. Phys. Rev. Lett.77(14):3009–3012

    1996

  40. [48]

    Lee SS. 2009. Phys. Rev. B80(16):165102

    2009

  41. [49]

    Mross DF, McGreevy J, Liu H, Senthil T. 2010. Phys. Rev. B82(4):045121

    2010

  42. [50]

    Holder T, Metzner W. 2015. Phys. Rev. B92(4):041112

    2015

  43. [51]

    Balatsky A V. 1993. Philosophical Magazine Letters68(4):251–256

    1993

  44. [52]

    Sudbø A. 1995. Phys. Rev. Lett.74(13):2575–2578

    1995

  45. [53]

    Yin L, Chakravarty S. 1996. International Journal of Modern Physics B10(07):805–845

    1996

  46. [54]

    Son DT. 1999. Phys. Rev. D59(9):094019

    1999

  47. [55]

    Abanov A, Chubukov A V, Finkel’stein AM. 2001. Europhysics Letters 54(4):488

    2001

  48. [56]

    Abanov A, Chubukov A V. 1999. Phys. Rev. Lett.83(8):1652–1655

    1999

  49. [57]

    Abanov A, Chubukov A V, Schmalian J. 2001. Europhysics Letters 55(3):369

    2001

  50. [58]

    Roussev R, Millis AJ. 2001. Phys. Rev. B63(14):140504

    2001

  51. [59]

    Abanov A, Chubukov A. 2004. Phys. Rev. Lett.93(25):255702

    2004

  52. [60]

    She JH, Zaanen J. 2009. Phys. Rev. B80(18):184518

    2009

  53. [61]

    Moon EG, Chubukov A. 2010. Journal of Low Temperature Physics161(1):263–281

    2010

  54. [62]

    Levchenko A, Vavilov MG, Khodas M, Chubukov A V. 2013. Phys. Rev. Lett.110(17):177003

    2013

  55. [63]

    Wang Y, Chubukov A. 2013. Phys. Rev. B88(2):024516

    2013

  56. [64]

    Wang Y, Chubukov A V. 2015. Phys. Rev. B92(12):125108

    2015

  57. [65]

    Varma CM. 2016. Reports on Progress in Physics79(8):082501

    2016

  58. [66]

    Khodas M, Dzero M, Levchenko A. 2020. Phys. Rev. B102(18):184505

    2020

  59. [67]

    Lederer S, Schattner Y, Berg E, Kivelson SA. 2015. Phys. Rev. Lett.114(9):097001

    2015

  60. [68]

    Metlitski MA, Mross DF, Sachdev S, Senthil T. 2015. Phys. Rev. B91(11):115111

    2015

  61. [69]

    Fitzpatrick AL, Kachru S, Kaplan J, Raghu S, Torroba G, Wang H. 2015. Phys. Rev. B 92(4):045118

    2015

  62. [70]

    Raghu S, Torroba G, Wang H. 2015. Phys. Rev. B92(20):205104

    2015

  63. [71]

    Mandal I. 2016. Phys. Rev. B94(11):115138

    2016

  64. [72]

    She JH, Overbosch BJ, Sun YW, Liu Y, Schalm KE, et al. 2011. Phys. Rev. B84(14):144527

    2011

  65. [73]

    Wang H, Raghu S, Torroba G. 2017. Phys. Rev. B95(16):165137

    2017

  66. [74]

    Wang H, Wang Y, Torroba G. 2018. Phys. Rev. B97(5):054502

    2018

  67. [75]

    Wang Y, Abanov A, Altshuler BL, Yuzbashyan EA, Chubukov A V. 2016. Phys. Rev. Lett. 117(15):157001

    2016

  68. [76]

    Wu YM, Abanov A, Wang Y, Chubukov A V. 2019. Phys. Rev. B99(14):144512

    2019

  69. [77]

    Abanov A, Chubukov A V. 2020. Phys. Rev. B102(2):024524 28 Esterlis and Schmalian

    2020

  70. [78]

    Wu YM, Abanov A, Wang Y, Chubukov A V. 2020. Phys. Rev. B102(2):024525

    2020

  71. [79]

    Wu YM, Abanov A, Chubukov A V. 2020. Phys. Rev. B102(9):094516

    2020

  72. [80]

    Wu YM, Zhang SS, Abanov A, Chubukov A V. 2021. Phys. Rev. B103(2):024522

    2021

  73. [81]

    Wu YM, Zhang SS, Abanov A, Chubukov A V. 2021. Phys. Rev. B103(18):184508

    2021

  74. [82]

