{"id":"75716b31-d5ca-429b-aaa5-2a188dae66ef","arxiv_id":"1908.04800","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Zero-sound collective modes in a clean or disordered Fermi liquid are mapped across the Pomeranchuk instability, including silent poles on unphysical Riemann sheets and a special l=1 current channel whose instability survives a vanishing residue.","lead":"This paper derives how collective sound-like modes in a two-dimensional Fermi liquid evolve as interactions are tuned toward a Pomeranchuk instability, where the Fermi surface spontaneously changes shape. It shows that, for current-type order in the l=1 channel, the static response stays finite while the dynamical mode still crosses into the unstable regime, so the ordered state forms but slowly.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The universal classification rests on the single-harmonic RPA ansatz of Eq. (7); a two-harmonic kinetic-equation check would show whether the F_l=-1 crossing survives without fine-tuning.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing premise: the single-harmonic quasiparticle susceptibility of Eq. (7). My stress-test agrees that this is the least secure condition for the central claim. The paper states the assumption explicitly and partially relaxes it for F_0 and F_1, but the universal arbitrary-l classification in Sec. II E is not protected against mixing of other Landau harmonics or against non-Fermi-liquid self-energy corrections near the instability. These are not internal inconsistencies; they are scope limitations that make the claimed universality conditional on a fine-tuned single-harmonic model. Since the reader already assigned CONDITIONAL with medium correctness risk, my analysis does not move the verdict. A concrete two-harmonic check would settle whether the F_l=-1 pole crossing is robust beyond the single-harmonic truncation.","tokens_in":44285,"tokens_out":8799,"duration_ms":106055,"concrete_test":"Use the Landau kinetic-equation / RPA expressions generalized to two comparable harmonics, e.g., F_2 and F_4 (or F_1 and F_3), and numerically follow the poles of the full susceptibility as F_2 approaches -1 at fixed small F_4. Check whether (i) the critical mode frequency still vanishes exactly at F_2=-1 and (ii) the longitudinal versus transverse, even versus odd classification of Sec. II E is unchanged. If the vanishing point shifts to the determinant condition, or the mode changes from relaxational to propagating, the Eq. (7)-based universality fails for generic parameters.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Central claim: for each l, in both charge and spin channels and in both longitudinal and transverse susceptibilities, a critical zero-sound mode has frequency zero at F_l=-1 and crosses into the upper frequency half-plane for F_l<-1. The engine of the proof is Eq. (7), chi_qp,l = nu_F chi_free,l/(1 + F_l chi_free,l), which is justified only when all Landau parameters except F_l are small. The pole equations and the even/odd parity classification in Sec. II E are derived entirely from this single-harmonic form. This is the load-bearing point: a Pomeranchuk instability in a rotationally invariant Fermi liquid is an eigenvalue condition of the full Landau matrix; if other F_l' are not negligible, the collective-mode pole is governed by a determinant that mixes harmonics, and the critical point is not generically at F_l=-1. The paper partially relaxes this only for F_0 and F_1 (Sec. II F), where the crossing is renormalized but survives; no analogous result is given for arbitrary l or for multiple non-neighboring harmonics. Separately, the disclaimer after Sec. I concedes that the Fermi-liquid range shrinks near the instability, so the free-fermion bubble in Eq. (7) misses singular self-energy corrections exactly in the regime where the universal classification is claimed. This does not invalidate the explicit l=0,1,2 calculations, but it means the 'for every l' universality is established only on a fine-tuned single-harmonic surface.