{"id":"761612b7-9394-4d09-a7bf-2cac08a2e2a3","arxiv_id":"2608.07714","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Tuning a single hopping parameter through a higher-order Van Hove singularity changes the predicted ordered state of the square-lattice Hubbard model, and ferromagnetism at Van Hove filling is destroyed by very small doping.","lead":"Using two numerical renormalization-group methods, this paper maps out which ordered states, magnetic, superconducting, or density-wave, a correlated lattice model prefers when the shape of its electronic density-of-states peak is tuned by one hopping parameter. A generalist might care because the predicted ground state turns out to be extremely sensitive to tiny electron doping, a fact relevant for designing new quantum materials.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed 0.001 t1 doping fragility of ferromagnetism is likely below the TUFRG k-grid energy resolution near the saddle point, so the FM suppression in Fig. 8(a) may be a discretization artifact.","rationale":"The paper has two central strands: (1) the type of VHS controls the weak-coupling ordered state, and (2) the ferromagnetic state at Van Hove filling is destroyed by very small doping. The first strand is supported by two independent RG methods with broad qualitative agreement in Figs. 5–7, and the pRG/TUFRG comparison is a genuine strength. The second strand, however, is supported only by TUFRG and involves a parameter range, ΔE = 0.001 t1, that appears to lie below the numerical resolution of the k-grid used for the loop integrals. Estimating the saddle-point curvature from Eq. (4) gives an energy scale of order 0.02–0.1 t1 for the stated 40×40 and 80×80 grids, so a 0.001 t1 shift is not resolved by the discretized band structure. The disappearance of FM in Fig. 8(a) could therefore be an artifact of the discrete DOS, which changes in a grid-dependent way as momentum points cross the Fermi surface, rather than a property of the continuum logarithmic DOS. This is a concrete, testable numerical issue, distinct from the broader one-loop truncation concern raised by the reader. If the finer-grid test shows FM persisting at 0.001 t1, the central doping claim would need to be weakened; if the suppression survives with adequate resolution, the claim is strengthened. Given that no code or raw data are provided, an independent resolution check is essential before the fragility claim can be accepted as quantitative. The phase diagrams at Van Hove filling are less affected by this concern and may well stand, which is why conditional acceptance, rather than rejection, is appropriate.","tokens_in":25747,"tokens_out":10512,"duration_ms":113648,"concrete_test":"Repeat the single-VHS TUFRG doping sweep (t2/t1 = 0.375, t3 = 0) for U/t1 in {0.5, 1, 2, 3} and ΔE_VHS/t1 in {0.0001, 0.0005, 0.001, 0.002, 0.005} using at least 160×160/80×80 and 240×120 k-grids. Also compute the non-interacting DOS on the same grids and compare N_grid(ΔE) with the analytic N(ε) ∝ ln(W/|ε|). If FM persists at ΔE = 0.001 on the finer grids, or if N_grid(0.001) is resolution-limited and flat, the FM boundary in Fig. 8(a) is a discretization artifact. A cheaper cross-check is to repeat the ΔE = 0.001 run on the existing grid with the k-grid shifted by half a spacing; a phase change under this shift would indicate lack of convergence.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim that the ferromagnetic state is unstable to very small shifts in the Fermi level rests entirely on the TUFRG doping sweeps in Fig. 8; pRG results are not presented away from Van Hove filling. In the single-VHS panel, FM is reported to be suppressed for detunings as small as ΔE_VHS/t1 = 0.001. But the TUFRG loop integrals use a 40×40 k-grid (Sec. III B). For t2/t1 = 0.375 and t3 = 0, the saddle-point mass from Eq. (4) is m_X^+ = (t1 − 2t2)/2 = 0.125 t1. With grid spacing Δk ≈ π/20 ≈ 0.16 (or π/40 ≈ 0.08 for the 80×80 momentum grid), the corresponding energy scale near the saddle is (Δk)^2/(2m_X^+) ≈ 0.1 t1, or ≈ 0.025 t1 with the finer grid. A detuning of 0.001 t1 is one to two orders of magnitude below this resolution. The discretized density of states cannot represent the continuum logarithmic divergence below that scale, so the reported disappearance of FM may reflect the finite-grid regularization rather than the analytic DOS. Because the doping sensitivity is a headline result and the basis for the proposed design rules, the quantitative '1 meV' statement is not supported by the numerics as presented.