{"id":"09ca3d7e-7867-4c95-aaf0-9c95910fd1f5","arxiv_id":"2501.07792","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"In a spatially random vector-coupling model with a magnetic field, linear-T resistivity persists while the Hall angle follows normal 1/T behavior rather than the strange-metal T^2 law.","lead":"This paper adds a magnetic field to a model of electrons randomly coupled to bosonic fields and computes the resulting Hall angle and resistivity. It finds the model keeps linear-in-temperature resistivity but produces a normal, not strange-metal, Hall angle, which helps narrow which theories can explain strange metals.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The normal-Hall-angle conclusion rests on the typical-Landau-level replacement; at the chosen parameters the suppression factor is only 1/(2n_t)≈0.1, so the full Landau sum must be checked before trusting the 1/T Hall angle.","rationale":"We agree with the reader's identification of the typical Landau level replacement as the weakest point. The central new result — a normal 1/T Hall angle — follows directly from the claim |Π_xy|≪Π_xx, and that claim is manufactured by the n_t substitution in Eq. (3.11). At the parameters used, the suppression is only a factor of ~1/(2n_t)=0.1, so even a modest failure of the substitution could change σ_xy by an order-one amount and alter the T-scaling of cot Θ_H. The same substitution is also used to justify ignoring D_xy and Σ_xy in the self-consistent loop, so the entire numerical solution depends on it. Independent checks are needed. We also note two secondary issues: the cancellation of MT diagrams is asserted without proof in Sec. 4, and Appendix B's data repository link resolves to the arXiv page rather than the promised Zenodo dataset, so the numerics are not independently reproducible as published. These do not change the verdict: the conclusion is plausible and consistent with normal Drude expectations, but it is conditional on the full Landau sum test. We therefore leave the reader's CONDITIONAL verdict unchanged.","tokens_in":16683,"tokens_out":6999,"duration_ms":70360,"concrete_test":"Evaluate the exact Landau sums for Π_xx and Π_xy, without substituting n_t, using the converged numerical G_n from the Dyson solution and the same cutoff n_±=W/(2ω_B). Specifically compute S_xx = Σ_{n,n'} (n+n'+2) G_n(iω)G_{n'}(i(ω-Ω)) and S_xy = Σ_{n,n'} G_n(iω)G_{n'}(i(ω-Ω)) for the Matsubara frequencies used in Fig. 6. If |S_xy/S_xx| is not small (≲0.1) at Ω∼T in the low-T window, then the D_xy, Σ_xy, and σ_xy used in Sec. 4 must be recomputed with the full sums; the resulting cot Θ_H should be compared with Fig. 13.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.1 introduces the 'typical Landau level' substitution: after deriving Eq. (3.11), the factor (n+n'+2) in Π_xx is replaced by its value at n≈n_t≈(k_Fℓ_B)^2/2, while Π_xy in Eq. (3.12) retains only the residual factor 1 after the cancellation of -n and n+1. This replacement is the entire basis for the claim |Π_xy|≪Π_xx, which in turn justifies dropping D_xy and Σ_xy in the Dyson loop (Sec. 3.2.1) and determines the small Hall conductivity in Sec. 4. For the simulation parameters (ω_B=0.1, k_F=1, m=1), ℓ_B^2=10 and n_t=5, so the naive suppression factor is 1/(2n_t)≈0.1 — not an order-of-magnitude separation. Because G_n is peaked near the Fermi level, the actual sums over n,n' could easily differ from the n_t substitution; a peak away from n_t, or interference between terms, could make Π_xy comparable to Π_xx. If that happens, the Hall angle would no longer follow the simple 1/T behavior claimed in Fig. 13. The assertion that Maki-Thompson diagrams cancel (Sec. 4, Fig. 9) is also not demonstrated, and it is needed to isolate the bubble (4.5); this is a second unverified link in the transport chain.