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Hall Angle of a Spatially Random Vector Model

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2501.07792 v4 pith:P3GRU2SG submitted 2025-01-14 hep-th cond-mat.str-el

Yi-Li Wang , Young-Kwon Han , Xian-Hui Ge , Sang-Jin Sin This is my paper · ORCID
classification hep-thcond-mat.str-el
keywords strangemetalHallanglelinearresistivityrandomcouplingLandaulevelsSchwinger-Dysonequationsvectormodelmagnetotransport
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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 $T^2$.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [Sec. 3.1, Eqs. (3.4), (3.11), (3.12)] 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.
  2. [Sec. 4, Fig. 9 and Eq. (4.5)] 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.
  3. [Sec. 3.2.2 and Fig. 12] 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.
minor comments (4)
  1. [Introduction and throughout] There are several typos, including 'starange metalicity' and 'Tt thus behooves'; please proofread the text carefully.
  2. [Fig. 12 caption] The caption labels panel (b) as the Hall conductivity sigma_xx; it should be sigma_xy.
  3. [Sec. 4] 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.
  4. [Reference [40]] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Hall angle and resistivity are forward outputs of the self-consistent Dyson-equation solution, not re-statements of the model inputs.

full rationale

The paper's derivation is forward-directed: it specifies the action, solves the Schwinger-Dyson equations numerically, and then computes conductivities via the Kubo formula. The central results—linear-in-T resistivity and a non-anomalous Hall angle with cot(Theta_H) ~ 1/T—are outputs of this calculation, not quantities fitted back into any parameter. The paper does rely on the authors' earlier vector model [12], but that prior work supplies the model and the zero-field linear-resistivity mechanism; the present magnetic-field transport and Hall-angle computation is carried out independently in this paper. The numerical strategy is attributed to [27], which is an external work by different authors, and no uniqueness theorem from the authors' own prior work is invoked to forbid alternatives. The 'typical Landau level' replacement n_t ~ (k_F ell_B)^2/2 is a physical approximation used to simplify Landau-level sums; it is a correctness or convergence question, not a circular one, because the final Hall angle is not assumed in that substitution but derived from it. Similarly, the asserted cancellation of Maki-Thompson diagrams is an unproven technical step, but it is not a reduction of the conclusion to its inputs. No equation in the paper is defined in terms of the quantity it is claimed to predict, and no fitted parameter is renamed as a prediction. Any weaknesses in the typical-Landau-level or MT-cancellation steps are substantive physics concerns, not circularity. Therefore the appropriate circularity score is 0.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The calculation depends on a small number of hand-picked scales and several domain assumptions inherited from the SYK-rised program. The most fragile input is the typical Landau level replacement, which directly controls the smallness of Pi_xy and therefore the central Hall angle negative result. The Maki-Thompson cancellation assertion is unproved in this text. No new entities are introduced.

free parameters (2)
  • Model scales and cutoffs (W, omega_B, k_F, mu, m, Lambda_q, K) = W=4, omega_B=0.1, k_F=1, mu=0, m=1, Lambda_q=2, K=1
    Chosen by hand, matching scalar-model parameters in ref. [27]; not fitted to data, but the numerical solutions and transport curves depend on these choices.
  • Potential disorder scattering rate Gamma = Gamma=0.2 for self-consistent solutions; Gamma=0.5, 1.0, 1.5, 2.0, 2.5 for transport
    Introduced to model impurities; varying Gamma changes conductivity magnitude and the temperature range of linear resistivity but not the qualitative Hall angle form.
assumptions (6)
  • standard math Large-N saddle point and replica trick justify the G-Sigma action (Eq. 2.7).
    Used in Sec. 2.1 to write Dyson equations (2.10)-(2.13); standard in the SYK literature.
  • domain assumption Spatial randomness removes momentum conservation at interaction vertices.
    Central mechanism inherited from refs. [8,12]; used everywhere, for example in the claim that AL diagrams vanish and in Eq. (5.1) where no (1-cos theta) correction appears.
  • domain assumption Only states near the Fermi surface contribute, so Landau levels can be replaced by the typical level n_t approximately (k_F * ell_B)^2/2 (Eq. 3.4).
    Load-bearing for suppressing Pi_xy relative to Pi_xx; authors state the critical value 'cannot be reliably estimated at this stage.'
  • domain assumption Electron self-energy is momentum independent and diagonal in the Landau basis.
    Assumed in Eq. (3.3) to make the problem tractable; later argued to be self-consistent, but the argument uses the typical-level approximation.
  • domain assumption Potential disorder contributes a constant -i Gamma/2 and suppresses Landau oscillations (Eq. 3.31).
    Used in Sec. 3.2.3 to approximate the full self-energy; verified only numerically for selected parameters.
  • ad hoc to paper Maki-Thompson vertex corrections exactly cancel the electron self-energy contribution to the conductivity.
    Asserted in Sec. 4 with no derivation or citation; it is essential for keeping only the bubble diagram in Fig. 11.

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Cite this review

Pith. "Pith review of Hall Angle of a Spatially Random Vector Model." pith.science (2026). https://pith.science/paper/P3GRU2SG

@misc{pith2026250107792,
  author       = {Pith},
  title        = {Pith review of: Hall Angle of a Spatially Random Vector Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P3GRU2SG}},
  note         = {Machine review of arXiv:2501.07792}
}
abstract

Strange metals exhibit linear resistivity and anomalous Hall transport, yet a comprehensive theory that accounts for both phenomena is still lacking. Recent studies have shown SYK-like spatially random couplings between a Fermi surface and a bosonic field, either scalar or vector type, can yield linear-$T$ resistivity. In this paper, we continue the investigation on a vector coupling in the presence of a magnetic field. We compute the fermion and boson propagators, along with the self-energy and polarization functions, and determine their dependence on the magnetic field. Although the Hall angle does not exhibit the signature of strange-metal, the linear-in-temperature resistivity remains at low temperatures. Results indicate that random interactions can robustly support linear transport, though additional ingredients may be required to capture the full phenomenology of strange metals.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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