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the t1-t2-t3 square-lattice Hubbard model with the Fermi level pinned to the Van Hove singularity (VHS), using hot-spot parquet RG (pRG) and truncated-unity functional RG (TUFRG). By varying t3, the authors tune the VHS from an ordinary logarithmic singularity through a higher-order (power-law) singularity at t3 = (t1 - 2t2)/4, and then into a split-VHS regime with two logarithmic singularities per X point. They map out (t2/t1, U/t1) and (t3/t1, U/t1) phase diagrams at Van Hove filling and compare pRG with TUFRG, reporting qualitative agreement with some differences. They then study doping, parametrized by an energy shift ΔE_VHS, and find that ferromagnetic order is extremely sensitive to detuning, disappearing for shifts as small as 0.001 t1 in the single-VHS case. The central claim is that the functional form of the VHS divergence controls the correlated ground state and that the ferromagnetic state at Van Hove filling is unstable to very small Fermi-level shifts.","tokens_in":26051,"tokens_out":6094,"duration_ms":65017,"significance":"If established, the paper would provide a concrete design principle: tuning the VHS type via third-neighbor hopping can select between d-wave superconductivity, ferromagnetism, p-wave superconductivity, and small-Q spin-density-wave order in a single microscopic model. The paper is also valuable as a head-to-head benchmark of hot-spot pRG against full-band-structure TUFRG for the same model, including an analytic derivation of inter-patch susceptibilities in Appendix A and reproduction of known single-VHS TUFRG results. The strengths include transparent reporting of truncation choices (frequency-independent vertex, no self-energy feedback, eight-unit-cell form-factor cutoff, 80×80/40×40 grids), the use of the public divERGe package, and the explicit acknowledgment of the high-U limitations of one-loop RG. However, a headline quantitative claim about doping fragility is not supported by the numerical resolution as presented, which is a load-bearing issue for the paper's central message.","major_comments":[{"comment":"The claim that ferromagnetism is suppressed for ΔE_VHS/t1 as small as 0.001 is not supported by the TUFRG numerics as presented. The loop susceptibilities are integrated on a 40×40 k-grid (Sec. III B), giving Δk ≈ π/40. With t2/t1 = 0.375 and t3 = 0, Eq. (4) gives m_X^+ = (t1 - 2t2)/2 = 0.125 t1, so the characteristic energy scale of the saddle-point neighborhood is (Δk)^2/(2m_X^+) ≈ 0.025 t1, about 25 times larger than the smallest detuning shown. A detuning of 0.001 t1 is therefore below the energy resolution with which the discretized density of states can represent the logarithmic singularity; the disappearance of the FM region may be a finite-grid artifact rather than a property of the continuum model. Since this result underlies the '1 meV' design statement and the asymmetry discussion, I ask for either (i) a convergence check with nkf = 80 or finer for the doping sweeps, (ii) a pRG doping calculation, where the DOS is treated analytically, to confirm the fragility, or (iii) a revised quantitative statement that is compatible with the grid resolution.","section":"§IV B, Fig. 8(a)"},{"comment":"The paper states in Sec. V that one-loop weak-coupling RG can be trusted only up to U of roughly half the bandwidth, i.e. U ≈ 4t1 for a bandwidth of 8t1, and that at larger U results are 'likely to be significantly modified by higher-loop terms'. Nevertheless, the main-text phase diagrams in Figs. 5, 7, and 8 extend to U = 5t1 and display phase regions in the range 4t1 < U < 5t1 as if they were predictions. These high-U regions are outside the stated domain of validity of the method. The authors should either restrict the main-text diagrams to the trusted range, or clearly hatched/mark the untrusted regions and refrain from drawing conclusions from them.","section":"Sec. V and Figs. 5, 7, 8"},{"comment":"The claimed 'broad qualitative agreement' between pRG and TUFRG is weakened by the complete absence in TUFRG of the large E_u (p-wave) superconducting region that pRG predicts in the split-VHS case, at the same t2/t1 used for the doping study. The authors attribute this to the pRG's neglect of the high-energy DOS step near the X-point band maxima, but no quantitative test of this explanation is provided. Since the paper uses pRG as an independent cross-check of the TUFRG phase diagrams, this qualitative discrepancy should be either resolved by a controlled comparison (e.g., including the step contribution in an extended pRG) or explicitly framed as a limitation that prevents relying on pRG for the split-VHS regime.","section":"§IV A 2 and Fig. 7"}],"minor_comments":[{"comment":"Typo: 'TURFG' should be 'TUFRG' in the sentence 'We find broad qualitative agreement between