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies a spatially random vector model with a magnetic field, continuing earlier work on SYK-like electron-boson couplings. The authors formulate the Dyson-Schwinger equations in a Landau-level basis, introduce potential disorder, and solve the electron and boson self-energies numerically using the typical-Landau-level approximation. They then compute sigma_xx, sigma_xy, rho_xx, and cot(theta_H) through the Kubo formula. The main findings are that linear-T resistivity survives at low temperatures in a magnetic field, but the Hall angle follows ordinary 1/T scaling rather than the strange-metal behavior cot(theta_H) ~ A + B T^2. The paper concludes that spatially random vector interactions robustly support linear transport but require additional ingredients to explain Hall anomalies.","tokens_in":16934,"tokens_out":5155,"duration_ms":52293,"significance":"The paper asks a sharp question: does the SYK-rised spatially random vector coupling, which gives T-linear resistivity, also reproduce the anomalous Hall angle of strange metals? It reports a useful negative result: in the computed saddle point, rho_xx remains T-linear while cot(theta_H) ~ 1/T. If the calculation is correct, this is an important boundary on a recently proposed mechanism and directs future work toward spin or other sectors. The numerical solution is self-consistent, the qualitative comparison with the zero-field solutions is explicit, and the authors make their data available, so the central claim is checkable. The paper does not fit its parameters to the Hall angle, so there is no circularity in the main output. The main reservations are quantitative: the typical Landau level approximation and the claimed Maki-Thompson cancellation are load-bearing and need stronger support.","major_comments":[{"comment":"The central suppression |Pi_xy| << Pi_xx is obtained by replacing n and n' by the typical Landau level n_t = (k_F ell_B)^2/2. With the listed parameters (omega_B = 0.1, k_F = 1, m = 1), ell_B^2 = 10 and n_t = 5, so the naive suppression factor is 1/(2 n_t) = 0.1, not an order of magnitude. Since G_n is peaked near the Fermi level and the sums over n and n' in (3.11) and (3.12) have different weight factors, the full Landau sum could render Pi_xy comparable to Pi_xx. This is load-bearing: D_xy and Sigma_xy are neglected in Secs. 3.1 and 3.2.1 on the basis of this inequality, and the small Hall conductivity in Sec. 4 follows from it. The text itself states right after (3.4) that the critical n_t 'cannot be reliably estimated at this stage.' I therefore request a numerical evaluation of the full Landau-level sums, or at least an explicit sensitivity study in n_t and in the Landau cutoff n_+, before the normal-Hall-angle claim can be considered established.","section":"Sec. 3.1, Eqs. (3.4), (3.11), (3.12)"},{"comment":"The transport calculation isolates the bubble (4.5) by asserting that the Maki-Thompson vertex corrections exactly cancel the electron self-energy contribution to the conductivity. For scalar coupling this cancellation is absent for a different reason, but here the paper states that for the vector coupling the MT graph is nonzero and cancels the Sigma_K contribution. No derivation or reference is given for this exact cancellation. Because this cancellation determines which diagrams contribute to both sigma_xx and sigma_xy, and hence to the Hall angle, it is a second load-bearing step that needs explicit justification, such as a Ward identity or a direct diagrammatic check in the large-N limit.","section":"Sec. 4, Fig. 9 and Eq. (4.5)"},{"comment":"The numerical results are presented without convergence checks: the text fixes W = 4, omega_B = 0.1, n_+ = 20 and n_t = 5, but does not report how the auxiliary propagators and the final conductivities depend on the Landau-level cutoff n_+, the frequency grid, or the analytic-continuation parameter eta. Since the Hall conductivity is obtained as a small difference after a sequence of approximations, a convergence test is necessary to show that the 1/T behavior in Fig. 13 is not an artifact of the truncations. Adding such a test would also directly address the sensitivity raised by the typical Landau level substitution.","section":"Sec. 3.2.2 and Fig. 12"}],"minor_comments":[{"comment":"There are several typos, including 'starange metalicity' and 'Tt thus behooves'; please proofread the text carefully.","section":"Introduction and throughout"},{"comment":"The caption labels panel (b) as the Hall conductivity sigma_xx; it should be sigma_xy.","section":"Fig. 12 caption"},{"comment":"The phrase 'cot(Theta_H) has no T^2-dependence' is imprecise because the shown scaling is 1/T; state