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REVIEW 2 major objections 4 minor 34 references

Spectral weight in holography with momentum relaxation

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Explicit momentum relaxation in a holographic superfluid suppresses anomalous low-energy spectral weight and enlarges the finite-momentum instability region, without eliminating Fermi shells.

desk verdict A solid transverse-channel extension of holographic superfluid spectral weight to axion momentum relaxation, but the longitudinal-channel claims rest on an underspecified cubic-root selection and two contradictory parameter-space statements. read the letter →

arxiv 1908.04312 v1 pith:SAIIU4MO submitted 2019-08-12 hep-th

Victoria L. Martin This is my paper · ORCID
classification hep-th
keywords holographicsuperconductorspectralweightmomentumrelaxationaxionFermishellsemi-localquantumliquidfinite-momentuminstabilityAdS/CFT
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 asks whether the peculiar zero-temperature spectral weight found in holographic superconductors survives when translations are no longer a symmetry. It answers by adding massless scalar fields linear in the spatial coordinates (called axions) to an Einstein-Maxwell-dilaton model with a condensate, and computing the low-energy scaling of the retarded Green's function. In the resulting theory, momentum relaxation does not remove the anomalous spectral weight: it strengthens the finite-momentum instability and suppresses the nested Fermi surfaces, with condensate charge and axion strength acting in the same direction in the transverse channel. The result matters because it tests whether these features are genuine signatures of strongly coupled matter or only artifacts of an overly symmetric toy model.

What carries the argument

The central object is the low-energy spectral weight $\sigma(k)=\lim_{\omega\to 0}\mathrm{Im}\,G^R_{OO}(\omega,k)/\omega$, whose infrared behaviour is $\sigma(k)\sim\omega^{2\nu_- - 1}$. The paper computes the scaling exponent $\nu_-$ by a power-law analysis of the linearised Einstein-Maxwell-dilaton-axion equations around the semi-local quantum liquid background (a hyperscaling-violating geometry with $z\to\infty$ and $\eta=-\theta/z$ fixed), with translation-breaking scalars $\psi_i=m x_i$ and a massive vector with $W_0=(1-\zeta)(\zeta+\eta)$. Whether the spectral weight is divergent (smeared Fermi surface), finite in a shell, or absent is decided by the sign of $2\nu_- - 1$. In the longitudinal channel $\nu_-$ comes from the roots of a cubic equation (4.20), which is why those results are numerical.

What would settle it

Take a point in the claimed stable positive-$\zeta$ region, for instance $\eta=1$, $m=1$, $\zeta=0.05$, solve the full radial perturbation equations numerically, and compute $\mathrm{Im}\,G^R_{OO}/\omega$ directly; if the resulting spectral weight or instability boundary disagrees with the $\nu_-=1/2$ contours in Figure 8, the branch selection for $\nu_-$ is wrong.

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Extended reading notes

Core claim

The paper's central claim is that explicit momentum relaxation changes, but does not erase, the anomalous low-energy spectral weight of holographic superfluids in semi-local quantum liquid geometries. In the transverse channel the Fermi surface size $k_*$, defined by where $\sigma(k)\sim\omega^{2\nu_- - 1}$ stops diverging, decreases with both the condensate charge $W_0$ and the axion strength $m$, and there is no instability. In the longitudinal channel the $U(1)$-breaking term $W_0$ is what creates the finite-momentum instability and the Fermi shells, bands $k_-<k<k_+$ of nonzero spectral weight; adding the axion enlarges the instability region so that it can reach positive $\zeta$, and suppresses the low-energy spectral weight for every $\zeta$, converting the Fermi shell into a smeared Fermi surface before killing it. The paper also finds that in the Einstein-Maxwell-dilaton-axion theory without a condensate there is no low-energy spectral weight and no instability at all.

Load-bearing premise

The paper's longitudinal-channel conclusions rest on the assumption that the numerically chosen root of the cubic equation for the scaling exponent is the physically relevant low-energy mode, and that the parameter scans in Figures 7 and 8 are complete.

Editorial extensions

If this is right

  • If the paper is right, the anomalous low-energy spectral weight seen in holographic superconductors is not an artifact of exact translation invariance; it persists for suitable parameters after momentum relaxation is added.
  • In the transverse channel, the condensate charge and the axion strength both shrink the Fermi surface size $k_*$, and their effects barely mix, so either parameter can be used to tune the smeared Fermi surface.
  • In the longitudinal channel, increasing $|m|$ enlarges the finite-momentum instability region and can push it into $\zeta>0$, making the semi-local quantum liquid ground state less stable toward spatially modulated order.
  • For a fixed $\zeta$, increasing $|m|$ suppresses the spectral weight and eventually removes the Fermi shell; at $\eta=1$ the no-spectral-weight region runs from about $\zeta=-1$ up to about $\zeta=-0.07$.

