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arxiv: 2604.24274 · v2 · submitted 2026-04-27 · ❄️ cond-mat.str-el

Umklapp correction to Landau damping and conditions for non-trivial modifications to quantum critical transport

Vibhu Mishra This is my paper

Pith reviewed 2026-05-08 01:51 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords umklapp scatteringLandau dampingIsing-nematic metalquantum critical transportparticle-hole bubblelinear resistivityFermi surfaceBrillouin zone boundary
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The pith

Umklapp scattering near the Brillouin zone boundary adds a square-root frequency term to the particle-hole bubble in two-dimensional Ising-nematic metals at high temperatures.

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

The paper calculates the polarization bubble for electrons in a two-dimensional Ising-nematic quantum critical metal when the Fermi surface nears the Brillouin zone boundary. Besides the usual antipodal term scaling as frequency over momentum, an umklapp term from zone-boundary electrons appears that scales as frequency to the power one-half at high temperatures or two-thirds at low temperatures. At high T where the square-root scaling holds, this term can potentially change the minimum temperature needed for linear-in-temperature resistivity from scaling as the cube of the distance to the boundary to the fourth power, though the authors identify only one very specific case where this occurs. In three dimensions the umklapp term remains linear in frequency and does not alter the temperature scale.

Core claim

When the critical Fermi surface approaches the Brillouin zone boundary in d=2, the umklapp piece gives Π_U(q≈Δ_q, iΩ) ∝ √Ω at high T, which could reduce the minimum temperature for linear/quasi-linear resistivity from T_U ∝ Δ_q^3 to T_U ∝ Δ_q^4 in one hyper-specific scenario; for d=3 the umklapp contribution gives Π_U ∼ Ω irrespective of T and T_U is not modified.

What carries the argument

The umklapp contribution Π_U to the particle-hole bubble at the minimum umklapp momentum q ≈ Δ_q from electrons near the zone boundary.

If this is right

  • The standard antipodal contribution continues to scale as Ω/q.
  • In three dimensions the umklapp term scales linearly with frequency regardless of temperature, leaving T_U unchanged.
  • The reduction in T_U to Δ_q^4 occurs only in one hyper-specific scenario.
  • Linear or quasi-linear resistivity can appear at lower temperatures than expected from z=3 criticality due to this √Ω term.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This effect might be observable in materials where the Fermi surface can be tuned close to the zone boundary.
  • Similar umklapp corrections could influence transport in other quantum critical systems with Fermi surfaces near Brillouin zone boundaries.
  • If realized experimentally, the mechanism would extend the temperature window for non-Fermi liquid transport in two-dimensional critical metals.

Load-bearing premise

The critical Fermi surface approaches the Brillouin zone boundary closely enough that the minimum umklapp momentum q≈Δ_q lies inside the relevant momentum window, and the high-T regime with α=1/2 is experimentally accessible without other scattering channels dominating.

What would settle it

A measurement in a two-dimensional Ising-nematic system showing the onset temperature for linear resistivity scaling as Δ_q to the fourth power rather than the third, in a regime where the Fermi surface sits near the zone boundary, would confirm or refute the proposed reduction.

Figures

Figures reproduced from arXiv: 2604.24274 by Vibhu Mishra.

Figure 1
Figure 1. Figure 1: FIG. 1. The two antipodal patches contributing to Π view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The scattering rate of the electron at view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The same plot for the naive large- view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The parameterization of the FS used to evaluate the scattering rate. The point view at source ↗
read the original abstract

We compute the particle--hole bubble for an Ising-nematic metal when the critical Fermi surface approaches the Brillouin zone boundary for $d=2$ dimensions. We find two qualitatively distinct contributions: i)~the standard antipodal piece, which gives $\Pi_{\rm{ATP}}(\mathbf{q}, i\Omega)\propto\Omega/q$ and ii)~an additional umklapp piece from electrons near the zone boundary, which gives $\Pi_{\rm{U}}(\mathbf{q}, i\Omega)\propto \Omega^\alpha$ at the minimum umklapp momentum $q\approx \Delta_q$ with $\alpha = 2/3 $ or $1/2$ depending on the temperature $T$. At high $T$ when $\alpha = 1/2$, the minimum $T$ for the activation of linear/quasi-linear in $T$ resistivity, which is expected to be $T_U \propto \Delta_q^3$ from $z=3$ criticality, could potentially get reduced to $T_U \propto \Delta_q^4$ due to the $\sqrt{\Omega}$ term and discuss why we find only one hyper-specific scenario where this possibility might be realized. For $d=3$ the umklapp contribution gives $\Pi_{\rm{U}}\sim \Omega$ irrespective of $T$ therefore $T_U$ is not modified in this case.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript computes the particle-hole bubble Π(q, iΩ) for an Ising-nematic quantum critical metal in d=2 when the critical Fermi surface approaches the Brillouin zone boundary. It identifies a conventional antipodal contribution Π_ATP ∝ Ω/q and an additional umklapp contribution Π_U(q≈Δ_q, iΩ) that yields Ω^α with α=2/3 (low T) or α=1/2 (high T). The authors argue that the high-T √Ω regime could reduce the minimum temperature T_U for linear/quasi-linear resistivity from the z=3 expectation ∝Δ_q^3 to ∝Δ_q^4 in one hyper-specific scenario; in d=3 the umklapp term is linear in Ω and produces no modification.