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper analyzes collective zero-sound modes associated with Pomeranchuk instabilities in a two-dimensional Fermi liquid. Starting from the single-harmonic RPA-like form of the quasiparticle susceptibility, Eq. (7), the authors derive explicit pole trajectories for l=0, 1, and 2 in both longitudinal and transverse channels, and for both charge and spin sectors, and then present a general classification for arbitrary l in Sec. II E. The central claim is that for each l there is a critical zero-sound mode whose frequency vanishes at the Pomeranchuk instability F_l^{c(s)}=-1 and which moves into the upper frequency half-plane for F_l^{c(s)}<-1. The paper also includes a treatment of finite disorder, a time-domain analysis of the response, and a special analysis of the l=1 charge-current and spin-current order parameters, where the Ward identity forces the static vertex to vanish as 1+F_1 so that the static susceptibility remains finite at F_1=-1 even though the dynamical susceptibility still has a pole that crosses into the upper half-plane. The l=1 claim is supported by a separate second-order perturbation calculation and by a Ginzburg-Landau functional argument.","tokens_in":44544,"tokens_out":6732,"duration_ms":75737,"significance":"If the central claims hold, the paper provides a useful and largely self-contained classification of collective modes near Pomeranchuk instabilities. The explicit l=0,1,2 pole equations, the detailed Riemann-sheet analysis, and the treatment of the l=1 current channel are valuable, and the paper contains no fitted parameters. The separate Ward-identity and O(U^2) perturbation-theory checks for the l=1 current case are a distinct strength, as they address an apparent paradox in the existing literature. The main limitation is that the 'for every l' universality is established only under the single-harmonic assumption of Eq. (7); if several Landau harmonics are comparable, the pole condition becomes a determinant condition that mixes harmonics, and the critical value is not generically F_l=-1. The paper's own Sec. II F shows this explicitly for the F_0/F_1 pair, and no analogous check is provided for general l.","major_comments":[{"comment":"The universal classification for arbitrary l in Sec. II E is derived entirely from the single-harmonic RPA expression Eq. (7), which the authors themselves state is valid only when all Landau parameters except one are small. In a rotationally invariant Fermi liquid, the collective-mode pole is governed by the full Landau matrix, and when two or more harmonics are comparable the denominator of the susceptibility is a determinant that mixes them, as demonstrated for F_0 and F_1 in Eq. (67). The manuscript gives a two-parameter check only for the F_0/F_1 pair and provides no analogous result for arbitrary l or for two distant harmonics. To sustain the 'for every l' claim in the abstract and conclusions, the authors should either prove that the F_l=-1 crossing is robust to generic multi-harmonic perturbations (for example, by a two-harmonic calculation for l and l+2) or explicitly restate the entire classification as conditional on single-harmonic dominance.","section":"Sec. II A, Eq. (7); Sec. II E; Sec. II F"},{"comment":"The concluding paragraph states, without qualification, that 'Right at the transition, the mode is located at omega=0, i.e. the static susceptibility diverges.' This contradicts the main result of Sec. V for the l=1 charge- and spin-current order parameters: there, the Ward identity makes the static vertex proportional to 1+F_1^{c(s)}, so the full static susceptibility in Eq. (118) remains finite at F_1^{c(s)}=-1 even though a dynamical pole crosses the real axis. The conclusion needs to be qualified to generic (non-current) order parameters, or the l=1 current case must be explicitly separated out.","section":"Sec. VI, Conclusions"},{"comment":"The disclaimer that the Fermi-liquid range shrinks near the Pomeranchuk instability is not quantified. Since the critical mode has s=omega/(v_F^* q) -> 0 exactly at F_l=-1, the poles of interest lie precisely in the regime where quasiparticle self-energy corrections are most singular, and the free-fermion bubble in Eq. (7) may receive non-Fermi-liquid corrections. The authors assume that at any fixed distance from the critical point one can choose omega and q small enough to remain in the Fermi-liquid regime, but no smallness parameter or estimate is provided. A quantitative statement, or a restriction of the universal pole-crossing claim to the regime where such corrections are controlled, would remove a correctness risk at the central point of the paper.","section":"End of Sec. I"}],"minor_comments":[{"comment":"The abstract contains a typographical error: 'susceptiblities' should be 'susceptibilities'.","section":"Abstract"},{"comment":"In the paragraph beginning 'For F_1 > 1/3' in the l=2 transverse channel subsection, the Landau parameter should be F_2^{c(s)}, not F_1^{c(s)}.","section":"Sec. II D.2"},{"comment":"The sentence 'At F_0 -> 0, s_{i,1} approx ...' appears in the l=1 