the pRG and TURFG phase diagrams'.","section":"§IV A 1"},{"comment":"The sentence 'This would accord with the observation that the ferromagnetic region in Fig. 1(c) moves to more negative ΔE_VHS as U increases' appears to refer to Fig. 8(c), not Fig. 1(c), which is a non-interacting DOS plot and contains no ferromagnetic region.","section":"§IV B"},{"comment":"The phrase '3200×3200 k-points in the Brillouin zone' is unclear: with a 80×80 bosonic q-grid and a 40×40 loop grid, the natural count is 80×40 = 3200 k-points, not 3200×3200. Please clarify the intended meaning.","section":"§III B"},{"comment":"The caption of Fig. D2(a) states 'U≳8 eV'; this should be 'U≳8 t1' to maintain dimensionless notation.","section":"Appendix D"},{"comment":"The notation '2K1-FFLO' in the split-VHS rate table is not defined in the text; please define K1 or introduce the notation before use.","section":"Appendix B"}],"recommendation":"major_revision","confidential_remarks":"The paper is potentially publishable, but the numerical-resolution issue with the doping-fragility claim is substantive and load-bearing. I am not recommending rejection because the issue is fixable by additional convergence checks or by softening the quantitative claim to a resolution-limited bound. The pRG/TUFRG discrepancy in the split-VHS regime also deserves more explicit treatment. No concerns about attribution or novelty disclosure beyond the points raised in the report."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth a look if you work on Van Hove physics. The paper does a careful pRG/TUFRG comparison for the t1-t2-t3 Hubbard model across single, higher-order, and split VHS regimes, and the authors are honest about their approximations. But the headline claim—that ferromagnetism dies for Fermi-level shifts as small as 0.001 t1—doesn't survive scrutiny of the numerics. The TUFRG loop integrals use a 40x40 k-grid. For t2/t1 = 0.375, the saddle-point mass is 0.125 t1, so the energy resolution near the saddle is (pi/40)^2/(2*0.125) ~ 0.025 t1, an order of magnitude larger than the claimed detuning. The discretized DOS cannot represent the logarithmic divergence below that scale. The FM suppression in Fig. 8(a) is therefore likely a grid artifact, not a physical effect. The authors even note that 3200x3200 k-points were needed to smoothen the split-VHS boundaries; the doping sweeps do not appear to have that resolution. Since the doping sensitivity is the paper's central design-rule claim, this is a load-bearing issue.\n\nWhat is genuinely new: the continuous t3 phase diagram, the pRG-vs-TUFRG comparison across all three regimes, and the electron/hole asymmetry in the HOVHS doping phase diagram. The flow equations are derived in appendices, and the two methods agree broadly at Van Hove filling—that is real independent support for the qualitative t2 and t3 dependence. The authors are also candid about the hot-spot approximation, the one-loop truncation, and the unclear SDW origin. That honesty counts.\n\nOther soft spots: no shipped code or data, no error bars on phase boundaries, and the high-U appendix results are presented despite being outside the stated validity range. Those are minor for a theory paper but add to the difficulty of pinning down whether the small-detuning behavior is real.\n\nBottom line: the paper deserves a serious referee. The comparison study is valuable and the phase diagrams at Van Hove filling are likely robust. But the doping-fragility result should be re-examined with higher k-resolution or backed by pRG detuning calculations before it becomes a design rule.","headline":"Solid pRG/TUFRG comparison of Van Hove regimes, but the headline doping-fragility of ferromagnetism is likely a k-grid artifact and should not be taken as a quantitative design rule.","tokens_in":26608,"tokens_out":2622,"would_cite":true,"duration_ms":25520,"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":"The type of Van Hove singularity — logarithmic, power-law, or split — selects the correlated ground state of a square-lattice Hubbard model, and tiny doping destroys the ferromagnetic state.","keywords":["Van Hove singularity","higher-order Van Hove singularity","Hubbard model","functional renormalization group","parquet renormalization group","ferromagnetism","unconventional superconductivity","spin-density wave"],"falsifier":"A concrete test: compute the Van-Hove-filling phase diagram with a method that includes self-energy feedback and frequency-dependent vertices, or with a much larger form-factor cutoff, and see whether the ferromagnetic region at $t_3=0$ still disappears when the Fermi level is shifted by $0.001\\,t_1$; if a quantitatively