    Zhang SS, Wu YM, Abanov A, Chubukov A V. 2021. Phys. Rev. B104(14):144509

    2021

  75. [83]

    Zhang SS, Chubukov A V. 2023. Phys. Rev. Lett.131(8):086502

    2023

  76. [84]

    Nosov PA, Burmistrov IS, Raghu S. 2023. Phys. Rev. B107(14):144508

    2023

  77. [85]

    Abanov A, Zhang SS, Chubukov A V. 2025. Phys. Rev. B111(7):075157

    2025

  78. [86]

    Berg E, Metlitski MA, Sachdev S. 2012. Science 338(6114):1606–1609

    2012

  79. [87]

    Schattner Y, Gerlach MH, Trebst S, Berg E. 2016. Phys. Rev. Lett.117(9):097002

    2016

  80. [88]

    Schattner Y, Lederer S, Kivelson SA, Berg E. 2016. Phys. Rev. X6(3):031028

    2016

  81. [89]

    Dumitrescu PT, Serbyn M, Scalettar RT, Vishwanath A. 2016. Phys. Rev. B94(15):155127

    2016

  82. [90]

    Gerlach MH, Schattner Y, Berg E, Trebst S. 2017. Phys. Rev. B95(3):035124

    2017

  83. [91]

    Lederer S, Schattner Y, Berg E, Kivelson SA. 2017. Proceedings of the National Academy of Sciences 114(19):4905–4910

    2017

  84. [92]

    Li ZX, Wang F, Yao H, Lee DH. 2017. Phys. Rev. B95(21):214505

    2017

  85. [93]

    Wang X, Schattner Y, Berg E, Fernandes RM. 2017. Phys. Rev. B95(17):174520

    2017

  86. [94]

    Xu XY, Sun K, Schattner Y, Berg E, Meng ZY. 2017. Phys. Rev. X7(3):031058

    2017

  87. [95]

    Wang X, Wang Y, Schattner Y, Berg E, Fernandes RM. 2018. Phys. Rev. Lett.120(24):247002

    2018

  88. [96]

    Berg E, Lederer S, Schattner Y, Trebst S. 2019. Annual Review of Condensed Matter Physics 10(1):63–84

    2019

  89. [97]

    Klein A, Chubukov A V, Schattner Y, Berg E. 2020. Phys. Rev. X10(3):031053

    2020

  90. [98]

    Xu XY, Klein A, Sun K, Chubukov A V, Meng ZY. 2020. npj Quantum Materials5(1):65

    2020

  91. [99]

    Lunts P, Albergo MS, Lindsey M. 2023. Nature Communications 14(1):2547

    2023

  92. [100]

    Patel AA, Lunts P, Albergo MS. 2024. Strange metals and planckian transport in a gapless phase from spatially random interactions

    2024

  93. [101]

    Esterlis I, Schmalian J. 2019. Phys. Rev. B100(11):115132

    2019

  94. [102]

    Wang Y. 2020. Phys. Rev. Lett.124(1):017002

    2020

  95. [103]

    Hauck D, Klug MJ, Esterlis I, Schmalian J. 2020. Annals of Physics417:168120

    2020

  96. [104]

    Wang Y, Chubukov A V. 2020. Phys. Rev. Research2(3):033084

    2020

  97. [105]

    Classen L, Chubukov A. 2021. Phys. Rev. B104(12):125120

    2021

  98. [106]

    Esterlis I, Guo H, Patel AA, Sachdev S. 2021. Phys. Rev. B103(23):235129

    2021

  99. [107]

    Guo H, Patel AA, Esterlis I, Sachdev S. 2022. Phys. Rev. B106(11):115151

    2022

  100. [108]

    Patel AA, Guo H, Esterlis I, Sachdev S. 2023. Science 381(6659):790–793

    2023

  101. [109]

    Valentinis D, Inkof GA, Schmalian J. 2023. Phys. Rev. B108(14):L140501

    2023

  102. [110]

    Valentinis D, Inkof GA, Schmalian J. 2023. Phys. Rev. Res.5(4):043007

    2023

  103. [111]

    Guo H, Valentinis D, Schmalian J, Sachdev S, Patel AA. 2024. Phys. Rev. B109(7):075162

    2024

  104. [112]

    Li C, Valentinis D, Patel AA, Guo H, Schmalian J, et al. 2024. Phys. Rev. Lett.133(18):186502

    2024

  105. [113]

    Pan G, Wang W, Davis A, Wang Y, Meng ZY. 2021. Phys. Rev. Res.3(1):013250

    2021

  106. [114]