transverse subsection and should refer to F_1, not F_0.","section":"Sec. III C.2"},{"comment":"The integrand 'cos x t^*' should be written as cos(x t^*), and the parentheses around the branch-cut discontinuity are missing; this makes the equation unnecessarily hard to read.","section":"Eq. (103)"},{"comment":"The word 'susceptibiliies' in the figure caption is a typo for 'susceptibilities'.","section":"Caption of Fig. 2"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the journal's scope and the explicit l=0,1,2 calculations are worth publishing after the load-bearing single-harmonic assumption is addressed. The main decision hinges on whether the 'for every l' universality claim can be either proven beyond the fine-tuned single-harmonic surface or appropriately downgraded in the abstract and conclusions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this one if you care about Fermi liquid collective modes or Pomeranchuk transitions. The genuinely new result is the l=1 current-order case: the static susceptibility does not diverge at F1=-1 because the vertex vanishes via a Ward identity, yet the dynamical susceptibility still has a pole that crosses into the upper half-plane, with residue vanishing as (1+F1)^2. The authors back this with an explicit O(U^2) perturbation-theory calculation, so it is not handwaving. This resolves a contested point in the literature.\n\nThe paper also maps the full pole evolution for l=0,1,2 in both channels, classifies even/odd parity for arbitrary l, and identifies 'silent' modes on unphysical Riemann sheets. The time-domain analysis is careful about causality when poles leave the lower half-plane. The authors explicitly acknowledge that some l=0,1,2 susceptibilities were computed before, so the novelty claim is honest: the uniform treatment, the Riemann-sheet analysis, and the l=1 resolution are the new parts.\n\nThe main soft spot is one the authors themselves flag. Eq. (7), the engine of the classification, assumes a single Landau harmonic dominates. For arbitrary l, the universal statements live on that single-harmonic surface. They relax this for F0 and F1, where the crossing survives with renormalized prefactors, but not for general l. That limits the 'universal' claim but does not invalidate the explicit l=0,1,2 results or the l=1 Ward-identity argument, which rests on separate grounds. The arbitrary-l section is more schematic, the finite-disorder l=2 results are stated without derivation, and the time-domain companion is unpublished. These are minor-to-moderate concerns for this type of paper. The disclaimer after Sec. I about the shrinking Fermi-liquid range near the instability is honest, but it does mean the F_l=-1 crossing statements are quasiparticle statements.\n\nWho is this for: researchers working on nematic or current order, zero sound, or quantum critical dynamics. It is a solid, honest paper that deserves a serious referee. The l=1 result is important enough that the field should decide whether it survives beyond the single-harmonic model; the tools to test that are all here. I would send it to review, with a request to tighten the arbitrary-l and disordered sections.","headline":"A solid, honest map of zero-sound poles near Pomeranchuk instabilities; the l=1 current-order resolution is the real news, and the stated single-harmonic scope is a caveat, not a fatal flaw.","tokens_in":45131,"tokens_out":3451,"would_cite":true,"duration_ms":34554,"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":"Every angular-momentum channel of a two-dimensional Fermi liquid has a critical zero-sound mode that freezes at the Pomeranchuk instability and then crosses into the unstable half-plane; for l=1 current-type order, the static…","keywords":["Fermi liquid","Pomeranchuk instability","zero sound","dynamical susceptibility","Landau parameters","Ward identity","collective modes","nematic order"],"falsifier":"A microscopic calculation of the retarded current-order susceptibility at second order in the interaction, with vertex corrections included beyond the one-harmonic kernel, would settle whether the $l=1$ pole really moves to the upper half-plane below $F_1^{c(s)}=-1$; a pump-probe measurement of the same channel across a clean two-dimensional nematic transition would provide an experimental check.","tokens_in":44064,"feed_emoji":"🌊","tokens_out":11351,"duration_ms":109229,"temperature":0.7,"pith_summary":"The paper shows that in a two-dimensional Fermi liquid every angular-momentum channel $l$ carries a critical collective mode that freezes at the Pomeranchuk instability $F_l^{c(s)}=-1$, and that below this value the mode's pole moves into the upper frequency half-plane, so any fluctuation of the corresponding order parameter grows exponentially. The character of the critical mode depends on parity and on the channel geometry: in even-$l$ longitudinal and odd-$l$ transverse susceptibilities the pole approaches along the imaginary axis (purely relaxational), while in odd-$l$ longitudinal and even-$l$ transverse channels it approaches with a real part much larger than the imaginary part (almost propagating). For the special $l=1$ order parameters with the symmetry of charge or spin current, the static susceptibility remains finite at $F_1^{c(s)}=-1$ because a Ward identity makes the vertex vanish, but the dynamical susceptibility still has a pole that crosses into the unstable half-plane, with residue proportional to $(1+F_1^{c(s)})^2$. A sympathetic reader should care because this gives a unified picture of collective-mode softening at rotational-symmetry-breaking transitions and corrects the earlier claim that current-type Pomeranchuk order cannot develop.","feed_headline":"Zero-sound mode freezes at every Pomeranchuk transition","feed_subtitle":"Even a current-type order whose static response stays finite still shows an unstable pole once F_1 < -1.","key_machinery":"The load-bearing object is the single-harmonic quasiparticle susceptibility $\\chi_{qp,l}^{c(s)}(s)=\\nu_F\\,\\chi_{free,l}(s)\\big/\\big(1+F_l^{c(s)}\\chi_{free,l}(s)\\big)$, whose free-fermion bubble $\\chi_{free,l}(s)$ is evaluated on the Fermi surface with $\\sqrt{2}\\cos l\\theta$ or $\\sqrt{2}\\sin l\\theta$ form factors depending on the longitudinal or transverse channel. Zero-sound collective modes are the poles of this function, and the critical mode is the pole whose velocity $s v_F^*$ tends to zero as $F_l^{c(s)}\\to -1$. The square-root branch cuts of the two-dimensional Lindhard function make the pole locations sheet-dependent, which explains the existence of silent modes; the Ward identity $\\Lambda_1^{c(s)}\\propto 1+F_1^{c(s)}$ for current-type order parameters supplies the extra vertex factor that keeps the static response finite while leaving the dynamical pole intact.","core_discovery":"The central claim is that the retarded quasiparticle susceptibility $\\chi_{qp,l}^{c(s)}(s)$, with $s=\\omega/(v_F^* q)$, has a pole at $s=0$ exactly when $F_l^{c(s)}=-1$, and that for $F_l^{c(s)}<-1$ this pole lies in the upper half-plane, making the symmetric Fermi liquid unstable. The pole trajectories have a parity structure: for even $l$ in the longitudinal channel and odd $l$ in the transverse channel the critical pole is purely imaginary, $s\\approx -i(1+F_l^{c(s)})/2$, whereas for odd $l$ longitudinal and even $l$ transverse channels it is almost real, $s\\approx \\pm\\big((1+F_l^{c(s)})/2l\\big)^{1/2}-i(1+F_l^{c(s)})/4$. Away from the critical point the poles move on a two-sheet Riemann surface, and some sink below the branch cut, so they are present mathematically but silent spectroscopically. For $l=1$ current-type order parameters, the Ward identity $\\Lambda_1^{c(s)}\\propto 1+F_1^{c(s)}$ keeps the static susceptibility finite at the transition, yet the same dynamical pole survives in the full susceptibility because its residue scales as $(1+F_1^{c(s)})^2$; the paper concludes that current-type Pomeranchuk order does develop, but its time evolution is slower.","pith_inferences":["The single-harmonic pole analysis suggests a general rule for weakly coupled channels: as long as no other Landau parameter is itself close to $-1$, the critical mode of the dominant harmonic survives with only its velocity renormalized by the other harmonics, by analogy with the paper's explicit two-parameter result.","The silent modes imply a sharp operational distinction between equilibrium spectroscopy and pump-probe response: a mode sitting below the branch cut will be invisible in the frequency-dependent loss but should still dominate the time-domain decay, so time-resolved experiments are a direct way to search for it.","A concrete testable extension is to apply an ultrafast perturbation with the symmetry of the $l=1$ current order in a clean two-dimensional electron system tuned across a nematic transition: the predicted response changes from decaying oscillations to $t\\sinh(\\sqrt{|1+F_1|/2}\\,t)$-type growth, with an amplitude suppressed by $(1+F_1)^2$.","The same pole-trajectory logic may carry over to other Fermi-surface instabilities whose low-energy response has the same free-fermion-bubble structure, in which case the parity