explicitly that the claimed result is cot(Theta_H) ~ 1/T rather than A + B T^2.","section":"Sec. 4"},{"comment":"Reference [40] is described as a Zenodo data set, but the DOI shown is an arXiv identifier; please update it to the actual data DOI.","section":"Reference [40]"}],"recommendation":"major_revision","confidential_remarks":"To the editor: I am, on balance, sympathetic to the paper's broad conclusion, and I find the released data and the direct negative test genuinely useful. My hesitation is not about novelty or scope but about the fact that the two controlling approximations - the typical Landau level replacement and the MT cancellation - are asserted rather than checked. Both can in principle be addressed in a revision; if they are, I would be satisfied. I do not see a citation or overlap concern."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper reports a useful negative result: in the spatially random vector Yukawa model, linear-T resistivity survives the addition of a magnetic field, but the Hall angle comes out normal, cot(Theta_H) ~ 1/T, not the anomalous A + B T^2 seen in cuprates. This is the first Hall-angle computation for this vector model, and it matters because it isolates what the random-coupling mechanism can and cannot explain. The authors do not overclaim; they explicitly conclude that this mechanism alone cannot reproduce the full strange-metal phenomenology.\n\nThe numerics follow the strategy of Guo et al. for the scalar model [27], but the vector case in a field is new. The paper is also honest about limitations, which counts for something.\n\nThe soft spots are real but addressable. First and most important: the central Hall result rests entirely on the \"typical Landau level\" replacement in Sec. 3.1. The authors replace n by n_t ~ (k_F l_B)^2/2 inside the sum, which makes Pi_xy smaller than Pi_xx by roughly 1/(2 n_t). With their parameters n_t = 5, that is only a factor of 0.1, not a wide separation. A full Landau-level sum could shift this ratio; if Pi_xy became comparable to Pi_xx, the 1/T Hall angle could change. The authors note that the critical n_t \"cannot be reliably estimated at this stage,\" which is honest but means the negative Hall result is conditional. Second, the cancellation of Maki-Thompson diagrams with the self-energy contribution in Sec. 4 is asserted, not demonstrated, and it is needed to isolate the bubble that produces the conductivity. Third, the data availability statement points to the arXiv page rather than an actual dataset, so the promised reproducibility is not currently delivered.\n\nDespite these issues, the qualitative conclusion is plausible: a single scattering rate gives linear rho and normal Hall behavior, consistent with the paper's numerical output. The fix is straightforward: perform the full Landau sum (or at least show convergence against n_t), and provide the code and data. The MT cancellation should also be shown explicitly.\n\nWho is this for? People working on SYK-like strange-metal theories and on whether spatially random coupling can explain cuprate transport. It deserves a serious referee: the question is important, the result is new, and the weaknesses are repairable rather than fatal. I would recommend sending it to peer review with the expectation of a revision.","headline":"Useful negative result: random vector coupling gives linear-T resistivity in a field but a normal Hall angle; the Hall conclusion rests on an unverified Landau-level approximation, but the paper is honest and worth a serious referee.","tokens_in":17535,"tokens_out":1421,"would_cite":true,"duration_ms":15483,"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":"In a spatially random vector model, a magnetic field preserves the linear-in-temperature resistivity while the Hall angle stays ordinary, cot Θ_H ~ 1/T instead of the strange-metal form A + B T^2.","keywords":["strange metal","Hall angle","linear resistivity","random coupling","Landau levels","Schwinger-Dyson equations","vector model","magnetotransport"],"falsifier":"Recompute the boson self-energy exactly, keeping the full Landau-level sums in Eqs. (3.11)-(3.12) instead of substituting the typical level, and check whether $|\\Pi_{xy}|/\\Pi_{xx}$ stays small at low temperature; if