    Wang W, Davis A, Pan G, Wang Y, Meng ZY. 2021. Phys. Rev. B103(19):195108

    2021

  107. [115]

    Guo H. 2024. Phys. Rev. B110(15):155130

    2024

  108. [116]

    Sutradhar J, Ruhman J, Klein A. 2024. Phys. Rev. Res.6(4):L042036

    2024

  109. [117]

    Sachdev S, Ye J. 1993. Phys. Rev. Lett.70(21):3339–3342

    1993

  110. [118]

    Georges A, Parcollet O, Sachdev S. 2000. Phys. Rev. Lett.85(4):840–843

    2000

  111. [119]

    Sachdev S. 2010. Phys. Rev. Lett.105(15):151602

    2010

  112. [120]

    Kitaev A. 2015. Talks at KITP, University of California, Santa Barbara, Entanglement in Strongly-Correlated Quantum Matter

    2015

  113. [121]

    Kitaev A. 2015. Talk 2

    2015

  114. [122]

    Sachdev S. 2015. Phys. Rev. X5(4):041025

    2015

  115. [123]

    Maldacena J, Stanford D. 2016. Phys. Rev. D94(10):106002

    2016

  116. [124]

    Polchinski J, Rosenhaus V. 2016. Journal of High Energy Physics2016(4):1 www.annualreviews.org • QC Eliashberg Theory 29

    2016

  117. [125]

    Fu W, Gaiotto D, Maldacena J, Sachdev S. 2017. Phys. Rev. D95(2):026009

    2017

  118. [126]

    Bi Z, Jian CM, You YZ, Pawlak KA, Xu C. 2017. Phys. Rev. B95(20):205105

    2017

  119. [127]

    Song XY, Jian CM, Balents L. 2017. Phys. Rev. Lett.119(21):216601

    2017

  120. [128]

    Chowdhury D, Werman Y, Berg E, Senthil T. 2018. Phys. Rev. X8(3):031024

    2018

  121. [129]

    Chowdhury D, Georges A, Parcollet O, Sachdev S. 2022. Rev. Mod. Phys.94(3):035004

    2022

  122. [130]

    Sachdev S. 2024. International Journal of Modern Physics B38(32)

    2024

  123. [131]

    Migdal A. 1958. Sov. Phys. JETP7(6):996–1001

    1958

  124. [132]

    Eliashberg G. 1960. Sov. Phys. JETP11(3):696–702

    1960

  125. [133]

    Marsiglio F. 2020. Annals of Physics417:168102

    2020

  126. [134]

    Maldacena JM. 1999. Int. J. Theor. Phys.38:1113–1133

    1999

  127. [135]

    Witten E. 1998. Advances in Theoretical and Mathematical Physics2(2):253–291

    1998

  128. [136]

    Gubser S, Klebanov I, Polyakov A. 1998. Physics Letters B428(1):105 – 114

    1998

  129. [137]

    Inkof GA, Schalm K, Schmalian J. 2022. npj Quantum Materials7(1)

    2022

  130. [138]

    Schmalian J. 2022. Holographic superconductivity of a critical fermi surface

    2022

  131. [139]

    Hosseinabadi H, Kelly SP, Schmalian J, Marino J. 2023. Phys. Rev. B108(10):104319

    2023

  132. [140]

    Kim J, Altman E, Cao X. 2021. Physical Review B103(8):L081113

    2021

  133. [141]

    Grunwald L, Passetti G, Kennes DM. 2024. Communications Physics 7(1):79

    2024

  134. [142]

    Cichutek N, R¨ uckriegel A, Hansen MO, Kopietz P. 2024. Phys. Rev. B109(15):155101

    2024

  135. [143]

    Bashan N, Tulipman E, Schmalian J, Berg E. 2024. Phys. Rev. Lett.132(23):236501

    2024

  136. [144]

    Tulipman E, Bashan N, Schmalian J, Berg E. 2024. Physical Review B110(15):155118

    2024

  137. [145]

    Tikhanovskaya M, Sachdev S, Patel AA. 2022. Phys. Rev. Lett.129(6):060601

    2022

  138. [146]

    Davis A, Wang Y. 2023. Phys. Rev. B107(20):205122

    2023

  139. [147]

    Wu YM, Nosov PA, Patel AA, Raghu S. 2023. Phys. Rev. Lett.130(2):026001

    2023

  140. [148]

    Bashan N, Tulipman E, Kivelson SA, Schmalian J, Berg E. 2025. arXiv preprint arXiv:2502.08699

