{"id":"191df963-df88-46a8-8512-02e3b8d49107","arxiv_id":"2607.07804","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":5,"one_line_summary":"An all-electronic FRG with retarded phonon-mediated interactions treats charge, spin, nematic, and pairing instabilities on equal footing and is illustrated for the square-lattice Hubbard model with acoustic phonons.","lead":"The authors build a functional renormalization-group method that puts electron-electron and electron-phonon interactions on the same footing by integrating out dispersive phonons into retarded electronic vertices. The approach is demonstrated on the square-lattice Hubbard model with acoustic phonons, recovering Peierls-type charge-bond orders and mapping their competition with magnetism and superconductivity.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified beyond the reader's already-flagged freezing of the retarded vertex.","rationale":"The reader's strongest claim accurately restates the paper's methodological contribution and its concrete illustration. The weakest assumption identified by the reader—freezing V_R—is indeed the softest internal link, but it is already flagged, is common practice for qualitative FRG phase diagrams, and is buttressed by the authors' own reference to a full-frequency study that found no qualitative change. The D-channel equivalence to phonon softening (Eqs. 39–43) is cleanly derived, the acoustic-phonon discontinuity is handled by a small regulator, and the reported phase diagrams are consistent with known SSH-Hubbard and pure-Hubbard limits. No stronger load-bearing flaw (incorrect channel projection, double-counting of DFPT screening, or internal contradiction) is present. Consequently the CONDITIONAL verdict with high confidence remains appropriate; no adjustment is required.","tokens_in":24949,"tokens_out":535,"duration_ms":6070,"concrete_test":"Re-run the (λ,α^{2}) FRG phase diagram of Fig. 6(c) at fixed U=0 with a single additional frequency-dependent update of the D-channel phonon propagator at each scale (or compare against the full-frequency Holstein FRG of Ref. [56] at the same parameters); if the CBO–dPom–AFM* boundaries shift by more than ~20 % in λ or α^{2} the static approximation is quantitatively fragile, otherwise the claim stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that an all-electronic static FRG with phonon-mediated retardation (Eqs. 20, 29–31) places phonon- and electron-mediated Fermi-liquid instabilities on equal footing and recovers Peierls-type charge-bond order for the square-lattice Hubbard model with acoustic phonons. The single most load-bearing technical choice is precisely the one the reader already isolates: the retarded vertex V_R is held fixed along the flow (Section III C). That approximation is standard for qualitative phase-boundary work, is supported by the cited full-frequency Hubbard–Holstein FRG comparison, and is required for the claimed computational equivalence to ordinary static TUFRG. No independent derivation error, channel inconsistency, or hidden double-counting appears in the D-channel equivalence proof (Section III F) or in the reported phase diagrams. The concern therefore does not rise above the reader's existing weakest-assumption statement.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript formulates a functional renormalization-group (FRG) scheme for lattice fermions that incorporates both electronic interactions and electron-phonon coupling (EPC) from dispersive phonon bands. Phonons are integrated out to produce a retarded density-density interaction that is inserted into the static truncated-unity FRG flow of the electronic two-particle vertex, with channel-dependent frequency projections that retain retardation effects (Eqs. 20, 29–31). The approach is illustrated on the square-lattice Hubbard model coupled to acoustic phonons, where pure D-channel FRG is shown to be equivalent to leading-order phonon softening (Eqs. 39–43), and full multi-channel FRG yields phase diagrams of Peierls-type charge-bond order competing with antiferromagnetism, d-wave Pomeranchuk order, and s-/d-wave superconductivity as functions of EPC strength, phonon anisotropy, Hubbard U and doping.","tokens_in":25215,"tokens_out":1259,"duration_ms":24011,"significance":"If the approximations hold, the work supplies a computationally practical, channel-unbiased framework that places phonon- and electron-mediated Fermi-liquid instabilities on equal footing and is already interfaced to existing ab-initio FRG codes. The analytic and numerical demonstration that D-channel FRG recovers the RPA phonon self-energy, together with the explicit (λ,α²) and (U,λ) phase diagrams, constitutes a concrete advance over earlier Holstein or SSH treatments limited to local or optical modes. The method’s scalability to multi-orbital, multi-phonon ab-initio Hamiltonians is a genuine strength for materials such as kagome metals and nickelates where charge, spin and lattice orders intertwine.","major_comments":[{"comment":"Section III C states that the retarded phonon-mediated vertex V_R is held fixed throughout the flow and is never renormalized. This is the central technical approximation that enables computational equivalence to static TUFRG. While the authors cite a full-frequency Hubbard–Holstein study [56] in support, that model has momentum-independent optical phonons; for the acoustic, dispersive, momentum-dependent EPC of the present work the frequency structure of vertex corrections differs. A quantitative estimate (or a limited frequency-dependent check) of the error incurred by freezing V_R is needed before the phase boundaries in Figs. 6 and 8 can be regarded as robust.","section":"Section III C"},{"comment":"Section III A and Fig. 2 introduce a finite lifetime regulator δ=10^{-7} that forces