the exact ratio is not tiny, the $1/T$ Hall angle would be an artifact of the approximation. A direct experimental version would be to measure $\\cot\\Theta_H$ at low temperature in a material whose linear resistivity is the cleanest realization of this random-coupling mechanism and see whether it tracks $1/T$ or the strange-metal $T^2$.","tokens_in":16395,"feed_emoji":"🧲","tokens_out":11819,"duration_ms":104240,"temperature":0.7,"pith_summary":"This paper tests whether the spatially random electron–boson coupling proposed to explain the linear-in-temperature resistivity of strange metals can also produce their anomalous Hall angle. The authors solve the vector version of the model in a magnetic field, computing Landau-level propagators, self-energies, and the Kubo conductivities numerically. They find that the linear-in-$T$ resistivity survives the magnetic field at low temperature, but the cotangent of the Hall angle grows as $1/T$, which is ordinary metal behavior rather than the $A + B T^2$ form seen in cuprates. The paper's conclusion is that random spatial couplings robustly generate linear transport, while the anomalous Hall effect calls for additional physics beyond this mechanism.","feed_headline":"Random couplings keep T-linear resistivity, miss Hall anomaly","feed_subtitle":"In a magnetic field, cot Θ_H scales as 1/T — normal metal behavior — while T-linear resistivity remains.","key_machinery":"The load-bearing device is the Landau-level Schwinger-Dyson computation with the 'typical Landau level' replacement: at every vertex of the boson self-energy the Landau index $n$ is set to $n_t \\simeq (k_F \\ell_B)^2/2$, the level of electrons at the Fermi surface (Eq. 3.4). Because the velocity matrix elements for the $x$ and $y$ components cancel against each other in the off-diagonal polarization, this substitution makes $|\\Pi_{xy}| \\ll \\Pi_{xx}$, and the off-diagonal boson propagator $D_{xy}$ can be dropped. With only the longitudinal channel active, the Hall conductivity is controlled by the same relaxation rate as the longitudinal one, which forces $\\cot\\Theta_H \\sim 1/T$. The numerical iteration that solves the coupled equations—guessing $\\bar{G}$, computing $\\Pi_{xx}(t)$, $\\bar{D}_{xx}(t)$, $\\Sigma(t)$, and feeding the result back into $\\bar{G}$—is the practical machinery that produces the conductivity curves.","core_discovery":"The paper shows that in the spatially random vector model, adding a perpendicular magnetic field leaves the zero-field transport story intact: the converged self-consistent solutions for the fermion and boson propagators are qualitatively the same as at $B = 0$ (up to Landau-level oscillations), and the longitudinal resistivity remains linear in temperature. The new feature is the Hall response. Because the velocity-matrix-element factors for the $x$ and $y$ directions partly cancel, the off-diagonal boson self-energy $\\Pi_{xy}$ is much smaller than $\\Pi_{xx}$ once each vertex is evaluated at the typical Landau level $n_t \\simeq (k_F \\ell_B)^2/2$; the paper then neglects $D_{xy}$, computes $\\sigma_{xx}$ and $\\sigma_{xy}$ by the Kubo formula, and finds $\\cot\\Theta_H \\sim 1/T$ at low temperature (Fig. 13). The model therefore shows no trace of the $A + B T^2$ Hall angle that characterizes many strange metals; in the paper's reading, the random-coupling mechanism is sufficient for the linear resistivity but not for the Hall anomaly, so a complete strange-metal theory needs extra ingredients such as spin degrees of freedom.","pith_inferences":["A decisive extension would be to keep the full Landau-level sum numerically and ask whether the near-cancellation in $\\Pi_{xy}$ survives; if it does not, the reported Hall angle is an artifact of the typical-level approximation, not a robust property of the mechanism.","The result hints at a general rule: any random-coupling model that removes momentum conservation at the vertices kills the small-angle $(1 - \\cos\\theta)$ suppression and therefore ties $\\cot\\Theta_H$ to the single relaxation rate; producing $T^2$ would require a scattering rate different from the transport rate, for instance from spin fluctuations with Curie-Weiss susceptibility, as the paper's di","One could extend the model by coupling the