Reading between the lines

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

  • The near-independence of $W_0$ and $m$ in the transverse channel suggests that the two symmetry-breaking effects act through separate bulk sectors; a testable extension would be to fix a physical observable such as the DC conductivity while varying $W_0$ and $m$, and see whether $k_*$ still tracks both.
  • Because the paper notes that top-down Fermi shells arise from two overlapping fermion species, a natural bottom-up check is to add two bulk fermion fields to the axion-plus-massive-vector model and compare the shell width and its $W_0$ and $m$ dependence with the two-species prediction.
  • The longitudinal-channel results depend on a root choice that the paper does not justify in closed form; an independent numerical solution of the full radial perturbation equations would both test that branch selection and map the claimed $\zeta>0$ stability region completely.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. This paper studies the zero-temperature, finite-momentum spectral weight of holographic superfluids when translational symmetry is additionally broken by axion fields. The author extends earlier work on the holographic superconductor by adding massless scalars linear in the boundary coordinates, and analyzes the transverse and longitudinal channels of the resulting scaling geometries. In the transverse channel, closed-form expressions are given for the scaling exponents and for the critical momentum k_* below which spectral weight survives, and it is shown that both the condensate charge W0 and the axion strength m suppress the effective Fermi surface. In the longitudinal channel, the paper reports numerical results indicating that axions enlarge the finite-momentum instability region and suppress Fermi-shell spectral weight, and that for positive ζ a stable region with smeared Fermi surfaces can appear. The paper also reviews the spectral-weight diagnostic and its relation to Pauli exclusion, and discusses the interpretation of Fermi shells as smeared nested Fermi surfaces.

Significance. If the results hold, the paper establishes a concrete bottom-up example in which explicit momentum relaxation changes but does not eliminate the anomalous low-energy spectral weight of holographic superfluids. The transverse-channel analysis is a genuine strength: the scaling exponents in Eqs. (3.10), (4.12), and (4.14) are given in closed form, reduce correctly to the m = 0 and W0 = 0 limits, and are parameter-free in the sense that no fitting to external data is involved. The qualitative predictions about how axion strength affects the instability region and Fermi-shell width are falsifiable and relevant to ongoing work on momentum relaxation in holographic quantum matter. The main weakness is that the longitudinal-channel conclusions rest on a numerical root analysis whose selection rule and scan details are not specified, making those figures currently non-reproducible.

major comments (2)
  1. [Sec. 4.2, Eqs. (4.19)-(4.21), Figs. 7-8] The longitudinal-channel scaling exponent nu_- is obtained as the square root of a combination of roots Y_i of the cubic (4.20), but the paper never states which root is selected, which branch is used, or how the remaining parameter space is scanned. The text explicitly says the closed form is 'too complicated to report', and the central claims of Section 4.2 -- that increasing |m| augments the instability region into zeta > 0 and that it converts Fermi shells into smeared Fermi surfaces -- are extracted numerically from this exponent. Without a stated root-selection rule (for example, the value that reduces to the known m = 0 result of Ref. [12], or the root corresponding to the dominant IR mode) and without the numerical method and scan discretization, Figures 7 and 8 cannot be independently reproduced, and the possibility that the qualitative results are an artifact of misidentifying the physical root cannot be excluded. The authors should specify the selection criterion and the numerical details before the longitudinal claims can be accepted.
  2. [Sec. 2.2, Eqs. (2.13)-(2.14), Fig. 3] There is an internal inconsistency in the definition of the instability and stability regions. Equation (2.13) states an instability region with zeta > 0 (0 < zeta < eta^2/2 or 0 < zeta < (1-eta)/2), yet the text says that it 'basically restricts zeta < 0'. Similarly, Eq. (2.14) gives the lower bound W0 > eta, but the text asserts that it 'restricts 0 < W0 < eta', and Figure 3 uses the range 0 < W0 < eta. These two ranges are not equivalent: for eta = 1/2, Eq. (2.14) gives W0 in (0.5, 0.546875), whereas Figure 3 uses W0 in (0.475, 0.5). The mismatch affects the baseline superfluid Fermi-shell results on which the axion comparison in Section 4.2 depends. The authors should correct the inequalities and clearly state whether Figure 3 is plotting the stability region or the instability region.
minor comments (4)
  1. [Eq. (1.2) and Appendix B, Eq. (B.13)] Equation (1.2) has the same symbol, Im G^R_OO(omega,k), on both sides of the proportionality. This appears to be a typo for a relation between the IR and UV Green's functions, as discussed in Appendix B. The notation should be corrected or the distinction explained.
  2. [Sec. 5, Discussion] In the Discussion, the EMD plus axion theory is described as 'spontaneously breaks translation symmetry'. The axion ansatz psi_i = m x_i explicitly breaks translational symmetry, as stated in the Introduction. The wording should be changed to 'explicitly breaks'.
  3. [Fig. 6 caption] The caption states 'Only the zeta_- root yields real results', but the paper does not define the two zeta roots or explain the selection. A sentence identifying zeta_- and why the other root is discarded would help the reader.
  4. [Sec. 4.2, discussion of Fig. 7] The text says the instability region 'still only exist for zeta > 0' while Figure 7 shows a new stability region for zeta > 0 when m is nonzero. The relationship between the instability region and the stability region in the same figure should be stated more precisely.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the axion and condensate-charge dependencies are derived from the stated action and scaling ansatz, not imported from the cited prior work.