reliable calculation finds ferromagnetism stable to that doping shift, the central claim about the fragility of the ferromagnetic state is wrong.","tokens_in":25515,"feed_emoji":"🧲","tokens_out":14045,"duration_ms":112307,"temperature":0.7,"pith_summary":"This paper argues that the functional form of a Van Hove singularity — ordinary logarithmic, higher-order power-law, or split into two ordinary saddle points — is a practical control knob for the correlated ground state of a two-dimensional Hubbard model. Tuning the third-neighbor hopping $t_3$ through $t_{3c}=(t_1-2t_2)/4$ changes the density-of-states divergence from $\\ln(W/|\\xi|)$ to $|\\xi|^{-1/4}$ and then back to logarithmic at two points, and the predicted weak-coupling phases rearrange accordingly: ferromagnetism, $d$-wave and $p$-wave superconductivity, and spin-density waves trade places. The paper further claims that the ferromagnetic state at Van Hove filling is extremely fragile, disappearing for Fermi-level shifts as small as $0.001\\,t_1$, with electron and hole doping selecting different ordered states. The reason to care is that it turns phase engineering into a band-structure problem: instead of fine-tuning interactions, one can tune the shape and position of Van Hove singularities.","feed_headline":"Van Hove singularity shape decides a Hubbard model's ground state","feed_subtitle":"Third-neighbor hopping switches the Van Hove divergence from logarithmic to power-law and back, flipping phases.","key_machinery":"The load-bearing object is the non-interacting dispersion $\\xi(\\mathbf{k})=-2t_1(\\cos k_x+\\cos k_y)-4t_2\\cos k_x\\cos k_y+2t_3(\\cos 2k_x+\\cos 2k_y)-\\mu$. Increasing $t_3$ to $t_{3c}=(t_1-2t_2)/4$ turns each quadratic saddle point at the X points into a higher-order (cusp $A_3$) saddle, giving a DOS divergence $|\\xi|^{-1/4}$; for $t_3>t_{3c}$ each X saddle splits into two ordinary saddle points at $\\mathbf{P}=(\\pi,\\pm k_P)$ with $k_P=\\arccos[(t_1-2t_2)/(4t_3)]$. The argument is carried by two one-loop weak-coupling RG schemes: a hot-spot parquet RG that flows a small set of interaction couplings from patches around the Van Hove points, and a truncated-unity functional RG that integrates the full Brillouin zone with a static vertex and a form-factor cutoff. The ordered phase is identified by which vertex combination diverges and by the associated susceptibility rates or gap symmetry.","core_discovery":"The central claim is that the type of Van Hove singularity at the Fermi level determines which weak-coupling ordering tendency dominates in the $t_1$-$t_2$-$t_3$ square-lattice Hubbard model. At Van Hove filling, the single logarithmic VHS produces competition between $B_{1g}$ ($d$-wave) superconductivity and ferromagnetism, with a sizeable Fermi-liquid region; the higher-order VHS at $t_3=t_{3c}$ stabilizes ferromagnetism over most of the phase diagram; and the split-VHS regime favours a triplet $E_u$ ($p$-wave) superconducting state competing with ferromagnetism, with a small-wavevector spin-density wave appearing in the full-Brillouin-zone calculation. Away from Van Hove filling, the ferromagnetic state is destroyed by tiny doping, while the other phases persist and acquire an energy asymmetry: in the HOVHS case, electron and hole doping select different ground states because the power-law density of states is asymmetric about the singularity.","pith_inferences":["An implication the authors leave implicit is a design rule for materials: if a single logarithmic VHS sits at the Fermi level, sub-meV chemical-potential control is required to realize ferromagnetism, whereas the HOVHS and split-VHS regimes tolerate larger doping windows for their non-ferromagnetic phases.","The electron-hole asymmetry near a HOVHS suggests that a real material with a higher-order saddle slightly off the Fermi level could be steered into either a $p$-wave superconducting state or a small-$\\mathbf{q}$ spin-density wave simply by changing the sign of the doping — a testable prediction for strained or gated quasi-2D metals.","A natural next calculation is to repeat the TUFRG flow with frequency-dependent vertices and self-energy feedback; if the ordering changes with $t_3$ survive, the design rule is not an artefact of the static-vertex truncation."],"forward_implications":["In the single-VHS regime, $d$-wave superconductivity and ferromagnetism compete, and the Fermi-liquid region at intermediate $t_2/t_1$ is widened in the full-Brillouin-zone calculation compared with pRG.","In the HOVHS regime, ferromagnetism dominates most of the $(t_2/t_1, U/t_1)$ phase diagram at Van Hove filling, consistently in both RG schemes.","In the split-VHS regime, $p$-wave ($E_u$) superconductivity competes with ferromagnetism, and a small-$\\mathbf{q}$ spin-density wave with $\\mathbf{Q}=(\\delta,\\delta)$, $\\delta\\lesssim\\pi/10$, appears only in the full-Brillouin-zone calculation.","The ferromagnetic state at Van Hove filling is destroyed by doping shifts of order $0.001\\,t_1$; the phases that replace it depend on whether the VHS sits above or below the Fermi level, especially in the HOVHS case.","The phase diagram is smooth in $t_3$, so the influence of the higher-order VHS persists over a range of $t_3$ values rather than only at the critical point."],"supporting_citations":[{"why":"supplies the parquet flow equations and logarithmic susceptibilities for the single-VHS square-lattice Hubbard model that this paper extends with $t_3$.","marker":"[16]"},{"why":"gives the expressions for $\\gamma_1$, $\\gamma_2$ and the type-II VHS susceptibilities used in the single- and split-VHS pRG.","marker":"[22]"},{"why":"provides the higher-order Van Hove susceptibilities and competing-orders pRG analysis, including $\\Pi_{pp}^0\\sim|\\Omega|^{-1/4}$, that the HOVHS case builds on.","marker":"[10]"},{"why":"introduces the truncated-unity functional renormalization group scheme used for the full-Brillouin-zone phase diagrams.","marker":"[35]"},{"why":"supplies the numerical implementation used for all TUFRG flows, vertex-divergence detection, and gap-symmetry analysis.","marker":"[39]"},{"why":"classifies the cusp ($A_3$) higher-order Van Hove singularity and its power-law DOS exponent.","marker":"[40]"},{"why":"provides the reference momentum-space fRG procedure and gap-equation analysis used to identify pairing symmetries.","marker":"[38]"}],"fun_headline_variants":["Van Hove type flips Hubbard model's ground state","Third-neighbor hopping reengineers correlated phases","Power-law VHS: ferromagnetism wins over superconductivity","Doping destroys ferromagnetism, switches ground states","VHS shape controls phase competition in Hubbard model"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The results depend on the assumption that the simplified renormalization-group schemes — which leave out some energy dependence and self-energy effects and, in the hot-spot version, keep only small patches of the Fermi surface — still predict the same winning ordered states as a fuller calculation at the interaction strengths used.","fun_headline_variants_meta":{"raw":{"variants":["Van Hove type flips Hubbard model's ground state","Third-neighbor hopping reengineers correlated phases","Power-law VHS: ferromagnetism wins over superconductivity","Doping destroys ferromagnetism, switches ground states","VHS shape controls phase competition in Hubbard model"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00019,"raw_usage":{"total_tokens":1397,"prompt_tokens":1058,"completion_tokens":339,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":674,"completion_tokens_details":{"reasoning_tokens":262}},"tokens_in":674,"tokens_out":339,"duration_ms":3667,"temperature":1.0,"reasoning_tokens":262,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T00:22:33.149826+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete test: compute the Van-Hove-filling phase diagram with a method that includes self-energy feedback and frequency-dependent vertices, or with a much larger form-factor cutoff, and see whether the ferromagnetic region at $t_3=0$ still disappears when the Fermi level is shifted by $0.001\\,t_1$; if a quantitatively reliable calculation finds ferromagnetism stable to that doping shift, the central claim about the fragility of the ferromagnetic state is wrong.","supporting_citations":[{"cited_title":"Classen, A","cited_arxiv_id":null,"evidence_quote":"supplies the parquet flow equations and logarithmic susceptibilities for the single-VHS square-lattice Hubbard model that this paper extends with $t_3$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the expressions for $\\gamma_1$, $\\gamma_2$ and the type-II VHS susceptibilities used in the single- and split-VHS pRG."},{"cited_title":"Benhabib, A","cited_arxiv_id":null,"evidence_quote":"provides the higher-order Van Hove susceptibilities and competing-orders pRG analysis, including $\\Pi_{pp}^0\\sim|\\Omega|^{-1/4}$, that the HOVHS case builds on."},{"cited_title":"Wu, Y.-M","cited_arxiv_id":null,"evidence_quote":"introduces the truncated-unity functional renormalization group scheme used for the full-Brillouin-zone phase diagrams."},{"cited_title":"Salmhofer and C","cited_arxiv_id":null,"evidence_quote":"supplies the numerical implementation used for all TUFRG flows, vertex-divergence detection, and gap-symmetry analysis."}],"review_version":1}