fermions to spin degrees of freedom and check whether $\\sigma_{xy} \\propto \\chi \\tau^2$ restores the anomalous Hall angle without destroying the linear resistivity."],"forward_implications":["If the paper is right, the linear-in-$T$ resistivity of spatially random couplings is robust in a magnetic field, so the zero-field linear transport found earlier is not an artifact of $B = 0$.","A normal $1/T$ Hall angle in this model means the random-coupling mechanism alone cannot explain the $A + B T^2$ Hall angle of cuprates; a separate scattering channel with its own temperature-dependent rate must be added.","The strong suppression of $\\Pi_{xy}$ makes magneto-transport longitudinal-dominated, so measurements in such systems would look like a normal metal with scattering rate $\\tau \\sim T$.","The linear resistivity holds up to a temperature $T_L$ that grows with impurity scattering $\\Gamma$, but fails at high temperature, limiting the model's reach for materials with linear resistivity persisting to the highest temperatures."],"supporting_citations":[{"why":"The scalar spatially random-coupling model that first produced linear-T resistivity; this paper's vector model tests whether the mechanism generalizes.","marker":"[8]"},{"why":"The authors' earlier vector model at zero field that produced linear-T resistivity; its action, notation, and Dyson equations are reused here.","marker":"[12]"},{"why":"The magnetic-field treatment of the scalar model that supplies the typical-Landau-level approximation, the numerical iterative strategy, and the parameter values used here.","marker":"[27]"},{"why":"The real-frequency iterative method for self-consistent many-particle diagrams, used to solve the Dyson equations numerically.","marker":"[29]"},{"why":"The large-N G-Sigma formalism and saddle-point/replica treatment on which the field-theoretic setup rests.","marker":"[11]"}],"fun_headline_variants":["Hall angle normal while resistivity stays linear","Random vectors give linear resistivity, classic Hall","Resistivity linear, but Hall angle says normal metal","No strange-metal Hall, only T-linear resistivity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole result rests on replacing every Landau-level index by one typical level $n_t \\simeq (k_F \\ell_B)^2/2$ near the Fermi surface, and the paper concedes that the critical value of $n_t$ cannot be reliably estimated at this stage; if the full sum over Landau levels were kept, the smallness of $\\Pi_{xy}$ relative to $\\Pi_{xx}$, and with it the $1/T$ Hall angle, could change.","fun_headline_variants_meta":{"raw":{"variants":["Hall angle normal while resistivity stays linear","Random vectors give linear resistivity, classic Hall","Resistivity linear, but Hall angle says normal metal","No strange-metal Hall, only T-linear resistivity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000269,"raw_usage":{"total_tokens":1612,"prompt_tokens":927,"completion_tokens":685,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":543,"completion_tokens_details":{"reasoning_tokens":627}},"tokens_in":543,"tokens_out":685,"duration_ms":7047,"temperature":1.0,"reasoning_tokens":627,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:35:38.567273+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the boson self-energy exactly, keeping the full Landau-level sums in Eqs. (3.11)-(3.12) instead of substituting the typical level, and check whether $|\\Pi_{xy}|/\\Pi_{xx}$ stays small at low temperature; if the exact ratio is not tiny, the $1/T$ Hall angle would be an artifact of the approximation. A direct experimental version would be to measure $\\cot\\Theta_H$ at low temperature in a material whose linear resistivity is the cleanest realization of this random-coupling mechanism and see whether it tracks $1/T$ or the strange-metal $T^2$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The magnetic-field treatment of the scalar model that supplies the typical-Landau-level approximation, the numerical iterative strategy, and the parameter values used here."},{"cited_title":"Schmalian, M","cited_arxiv_id":null,"evidence_quote":"The real-frequency iterative method for self-consistent many-particle diagrams, used to solve the Dyson equations numerically."}],"review_version":1}