full rationale

The paper's central new results are obtained by a self-contained scaling analysis of the stated bulk action. The transverse-channel exponent and Fermi-surface momentum are given in closed form (Eqs. 4.10-4.15), and the longitudinal channel is analyzed numerically from the cubic (Eqs. 4.19-4.21); neither involves fitting to the claimed output. The parameters W0 and m are scanned inputs, not fitted parameters renamed as predictions. The citations to the author's prior work [7, 12] provide the baseline m=0 and W0=0 results and the spectral-weight diagnostic, but the new m-dependence is derived here rather than taken from those references. The statement that setting m=0 reproduces Ref. [12] is a consistency check, not a circular reduction. The substantive concerns raised by the manuscript text, such as the unspecified root selection for the longitudinal cubic, the lack of numerical scan details, and the inconsistent inequalities in Eqs. (2.13)-(2.14), are reproducibility and correctness issues, not circularity: they do not show that a prediction equals an input by construction. Therefore no circular step is exhibited, and the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim is built on standard holographic tools (IR scaling exponents, IR/UV matching) and on the model parameters η, ζ (or equivalently W0) and m, which are scanned over physical ranges rather than fitted to data. The paper introduces no new bulk fields beyond the known axion and massive vector constructions, and no new entities. The main ingredients that are imported without re-derivation are the near-horizon matching and the interpretation of divergent IR spectral weight as a Fermi surface diagnostic; both are stated in the paper but not proven there.

assumptions (4)
  • domain assumption The low-energy boundary spectral weight σ(k) is determined by the near-horizon IR scaling exponent ν− via σ(k) ∝ lim_{ω→0} ω^{2ν−−1}.
    Invoked throughout Sections 2 to 4 (e.g., Eqs. 2.6, 3.7, 4.13). Relies on the matching argument in Appendix B.3 (Eqs. B.12 and B.13) that IR and UV Green's functions differ by real ω-independent constants, and on the spectral decomposition in Eq. 1.3 connecting finite-k spectral weight to Pauli exclusion.
  • domain assumption The η-geometry, ds² = r^{−η}(−dt² + dr²/r² + dx² + dy²), with z → ∞ and η = −θ/z fixed, is the correct holographic background for the semi-local quantum liquid regime of these theories.
    Used as the background for all three models (Eqs. 2.3 and 4.3). Inherited from prior work [10,11,12]; the paper does not re-derive why this limit captures the low-energy physics.
  • ad hoc to paper The axion ansatz ψi = m xi with Y(φ) = e^{λφ} and λ = 0 consistently breaks translations while preserving the scaling solution.
    Modeling choice in Sections 3 and 4 (Eqs. 3.2, 3.3 and the constraint λ = 0 in Eq. 3.4). It is the standard momentum relaxation mechanism of [17,18], but it is not derived from a top-down string construction here.
  • domain assumption The exponential IR scaling forms V = V0 e^{−δφ}, Z = Z0 e^{γφ}, W = W0 e^{ϵφ}, Y = Y0 e^{λφ} with the stated constraints provide the correct low-energy description of the holographic superfluid.
    Imposed in Eqs. 2.2, 3.3 and 4.2, motivated by top-down realizations [20-25]. This is the standard bottom-up assumption that the deep IR is governed by these scaling forms; the paper does not justify them from a specific string construction.