classification of critical dynamics is not specific to Pomeranchuk transitions."],"forward_implications":["At every Pomeranchuk transition the collective-mode velocity vanishes, so long-wavelength order-parameter fluctuations become soft and slow as $F_l^{c(s)}$ approaches $-1$ from above.","The parity rule predicts distinct time-domain signatures: in even-$l$ longitudinal and odd-$l$ transverse channels the order parameter relaxes monotonically near the transition, while in the complementary channels it oscillates with decaying amplitude and a period that diverges at the transition.","For $l=1$ current-type order a static measurement of the susceptibility will not show a divergence at $F_1^{c(s)}=-1$, but the system still becomes unstable below this value; the growth of order after an instantaneous perturbation is real but delayed by the $(1+F_1^{c(s)})^2$ residue.","For positive $F_l^{c(s)}$ with $l>0$, zero-sound poles can move to an unphysical Riemann sheet beyond a threshold, so the spectral peak in the imaginary part of the susceptibility may be absent or anomalously broad even though a collective mode exists; the long-time response is then governed by the branch-cut endpoint.","Impurity scattering overdamps the critical modes and destroys the clean threshold behavior, so the clean parity classification applies only to sufficiently clean Fermi liquids."],"supporting_citations":[{"why":"Supplies the Landau quasiparticle formalism and the definition of Landau parameters used throughout.","marker":"[7]"},{"why":"Gives the general decomposition of the full susceptibility into quasiparticle, side-vertex, and incoherent parts used for all l.","marker":"[29]"},{"why":"Found that the static l=1 current/spin susceptibility does not diverge at F_1=-1, the result the paper extends to dynamics.","marker":"[26]"},{"why":"Derives the Ward identity Lambda_1^{c(s)} proportional to 1+F_1^{c(s)} that controls the l=1 current-order vertex.","marker":"[28]"},{"why":"Earlier analysis of zero-sound modes near the l=0 instability, including the causality contour used below the transition.","marker":"[30]"},{"why":"Supplies the coupled RPA susceptibility with two Landau parameters, used for the F0/F1 case and for the general pole structure.","marker":"[35]"},{"why":"Shows the threshold condition for propagating transverse l=1 zero-sound at positive F_1.","marker":"[36]"},{"why":"Provides the disorder vertex-correction scheme that yields the conserved l=0 and nonconserved l>0 free-fermion susceptibilities.","marker":"[44]"}],"fun_headline_variants":["Zero-sound mode freezes at each Pomeranchuk transition","Critical mode's velocity hits zero at F_l = -1","Parity controls whether critical mode relaxes or propagates","Current order still hosts unstable mode at Pomeranchuk"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The assumption that the full low-energy dynamics is captured by the single-harmonic quasiparticle form $\\chi_{qp,l} = \\nu_F\\chi_{free,l}/(1+F_l\\chi_{free,l})$ with one Landau parameter dominating is load-bearing; if other harmonics are not negligible near the instability, or if non-Fermi-liquid corrections alter the kernel, the specific pole trajectories and parity classification could fail.","fun_headline_variants_meta":{"raw":{"variants":["Zero-sound mode freezes at each Pomeranchuk transition","Critical mode's velocity hits zero at F_l = -1","Parity controls whether critical mode relaxes or propagates","Current order still hosts unstable mode at Pomeranchuk"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00069,"raw_usage":{"total_tokens":3265,"prompt_tokens":1224,"completion_tokens":2041,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":840,"completion_tokens_details":{"reasoning_tokens":1973}},"tokens_in":840,"tokens_out":2041,"duration_ms":14742,"temperature":1.0,"reasoning_tokens":1973,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:32:38.421069+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A microscopic calculation of the retarded current-order susceptibility at second order in the interaction, with vertex corrections included beyond the one-harmonic kernel, would settle whether the $l=1$ pole really moves to the upper half-plane below $F_1^{c(s)}=-1$; a pump-probe measurement of the same channel across a clean two-dimensional nematic transition would provide an experimental check.","supporting_citations":[{"cited_title":"30 and 46","cited_arxiv_id":null,"evidence_quote":"Supplies the Landau quasiparticle formalism and the definition of Landau parameters used throughout."}],"review_version":1}