{"id":"1c8b31a7-afa0-49a0-8da2-ba5cf01dc0e3","arxiv_id":"1908.04312","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In holographic superfluids with axion-induced momentum relaxation, the finite-momentum instability is strengthened and low-energy spectral weight, including Fermi shells, is suppressed.","lead":"This paper computes how momentum relaxation, modeled by axion fields, changes the low-energy spectral weight of a holographic superfluid with a broken U(1) symmetry. It finds that adding momentum relaxation shrinks the effective Fermi surface and Fermi shell widths, and enlarges the region of parameter space with a finite-momentum instability.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Longitudinal spectral-weight claims in Sec 4.2 rest on an unspecified root of cubic (4.20); without the ν- selection rule and scan details, Figs 7–8 are not independently checkable.","rationale":"The paper's transverse-channel formulas are explicit and reduce to known limits, giving some independent support to the claim that m and W0 suppress k*. However, the most novel longitudinal claims are computed from a cubic whose relevant root is never identified. The reader's weakest assumption identifies precisely this gap: the numerical root analysis for ν- in Section 4.2 is underspecified. My reading of the paper confirms that no root-selection rule, numerical method, or scan details are given, and the figures are therefore not reproducible from the text alone. The Section 2.2 parameter-space inequalities are internally inconsistent, which further undermines the baseline comparison but is secondary to the cubic-root issue. Because the concern is about reproducibility and potential root misidentification rather than a demonstrated error, the appropriate verdict remains conditional: the central claims should be accepted only after the numerical procedure is specified and the figures are independently reproduced. No adjustment to the reader's verdict is needed.","tokens_in":16084,"tokens_out":7155,"duration_ms":67707,"concrete_test":"Independently solve the cubic (4.20) with coefficients (4.21) on a dense (η,ζ,m^2,k) grid covering the ranges in Figs. 7–8, e.g., η=1, ζ∈[-0.2,0.2], m^2∈[0,2). Identify the root whose real part gives 2ν- matching (i) the m=0 limit of [12] and (ii) the W0=0 limit of Eqs. (3.15)–(3.16). Then recompute 2ν- -1 and the instability condition and overplot Figs. 7 and 8. If the selected root or the scan boundaries differ, the claimed enhancement of the instability region and suppression of spectral weight are not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central new longitudinal results—that axion strength m enlarges the finite-k instability region into ζ>0 (Fig. 7) and suppresses or reshapes low-energy spectral weight (Fig. 8)—are obtained numerically from the scaling exponent ν-, but the paper never states which of the three roots of the cubic (4.20)–(4.21) is ν-, nor the branch choices, root-finding algorithm, scan resolution, or how the allowed parameter space was discretized. Without this, a reader cannot reproduce Fig. 7 or Fig. 8, and the qualitative claims could be artifacts of root misidentification or an incomplete scan. The same section says the closed form is 'too complicated to report', which makes the missing numerical specification decisive. A secondary internal inconsistency compounds the issue: Eq. (2.13) restricts ζ to positive values while the text says it 'basically restricts ζ<0', and Eq. (2.14) has lower bound η<W0 while the text says '0<W0<η'; Figure 3 uses the latter. This ambiguity affects the baseline superfluid results that the axion comparison depends on.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":16284,"tokens_out":6784,"duration_ms":67432,"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":[{"comment":"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.","section":"Sec. 4.2, Eqs. (4.19)-(4.21), Figs. 7-8"},{"comment":"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.","section":"Sec. 2.2, Eqs. (2.13)-(2.14), Fig. 3"}],"minor_comments":[{"comment":"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.","section":"Eq. (1.2) and Appendix B, Eq. (B.13)"},{"comment":"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'.","section":"Sec. 5, Discussion"},{"comment":"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.","section":"Fig. 6 caption"},{"comment":"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.","section":"Sec. 4.2, discussion of Fig. 7"}],"recommendation":"major_revision","confidential_remarks":"The paper is an honest extension of the author's earlier work, and the transverse-channel results are analytically solid. The main barrier to acceptance is the irreproducibility of the longitudinal numerical analysis: the unspecified root of the cubic in Eq. (4.20) is load-bearing for the qualitative claims in Figures 7 and 8. If the authors add the root-selection rule and numerical scan details, the paper would be suitable for publication. The internal inconsistency in Section 2.2 should also be fixed, as it undermines the baseline comparison."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The take-home: the transverse-channel part of this paper is solid and explicit, and worth taking seriously; the longitudinal-channel part, which carries the more interesting claims, is not independently checkable as written, because the paper never says which root of the cubic (4.20) is ν−, nor how Figures 7 and 8 were generated.