    2025

  141. [149]

    Wang X, Chowdhury D. 2023. Phys. Rev. B107(12):125157

    2023

  142. [150]

    Wang X, Moessner R, Chowdhury D. 2024. Phys. Rev. B109(12):L121102

    2024

  143. [151]

    Patel AA, Lawler MJ, Kim EA. 2018. Phys. Rev. Lett.121(18):187001

    2018

  144. [152]

    Gnezdilov NV. 2019. Phys. Rev. B99(2):024506

    2019

  145. [153]

    Chowdhury D, Berg E. 2020. Phys. Rev. Res.2(1):013301

    2020

  146. [154]

    Wang H, Chudnovskiy AL, Gorsky A, Kamenev A. 2020. Phys. Rev. Res.2(3):033025

    2020

  147. [155]

    Lantagne-Hurtubise E, Pathak V, Sahoo S, Franz M. 2021. Phys. Rev. B104(2):L020509

    2021

  148. [156]

    Choi W, Tavakol O, Kim YB. 2022. SciPost Phys. 12:151

    2022

  149. [157]

    Chudnovskiy AL, Kamenev A. 2022. Phys. Rev. Lett.129(26):266601

    2022

  150. [158]

    Gnezdilov NV, Wang Y. 2022. Phys. Rev. B106(9):094508

    2022

  151. [159]

    Li C, Sachdev S, Joshi DG. 2023. Phys. Rev. Res.5(1):013045

    2023

  152. [160]

    Abrikosov A, Gor’kov L. 1961. Sov. Phys. JETP12:1243

    1961

  153. [161]

    Georges A, Parcollet O, Sachdev S. 2001. Phys. Rev. B63(13):134406

    2001

  154. [162]

    Schmalian J, Langer M, Grabowski S, Bennemann K. 1996. Computer Physics Communica- tions 93(2):141–151

    1996

  155. [163]

    Anderson P. 1959. Journal of Physics and Chemistry of Solids11(1):26–30

    1959

  156. [164]

    Abrikosov A, Gorkov L. 1959. Sov. Phys. JETP8(6):1090–1098

    1959

  157. [165]

    Abrikosov A, Gor’Kov L. 1959. Sov. Phys. JETP9(1):220–221

    1959

  158. [166]

    Potter AC, Lee PA. 2011. Phys. Rev. B83(18):184520

    2011

  159. [167]

    Kang J, Fernandes RM. 2016. Phys. Rev. B93(22):224514

    2016

  160. [168]

    Millis AJ, Sachdev S, Varma CM. 1988. Phys. Rev. B37(10):4975–4986

    1988

  161. [169]

    Abanov A, Chubukov A V, Norman MR. 2008. Phys. Rev. B78(22):220507

    2008

  162. [170]

    Combescot R. 1995. Phys. Rev. B51(17):11625–11634

    1995

  163. [171]

    Kaplan DB, Lee JW, Son DT, Stephanov MA. 2009. Phys. Rev. D80(12):125005

    2009

  164. [172]

    unpublished

    Esterlis I. unpublished

  165. [173]

    Mahan GD. 2013. Many-particle physics. Springer Science & Business Media 30 Esterlis and Schmalian

    2013

  166. [174]

    Chubukov A V, Abanov A, Esterlis I, Kivelson SA. 2020. Annals of Physics417:168190Eliash- berg theory at 60: Strong-coupling superconductivity and beyond

    2020

  167. [175]

    Chowdhury D, Berg E. 2020. Annals of Physics 417:168125Eliashberg theory at 60: Strong- coupling superconductivity and beyond

    2020

  168. [176]

    Aldape EE, Cookmeyer T, Patel AA, Altman E. 2022. Phys. Rev. B105(23):235111

    2022

  169. [177]

    Hertz JA. 1976. Phys. Rev. B14(3):1165–1184

    1976

  170. [178]

    Paul I, Garst M. 2017. Physical Review Letters118(22):227601

    2017

  171. [179]

    Michon B, Berthod C, Rischau CW, Ataei A, Chen L, et al. 2023. Nature Communications 14(1):3033

    2023

  172. [180]

    Patel AA, Lunts P, Sachdev S. 2024. Proceedings of the National Academy of Sciences 121(14):e2402052121

    2024

  173. [181]

    Dell’Anna L, Metzner W. 2006. Phys. Rev. B73(4):045127

    2006

  174. [182]

    2020.Phys

    Damia JA, Sol ´ ıs M, Torroba G. 2020.Phys. Rev. B102(4):045147

    2020

  175. [183]