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Pith. "Pith review of Spectral weight in holography with momentum relaxation." pith.science (2026). https://pith.science/paper/SAIIU4MO

@misc{pith2026190804312,
  author       = {Pith},
  title        = {Pith review of: Spectral weight in holography with momentum relaxation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SAIIU4MO}},
  note         = {Machine review of arXiv:1908.04312}
}
read the original abstract

Holographic low-energy spectral weight at zero temperature and finite momenta indicates the presence of a strongly coupled remnant of Pauli exclusion. Building upon previous work, we study the spectral weight of a bottom-up holographic superfluid model with spontaneously broken translational symmetry. We determine the effect of this symmetry breaking on the previously known attributes of the holographic superconductor spectral weight: 1) an instability at finite momenta and 2) the presence of nested Fermi surfaces (sometimes called Fermi shells). We find that the symmetry breaking seems to strengthen the former and suppress the latter, in a way that we describe.

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Works this paper leans on

34 extracted references · 17 canonical work pages

  1. [12]

    Spectral weight and spatially modulated instabilities in holographic superfluids

    B. Gout´ eraux and V. L. Martin,Spectral weight and spatially modulated instabilities in holographic superfluids, 1612.03466

    work page Pith review arXiv

  2. [1]

    Maldacena, The large n limit of superconformal field theories and supergravity , in AIP Conference Proceedings CONF-981170, vol

    J. Maldacena, The large n limit of superconformal field theories and supergravity , in AIP Conference Proceedings CONF-981170, vol. 484, pp. 51–63, AIP, 1999

    work page 1999

  3. [2]

    Varma, P

    C. Varma, P. B. Littlewood, S. Schmitt-Rink, E. Abrahams and A. Ruckenstein, Phenomenology of the normal state of cu-o high-temperature superconductors , Physical Review Letters 63 (1989) 1996

    work page 1989

  4. [3]

    R. A. Davison, K. Schalm and J. Zaanen, Holographic duality and the resistivity of strange metals, Phys. Rev. B89 (2014) 245116 [ 1311.2451]

    arXiv 2014

  5. [4]

    S. A. Hartnoll, A. Lucas and S. Sachdev, Holographic quantum matter, 1612.07324

    arXiv

  6. [5]

    Armitage, F

    N. Armitage, F. Ronning, D. Lu, C. Kim, A. Damascelli, K. Shen et al., Doping dependence of an n-type cuprate superconductor investigated by angle-resolved photoemission spectroscopy, Physical Review Letters 88 (2002) 257001

    work page 2002

  7. [6]

    Iqbal, H

    N. Iqbal, H. Liu and M. Mezei, Lectures on holographic non-Fermi liquids and quantum phase transitions, in Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 2010). String Theory and Its Applications: From meV to the Planck Scale: Boulder, Colorado, USA, June 1-25, 2010 , pp. 707–816, 2011, 1110.3814, DOI

    arXiv 2010

  8. [7]

    V. L. Martin and N. Monga, Spectral weight in Chern-Simons theory with symmetry breaking, 1905.07417

    work page Pith review arXiv 1905

Show all 34 references
  1. [8]

    S. A. Hartnoll and E. Shaghoulian, Spectral weight in holographic scaling geometries , JHEP 07 (2012) 078 [ 1203.4236]

    2012 arXiv

  2. [9]

    Keeler, G

    C. Keeler, G. Knodel and J. T. Liu, Hidden horizons in non-relativistic AdS/CFT , JHEP 08 (2014) 024 [ 1404.4877]. 5HereA(ω) is the electric potential and should not be confused with the spectral function A(ω,k )! – 24 –

    2014 arXiv

  3. [10]

    R. J. Anantua, S. A. Hartnoll, V. L. Martin and D. M. Ramirez, The Pauli exclusion principle at strong coupling: Holographic matter and momentum space , JHEP 03 (2013) 104 [ 1210.1590]

    2013 arXiv

  4. [11]

    S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, Building a Holographic Superconductor, Phys. Rev. Lett. 101 (2008) 031601 [ 0803.3295]

    2008 arXiv

  5. [13]

    T. Wu, H. Mayaffre, S. Kr¨ amer, M. Horvati´ c, C. Berthier, W. Hardy et al., Magnetic-field-induced charge-stripe order in the high-temperature superconductor yba 2 cu 3 o y , Nature 477 (2011) 191

    2011

  6. [14]

    Nakamura, H

    S. Nakamura, H. Ooguri and C.-S. Park, Gravity Dual of Spatially Modulated Phase , Phys. Rev. D81 (2010) 044018 [ 0911.0679]

    2010 arXiv

  7. [15]

    DeWolfe, S

    O. DeWolfe, S. S. Gubser and C. Rosen, Fermi surfaces in N=4 Super-Yang-Mills theory, Phys. Rev. D86 (2012) 106002 [ 1207.3352]