\n\nWhat's new: for the Einstein-Maxwell-dilaton-axion theory with a massive vector, the closed-form transverse exponent (4.12) and critical momentum (4.14) are new, and they reduce cleanly to the known m=0 and W0=0 limits. The observation that the axion parameter m and the condensate charge W0 suppress k* in the same way in the transverse channel, but behave differently in the longitudinal channel, is a useful classification for bottom-up model builders. The paper is also honest: it uses a scaling ansatz with no fitting, and it hedges appropriately with 'seems'.\n\nSoft spots, in proportion:\n\nFirst, the parameter space text in Section 2.2 is internally contradictory. Equation (2.13) gives instability for 0<ζ<... (positive ζ), but the sentence right after says it 'basically restricts ζ<0'. Equation (2.14) has lower bound η<W0, but the text says it restricts 0<W0<η. That matters: Figure 3 and the Fermi-shell-width discussion depend on which inequality you believe. This is a fixable error, but it needs fixing.\n\nSecond, and more serious, the longitudinal channel results in Section 4.2 rest on the roots Yi of the cubic (4.20), but no branch selection is stated. The paper says the closed form is too complicated and then proceeds numerically. For a reader to check Figures 7 and 8, we need to know which of the three roots is ν−, the branch choice, the root-finding method, and the scan resolution. Without that, the claims that axions enlarge the instability region into ζ>0 and convert Fermi shells into smeared surfaces are not reproducible. This is the load-bearing issue for the paper's central message.\n\nMinor: the abstract says 'spontaneously broken translational symmetry', but axions ψ=mx break translations explicitly, as the introduction correctly says. That should be corrected.\n\nBottom line: this is a competent paper for the holographic spectral-weight community. The transverse formulas are clean enough to be useful, and the qualitative longitudinal picture is plausible. But the longitudinal claims need the missing numerical specifications, and the parameter-space text needs a fix, before a reader can trust them. I'd send it to a serious referee, not desk-reject it; the issues are substantive but not fatal.","headline":"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.","tokens_in":16832,"tokens_out":3674,"would_cite":false,"duration_ms":35398,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Explicit momentum relaxation in a holographic superfluid suppresses anomalous low-energy spectral weight and enlarges the finite-momentum instability region, without eliminating Fermi shells.","keywords":["holographic superconductor","spectral weight","momentum relaxation","axion","Fermi shell","semi-local quantum liquid","finite-momentum instability","AdS/CFT"],"falsifier":"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.","tokens_in":15803,"feed_emoji":"⚛️","tokens_out":13131,"duration_ms":123127,"temperature":0.7,"pith_summary":"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.","feed_headline":"Broken translations shrink Fermi surfaces and widen instabilities","feed_subtitle":"Momentum relaxation in a holographic superfluid suppresses spectral weight but enlarges the instability region.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $\\zeta=-0.07$."],"supporting_citations":[{"why":"Supplies the holographic superfluid spectral-weight analysis, Fermi-shell region, and finite-k instability that this paper extends to broken translations.","marker":"[12]"},{"why":"Establishes low-energy spectral weight in the semi-local quantum liquid/EMD geometry, the baseline for all models considered here.","marker":"[10]"},{"why":"Provides the axion/EMDA background and parameter space used to model momentum dissipation.","marker":"[18]"},{"why":"Companion analysis showing the pervasiveness of Fermi shells in eta-geometries and giving higher-dimensional context.","marker":"[7]"},{"why":"Introduces the holographic superconductor model whose U(1) breaking is combined with axions.","marker":"[11]"},{"why":"Gives the IR/UV matching relation connecting the IR scaling exponent to the UV spectral weight.","marker":"[6]"}],"fun_headline_variants":["Axion shrinks Fermi shells, widens instabilities","Broken translations suppress holographic Fermi surfaces","Momentum relaxation enlarges superfluid instabilities","Axion smears Fermi shells in holographic superfluid"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Axion shrinks Fermi shells, widens instabilities","Broken translations suppress holographic Fermi surfaces","Momentum relaxation enlarges superfluid instabilities","Axion smears Fermi shells in holographic superfluid"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000242,"raw_usage":{"total_tokens":1478,"prompt_tokens":849,"completion_tokens":629,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":465,"completion_tokens_details":{"reasoning_tokens":566}},"tokens_in":465,"tokens_out":629,"duration_ms":7260,"temperature":1.0,"reasoning_tokens":566,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:46:09.311534+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Spectral weight and spatially modulated instabilities in holographic superfluids","cited_arxiv_id":"1612.03466","evidence_quote":"Supplies the holographic superfluid spectral-weight analysis, Fermi-shell region, and finite-k instability that this paper extends to broken translations."},{"cited_title":"The Pauli exclusion principle at strong coupling: Holographic matter and momentum space","cited_arxiv_id":"1210.1590","evidence_quote":"Establishes low-energy spectral weight in the semi-local quantum liquid/EMD geometry, the baseline for all models considered here."},{"cited_title":"Spectral weight in Chern-Simons theory with symmetry breaking","cited_arxiv_id":"1905.07417","evidence_quote":"Companion analysis showing the pervasiveness of Fermi shells in eta-geometries and giving higher-dimensional context."}],"review_version":1}