V(q=0,ω_n=0)=0 in order to remove the directional discontinuity of acoustic phonons. The d-wave Pomeranchuk instability reported in the large-α² FRG phase diagram is an asymptotic q→0 order. The manuscript does not demonstrate that the location or critical scale of this instability is stable under variation of δ (or under a soft infrared cutoff). Such a check is required to confirm that the dPom phase is physical rather than an artifact of the regulator.","section":"Section III A, Fig. 2"},{"comment":"The equivalence between phonon softening and electronic D-RPA is established rigorously only for U=0 (Section III F and Fig. 5). Once a finite Hubbard U is present, the full multi-channel FRG generates additional self-energy and vertex corrections that have no direct counterpart in the phonon Dyson equation used in DFPT. The manuscript should clarify to what extent the “phononic picture” remains valid in the (U,λ) diagrams of Fig. 8, or explicitly restrict the phonon-softening interpretation to the U=0 sector.","section":"Section III F, Fig. 8"}],"minor_comments":[{"comment":"Several section headings contain spurious spaces (“F unctional Renormalization”, “T runcated Unity FRG”, “RESUL TS”, “SUMMAR Y & OUTLOOK”), presumably from PDF extraction; these should be corrected for the final version.","section":"Throughout"},{"comment":"In the Introduction the phrase “charge-bond order from the electronic picture condify the same transition” appears to be a typographical error for “codify” or “confirm”.","section":"Section I"},{"comment":"Figure 1(d) caption refers to “red line thickness corresponding to its strength s(ν)_q”; the definition of s(ν)_q is given only later in Eq. (16). A forward reference would improve readability.","section":"Fig. 1"},{"comment":"The parameter α² is defined in Eq. (45) as K∥/(K∥+K⊥), yet the phase diagrams of Fig. 6 are plotted versus α² while the text occasionally refers to “α”. Consistent notation would avoid confusion.","section":"Eq. (45), Fig. 6"},{"comment":"The outlook lists eight material platforms; a brief remark on which of them already possess publicly available cDFPT or EPC matrix elements would help readers assess immediate applicability.","section":"Section V"}],"recommendation":"minor_revision","confidential_remarks":"The methodological core is solid and the phase diagrams are interesting, but the work is primarily a methods paper with a single-model illustration. It is better suited to Phys. Rev. B than to a higher-impact venue that expects either a decisive materials prediction or a fully frequency-dependent benchmark. The frozen-V_R approximation is standard yet remains the weakest link; if the authors can add even a limited frequency-dependent test or a clear error estimate, the paper will be ready for acceptance after minor revision."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing worth knowing is that this is a practical, ab-initio-ready extension of static TUFRG that puts phonon-mediated and electron-mediated channels on the same footing without inventing a new bosonic FRG. They integrate out dispersive acoustic phonons to a retarded density-density vertex, keep only the static electronic flow, and show analytically and numerically that pure D-channel FRG is exactly the leading phonon-softening RPA. That equivalence is clean (Eqs. 39–43) and the phase diagrams for the square-lattice Hubbard model plus acoustic springs are new relative to the Holstein/SSH literature they cite.\n\nWhat they do well: the derivation from IFCs through EPC to the channel-projected flow (Eqs. 20, 29–31) is internally consistent, the q→0 regulator is handled carefully, and the (λ,α²) and (U,λ) diagrams map the expected competition—X-type vs M-type charge-bond order, the O(4) AFM*/sSC/CDW degeneracy at U=0 half-filling, and the doping-driven switch to dSC or sSC—without overclaiming quantitative accuracy. Cross-channel feedback and the hierarchy set by ω/Λc are explained with the right diagrams. Citations to prior FRG and DFPT work are honest; self-cites are to their own code base and are not load-bearing.\n\nThe soft spot is exactly the one the reader flagged: the retarded piece is frozen along the flow. That is a standard qualitative approximation, backed by the full-frequency Holstein comparison they cite, and it is what keeps the cost identical to ordinary static TUFRG. It is not a derivation error and does not break the D-channel equivalence. No code for the extension is shipped, which is a minor practical annoyance. Everything else (regulator, form-factor truncation, free parameters) is transparent.\n\nThis is for people who already run FRG or DFPT on quantum materials and want a single framework for intertwined charge/spin/lattice order. It deserves a serious referee; the math and the phase diagrams are solid enough that the frozen-vertex choice can be debated in review rather than used as a desk-reject reason. I would bring it to reading group and would cite the method if I needed acoustic phonons in an FRG calculation.","headline":"Clean, usable FRG that folds dispersive acoustic phonons into the static electronic vertex and recovers Peierls CBOs competing with AFM and s/d-wave SC; the frozen-retarded-vertex choice is the only real soft spot and is already standard.","tokens_in":25827,"tokens_out":607,"would_cite":true,"duration_ms":13670,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["71.10.Fd","71.38.