    2021.Phys

    Damia JA, Sol ´ ıs M, Torroba G. 2021.Phys. Rev. B103(15):155161

    2021

  176. [184]

    Varma CM, Littlewood PB, Schmitt-Rink S, Abrahams E, Ruckenstein AE. 1989. Phys. Rev. Lett. 63(18):1996–1999

    1989

  177. [185]

    Khurana A. 1990. Phys. Rev. Lett.64(16):1990–1990

    1990

  178. [186]

    Sachdev S. 2000. Quantum Phase Transitions. Cambridge: Cambridge University Press

    2000

  179. [187]

    Varma CM. 2020. Rev. Mod. Phys.92(3):031001

    2020

  180. [188]

    Mitrano M, Husain AA, Vig S, Kogar A, Rak MS, et al. 2018. Proceedings of the National Academy of Sciences115(21):5392–5396

    2018

  181. [189]

    Husain AA, Mitrano M, Rak MS, Rubeck S, Uchoa B, et al. 2019. Phys. Rev. X9(4):041062

    2019

  182. [190]

    forthcoming

    Klein A, Schmalian J. forthcoming

  183. [191]

    De Haro S, Skenderis K, Solodukhin SN. 2001. Communications in Mathematical Physics 217(3):595–622

    2001

  184. [192]

    Skenderis K. 2002. Classical and Quantum Gravity19(22):5849–5876

    2002

  185. [193]

    Donos A, Gauntlett JP. 2014. Journal of High Energy Physics2014(11):81

    2014

  186. [194]

    Hartnoll SA, Kovtun PK, M¨ uller M, Sachdev S. 2007. Phys. Rev. B76(14):144502

    2007

  187. [195]

    Casalderrey-Solana J, Liu H, Mateos D, Rajagopal K, Wiedemann UA. 2014. Gauge/string duality, hot qcd and heavy ion collisions. Cambridge University Press

    2014

  188. [196]

    Kovtun PK, Son DT, Starinets AO. 2005. Physical Review Letters94(11):23–26

    2005

  189. [197]

    Policastro G, Son DT, Starinets AO. 2001. Phys. Rev. Lett.87(8):081601

    2001

  190. [198]

    Gubser SS. 2008. Physical Review D - Particles, Fields, Gravitation and Cosmology78(6)

    2008

  191. [199]

    2008 (Dc)

    Hartnoll SA, Herzog CP, Horowitz GT. 2008 (Dc)

    2008

  192. [200]

    Hartnoll SA, Herzog CP, Horowitz GT. 2008. Journal of High Energy Physics2008(12):015– 015

    2008

  193. [201]

    Donos A, Gauntlett JP. 2011. Journal of High Energy Physics2011(8)

    2011

  194. [202]

    2017.Phys

    Delacr´ etaz L V, Gout´ eraux B, Hartnoll SA, Karlsson A. 2017.Phys. Rev. B96(19):195128

    2017

  195. [203]

    2018.Physical Review D97(8):1–18

    Amoretti A, Are´ an D, Gout´ eraux B, Musso D. 2018.Physical Review D97(8):1–18

    2018

  196. [204]

    Breitenlohner P, Freedman DZ. 1982. Physics Letters B115(3):197–201

    1982

  197. [205]

    Inkof GA. 2022. PhD-thesis, Karlsruhe Institute of Technology

    2022

  198. [206]

    Das SR, Ghosh A, Jevicki A, Suzuki K. 2018. JHEP 07:184

    2018

  199. [207]

    Sachdev S. 2019. Journal of Mathematical Physics60(5):0–23

    2019

  200. [208]

    Luttinger JM. 1960. Phys. Rev. 119(4):1153–1163

    1960

  201. [209]

    Gu Y, Kitaev A, Sachdev S, Tarnopolsky G. 2020. Journal of High Energy Physics2020(2):157

    2020

  202. [210]

    Faulkner T, Iqbal N, Liu H, McGreevy J, Vegh D. 2011

    2011

  203. [211]

    Gor’kov LP. 1959. Sov. Phys. JETP9(6):1364–1367

    1959

  204. [212]

    Ginzburg VL, Ginzburg VL, Landau L. 2009. On the theory of superconductivity. Springer

    2009

  205. [213]

    Zhang SS, Raines ZM, Chubukov A V. 2024. Phys. Rev. B109(24):245132

    2024

  206. [214]

    Hardy A, Parcollet O, Georges A, Patel AA. 2025. Physical Review Letters134(3) www.annualreviews.org • QC Eliashberg Theory 31

    2025

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.