    2012 arXiv

  8. [16]

    DeWolfe, O

    O. DeWolfe, O. Henriksson and C. Rosen, Fermi surface behavior in the ABJM M2-brane theory, Phys. Rev. D91 (2015) 126017 [ 1410.6986]

    2015 arXiv

  9. [17]

    R. A. Davison and B. Gout´ eraux,Momentum dissipation and effective theories of coherent and incoherent transport, JHEP 01 (2015) 039 [ 1411.1062]

    2015 arXiv

  10. [18]

    Gout´ eraux,Charge transport in holography with momentum dissipation , JHEP 04 (2014) 181 [ 1401.5436]

    B. Gout´ eraux,Charge transport in holography with momentum dissipation , JHEP 04 (2014) 181 [ 1401.5436]

    2014 arXiv

  11. [19]

    Y. Ling, Z. Xian and Z. Zhou, Power Law of Shear Viscosity in Einstein-Maxwell-Dilaton-Axion model, Chin. Phys. C41 (2017) 023104 [ 1610.08823]

    2017 arXiv

  12. [20]

    S. S. Gubser, C. P. Herzog, S. S. Pufu and T. Tesileanu, Superconductors from Superstrings, Phys. Rev. Lett. 103 (2009) 141601 [ 0907.3510]

    2009 arXiv

  13. [21]

    J. P. Gauntlett, J. Sonner and T. Wiseman, Holographic superconductivity in M-Theory, Phys. Rev. Lett. 103 (2009) 151601 [ 0907.3796]

    2009 arXiv

  14. [22]

    J. P. Gauntlett, J. Sonner and T. Wiseman, Quantum Criticality and Holographic Superconductors in M-theory, JHEP 02 (2010) 060 [ 0912.0512]

    2010 arXiv

  15. [23]

    Bobev, A

    N. Bobev, A. Kundu, K. Pilch and N. P. Warner, Minimal Holographic Superconductors from Maximal Supergravity, JHEP 03 (2012) 064 [ 1110.3454]

    2012 arXiv

  16. [24]

    DeWolfe, S

    O. DeWolfe, S. S. Gubser, O. Henriksson and C. Rosen, Fermionic Response in Finite-Density ABJM Theory with Broken Symmetry , Phys. Rev. D93 (2016) 026001 [1509.00518]

    2016 arXiv

  17. [25]

    DeWolfe, S

    O. DeWolfe, S. S. Gubser, O. Henriksson and C. Rosen, Gapped Fermions in Top-down Holographic Superconductors, 1609.07186. – 25 –

    arXiv

  18. [26]

    Huijse, S

    L. Huijse, S. Sachdev and B. Swingle, Hidden Fermi surfaces in compressible states of gauge-gravity duality, Phys. Rev. B85 (2012) 035121 [ 1112.0573]

    2012 arXiv

  19. [27]

    S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, Holographic Superconductors, JHEP 12 (2008) 015 [ 0810.1563]

    2008 arXiv

  20. [28]

    Iqbal, H

    N. Iqbal, H. Liu and M. Mezei, Semi-local quantum liquids, JHEP 04 (2012) 086 [1105.4621]

    2012 arXiv

  21. [29]

    Abrahams and C

    E. Abrahams and C. Varma, What angle-resolved photoemission experiments tell about the microscopic theory for high-temperature superconductors, Proceedings of the National Academy of Sciences 97 (2000) 5714

    2000

  22. [30]

    Lee, A Non-Fermi Liquid from a Charged Black Hole: A Critical Fermi Ball , Phys

    S.-S. Lee, A Non-Fermi Liquid from a Charged Black Hole: A Critical Fermi Ball , Phys. Rev. D79 (2009) 086006 [ 0809.3402]

    2009 arXiv

  23. [31]

    H. Liu, J. McGreevy and D. Vegh, Non-Fermi liquids from holography, Phys. Rev. D83 (2011) 065029 [ 0903.2477]

    2011 arXiv

  24. [32]

    Cubrovic, J

    M. Cubrovic, J. Zaanen and K. Schalm, String Theory, Quantum Phase Transitions and the Emergent Fermi-Liquid , Science 325 (2009) 439 [ 0904.1993]

    2009 arXiv

  25. [33]

    Cremonini, L

    S. Cremonini, L. Li and J. Ren, Holographic Fermions in Striped Phases , JHEP 12 (2018) 080 [ 1807.11730]

    2018 arXiv

  26. [34]

    Ammon and J

    M. Ammon and J. Erdmenger, Gauge/gravity duality. Cambridge University Press, Cambridge, 2015. – 26 –

    2015

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