-k","74.20.Mn","71.45.Lr"],"model":"grok-4.5","headline":"A single FRG flow of the electronic vertex treats phonon- and electron-driven Fermi-liquid instabilities on equal footing, recovering Peierls charge-bond order and its competition with magnetism and superconductivity.","keywords":["functional renormalization group","electron-phonon coupling","Peierls transition","charge-bond order","Fermi-liquid instabilities","Hubbard model","acoustic phonons","truncated-unity FRG"],"falsifier":"A frequency-dependent FRG or determinant quantum Monte Carlo calculation on the same square-lattice acoustic model that finds a different leading instability or a substantially shifted critical scale for the same (U,λ,α²) parameters.","tokens_in":25851,"feed_emoji":"⚛️","tokens_out":968,"duration_ms":12566,"temperature":0.7,"pith_summary":"The paper sets out a practical way to put electron-electron repulsion and electron-phonon coupling inside one renormalization-group calculation. Phonons are integrated out once, leaving a retarded density-density interaction that is then fed into the usual all-electronic functional renormalization group for the two-particle vertex. Because every diagrammatic channel is kept, charge-bond order, antiferromagnetism, nematic Pomeranchuk order and both s- and d-wave pairing compete on the same footing. On the square-lattice Hubbard model with acoustic phonons the method recovers the expected Peierls charge-bond orders (and shows they are equivalent to phonon softening), then maps how those orders give way to magnetic and superconducting states once a Hubbard U is turned on or the system is doped. The calculation is designed so that realistic ab-initio band structures and phonon dispersions can be dropped in without changing the numerical cost of ordinary static FRG. A sympathetic reader therefore obtains a single, unbiased tool for deciding which electronic order wins when lattice and correlation effects are intertwined.","feed_headline":"One FRG flow unifies phonon and electronic Fermi-liquid orders","feed_subtitle":"Peierls bond order, magnetism and s/d-wave pairing compete inside a single static vertex calculation","key_machinery":"The phonon-mediated retarded interaction V (Eq. 20) projected into the static FRG channels according to Eqs. 29–31, so that the same vertex flow simultaneously generates charge-bond, spin and pairing instabilities.","core_discovery":"Integrating out dispersive phonons produces a retarded electron-electron interaction that can be inserted into a static truncated-unity FRG flow of the two-particle vertex; the resulting channel-decomposed flow treats phonon-mediated charge-bond order and purely electronic instabilities on equal footing and, for the square-lattice Hubbard model with acoustic phonons, yields Peierls-type transitions whose competition with AFM and s/d-wave superconductivity is fully mapped.","pith_inferences":["The same static-flow construction immediately extends to optical phonons, multi-orbital models and systems with spin-orbit coupling, giving a uniform route to electron-phonon problems that currently require separate Eliashberg or DFPT treatments.","Because the method already interfaces to existing ab-initio codes, the first concrete applications are likely to be materials in which DFPT already predicts soft modes while electronic correlations are known to be strong (kagome metals, nickelates, twisted graphene).","Holding the retarded interaction fixed is the price paid for numerical scalability; a controlled test that re-introduces selected frequency dependence only in the softest phonon modes would quantify how much the present phase boundaries move."],"forward_implications":["Peierls charge-bond order and phonon softening become two descriptions of one and the same transition inside a single electronic calculation.","Competition between conventional s-wave and unconventional d-wave pairing can be read off directly from the same FRG flow once both U and electron-phonon coupling are present.","Ab-initio electronic and phononic Hamiltonians can be fed into the existing static FRG pipeline without extra numerical cost, enabling material-specific phase diagrams for intertwined lattice and correlation instabilities.","Cross-channel vertex corrections lift artificial degeneracies (such as the O(4) AFM/sSC/CDW manifold) and generate mixed CDW+CBO order parameters that pure RPA cannot produce."],"fun_headline_variants":["Unified FRG flow for phonon and electronic Fermi liquid instabilities","Phonon Peierls order competes with AFM and pairing in one FRG","Electron-phonon FRG treats bond order and pairing equally","Single vertex flow maps Peierls against magnetism and SC","Retarded EPC joins electronic interactions in static FRG"],"cache_read_input_tokens":7936,"weakest_assumption_plain":"The phonon-induced retarded piece of the interaction is frozen and never itself renormalized during the flow; if frequency-dependent corrections to the electron-phonon coupling grow large, the phase boundaries can shift.","fun_headline_variants_meta":{"raw":{"variants":["Unified FRG flow for phonon and electronic Fermi liquid instabilities","Phonon Peierls order competes with AFM and pairing in one FRG","Electron-phonon FRG treats bond order and pairing equally","Single vertex flow maps Peierls against magnetism and SC","Retarded EPC joins electronic interactions in static FRG"]},"model":"grok-4.5","effort":"low","cost_usd":0.006242,"raw_usage":{"total_tokens":1541,"prompt_tokens":703,"num_sources_used":0,"completion_tokens":70,"cost_in_usd_ticks":62420000,"prompt_tokens_details":{"text_tokens":703,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":768,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":703,"tokens_out":70,"duration_ms":7297,"temperature":1.0,"reasoning_tokens":768,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-10T17:41:35.837933+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"A frequency-dependent FRG or determinant quantum Monte Carlo calculation on the same square-lattice acoustic model that finds a different leading instability or a substantially shifted critical scale for the same (U,λ,α²) parameters.","supporting_citations":[],"review_version":1}