Significance. If the asserted link to transport holds, the result supplies a controlled microscopic mechanism by which umklapp processes near the zone boundary can alter the infrared Landau damping and thereby shift the onset of non-Fermi-liquid resistivity in a material-specific way. The explicit temperature-dependent exponent in the umklapp bubble is a technically useful addition to the literature on quantum critical transport, even if its quantitative impact on resistivity remains to be demonstrated.

major comments (2)
  1. [Abstract and concluding discussion] The central claim that the high-T Π_U ∝ √Ω term reduces T_U from ∝Δ_q^3 to ∝Δ_q^4 is asserted in the abstract and conclusion without an explicit transport calculation. The manuscript evaluates only the bubble diagram but does not insert the modified damping into the Kubo formula, memory-function expression, or Boltzmann equation to re-derive the T-dependence of ρ(T) at the relevant momenta. This step is load-bearing for the stated implication on resistivity scaling.
  2. [Discussion of the high-T regime] The hyper-specific scenario in which the umklapp term dominates transport is not quantified. No estimate is given for the momentum window around q≈Δ_q over which Π_U controls the scattering rate, nor for the temperature range in which α=1/2 remains visible before impurity or other channels intervene (see weakest assumption in the reader's report).
minor comments (2)
  1. [Section on bubble evaluation] The notation for the minimum umklapp momentum (Δ_q) and the precise definition of the high-T versus low-T regimes should be stated explicitly in the main text with a figure or equation reference.
  2. [Results for d=2] A brief comparison of the computed Π_U with the standard z=3 Landau-damping form (e.g., via a plot of the frequency dependence) would help readers assess the magnitude of the correction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and insightful report. We provide point-by-point responses to the major comments and will revise the manuscript to incorporate clarifications on the transport implications and the high-temperature regime.

read point-by-point responses

  1. Referee: [Abstract and concluding discussion] The central claim that the high-T Π_U ∝ √Ω term reduces T_U from ∝Δ_q^3 to ∝Δ_q^4 is asserted in the abstract and conclusion without an explicit transport calculation. The manuscript evaluates only the bubble diagram but does not insert the modified damping into the Kubo formula, memory-function expression, or Boltzmann equation to re-derive the T-dependence of ρ(T) at the relevant momenta. This step is load-bearing for the stated implication on resistivity scaling.

    Authors: We agree that an explicit transport calculation is not performed in the manuscript. Our central result is the evaluation of the particle-hole bubble, including the umklapp contribution. The discussion of the possible reduction in T_U is presented as a potential consequence based on the standard connection between the Landau damping and the critical transport in Ising-nematic models. To address this comment, we will revise the abstract and conclusion to more clearly state that this is a speculative implication and add a brief explanation of how the modified damping term would affect the resistivity scaling in the memory-function formalism. We maintain that the bubble calculation stands on its own as a technical contribution. revision: partial

  2. Referee: [Discussion of the high-T regime] The hyper-specific scenario in which the umklapp term dominates transport is not quantified. No estimate is given for the momentum window around q≈Δ_q over which Π_U controls the scattering rate, nor for the temperature range in which α=1/2 remains visible before impurity or other channels intervene (see weakest assumption in the reader's report).

    Authors: We concur that quantitative estimates for the momentum window and temperature range are not provided. In the revised manuscript, we will add estimates for the momentum range Δq around q ≈ Δ_q where the umklapp term Π_U could dominate over the antipodal term, based on comparing their magnitudes as functions of q and Ω. Additionally, we will discuss the temperature window for the √Ω regime, considering the crossover to other scattering channels such as impurities, and explain the conditions under which this hyper-specific scenario could be realized in a material. This will help delineate the applicability of the potential modification to transport. revision: yes

Circularity Check

0 steps flagged

Direct diagrammatic evaluation of umklapp bubble with no reduction to inputs

full rationale

The paper's derivation consists of an explicit computation of the particle-hole bubble diagram Π(q, iΩ) in the umklapp channel for an Ising-nematic metal when the critical Fermi surface nears the Brillouin zone boundary. This yields the stated forms Π_U(q≈Δ_q, iΩ) ∝ Ω^α (with α=1/2 at high T) and the antipodal piece ∝ Ω/q through momentum integration over the model dispersion and critical fluctuations. The discussion of possible shifts in the linear-resistivity threshold T_U references the standard z=3 Landau-damping scaling as an external benchmark but does not insert the new Π_U into a Kubo or memory-function transport formula, nor does it fit parameters or invoke self-citations to close the loop. No step equates the output to the input by construction, renames a known result, or smuggles an ansatz; the central result is an additive correction computed from the assumed Hamiltonian and remains independent of the speculative transport implications.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The claim rests on the standard Landau-damping framework for Ising-nematic criticality (z=3), the assumption that the Fermi surface can be tuned arbitrarily close to the zone boundary, and the separation into antipodal versus umklapp momentum windows; no new particles or forces are introduced.

free parameters (1)
  • Δ_q
    Minimum umklapp momentum set by how close the critical Fermi surface approaches the Brillouin zone boundary; treated as a tunable parameter that controls the scale of the new contribution.
axioms (2)
  • domain assumption The system is described by z=3 Ising-nematic quantum criticality in the standard Hertz-Millis framework.
    Invoked to motivate the expected T_U ∝ Δ_q^3 scaling that the umklapp term might modify.
  • domain assumption Only the particle-hole bubble at the minimum umklapp momentum q≈Δ_q needs to be computed; other momentum transfers are sub-dominant.
    Used to isolate the new Π_U term.

pith-pipeline@v0.9.0 · 5547 in / 1679 out tokens · 69026 ms · 2026-05-08T01:51:03.739878+00:00 · methodology

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Reference graph

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