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arxiv: 2503.01381 · v2 · pith:UTCIL7A7new · submitted 2025-03-03 · 🧮 math-ph · cond-mat.mes-hall· math.MP

Longitudinal conductivity at integer quantum Hall transitions

Giovanna Marcelli , Lorenzo Pigozzi , Marcello Porta This is my paper

Pith reviewed 2026-05-23 01:54 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.mes-hallmath.MP
keywords longitudinal conductivityquantum Hall transitionsconical intersectionsKubo formulatight-binding modelsDirac coneslinear responseinteger quantum Hall effect
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The pith

Longitudinal conductivity at integer quantum Hall transitions is explicitly determined by the number and shapes of conical intersections.

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

The paper establishes an explicit formula for the longitudinal conductivity using the Kubo formula in two-dimensional tight-binding models that feature conical intersections of Bloch bands at the Fermi level. These models encompass generic transitions between integer quantum Hall phases. The conductivity depends only on the number of these intersections and the detailed shape of each cone. A reader would care because this gives a simple way to compute transport from local band features near crossings, and it confirms the Kubo approach remains valid by deriving it from the time-dependent Schrödinger equation under weak electric fields.

Core claim

We obtain an explicit expression for the longitudinal conductivity, completely determined by the number of conical intersections and by the shape of the cones. In particular, the formula reproduces the known quantized values found for graphene and for the critical Haldane model. For electric fields which are weak and slowly varying in space and in time, we prove the validity of linear response from quantum dynamics.

What carries the argument

The Kubo formula applied to models with conical intersections, yielding conductivity as a function of intersection count and cone shapes.

If this is right

  • The longitudinal conductivity takes specific values determined by cone geometry at transitions between quantum Hall phases.
  • The result holds for a broad class of tight-binding models with Dirac points at the Fermi energy.
  • Linear response theory applies directly from the time-dependent Schrödinger equation when the electric field is weak and adiabatic.
  • The known conductivity values in graphene and the critical Haldane model are recovered exactly.

Where Pith is reading between the lines

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

  • This approach could be used to design lattice models where conductivity is tuned by adjusting the number or anisotropy of conical intersections.
  • Similar expressions might apply to other transport coefficients near band crossings.
  • Connections to effective low-energy Dirac theories could be explored to interpret the shape dependence.

Load-bearing premise

The Bloch bands must exhibit conical intersections precisely at the Fermi level, and the electric field must remain weak and slowly varying to justify the linear response derivation.

What would settle it

Compute the longitudinal conductivity numerically in the critical Haldane model and check whether it matches the value predicted from the number and shape of its conical intersections.

Figures

Figures reproduced from arXiv: 2503.01381 by Giovanna Marcelli, Lorenzo Pigozzi, Marcello Porta.

Figure 1
Figure 1. Figure 1: Path used in the complex-time deformation. [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Graphs of the smooth cut-off functions χ, 1−χ. 3.3 Singular and regular parts of ˜fj j First we single out from ˜fj j the singular part, denoted by ˜f sing j j , which is due to the energies close to the Fermi energy µ. We proceed as follows. (i) Let δ > 0 small enough, and let χ ∈C ∞(R+) such that χ(x) = 1 if x ≤ 1 and χ(x) = 0 if x ≥ 2, see e. g [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Graphical representation of the energy cutoff. The colored regions represent the re [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Complex path used to prove Lemma 3.2. We denote the minimum of the spectrum of [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗
read the original abstract

We consider a class of two-dimensional tight binding models displaying conical intersections of the Bloch bands at the Fermi level. The setting includes the case of generic transitions between quantum Hall phases. We consider the longitudinal conductivity, as given by Kubo formula, describing the variation of the current after introducing a space-homogeneous electric field, in an adiabatic way. We obtain an explicit expression for the longitudinal conductivity, completely determined by the number of conical intersections and by the shape of the cones. In particular, the formula reproduces the known quantized values found for graphene and for the critical Haldane model. Furthermore, we discuss the validity of Kubo formula in presence of conical intersections in the spectrum, starting from the time-dependent Schr\"odinger equation. For electric fields which are weak and slowly varying in space and in time, we prove the validity of linear response from quantum dynamics.

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

0 major / 2 minor

Summary. The manuscript examines a class of two-dimensional tight-binding models that exhibit conical intersections of Bloch bands at the Fermi level, which includes generic transitions between integer quantum Hall phases. The authors apply the Kubo formula to compute the longitudinal conductivity in response to a space-homogeneous electric field introduced adiabatically. They derive an explicit expression for this conductivity that depends only on the number of conical intersections and the local shape of the cones. This formula is shown to match known quantized values in graphene and the critical Haldane model. The paper also establishes the validity of the Kubo formula by proving linear response from the time-dependent Schrödinger equation for weak and slowly varying electric fields.

Significance. Should the derivation be correct, the result is significant as it provides a direct, explicit formula for longitudinal conductivity at quantum Hall transitions based on band structure features, without additional parameters. This unifies results across different models and offers a way to predict conductivity from the geometry of band crossings. The proof of linear response from quantum dynamics is a valuable contribution, addressing potential concerns about the applicability of Kubo formula near degeneracies.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'the shape of the cones' is used without indicating how the local geometry (e.g., opening angle or tilt parameters) enters the final conductivity formula; a one-sentence clarification would help readers connect the claim to the derivation.
  2. The manuscript should explicitly state the precise conditions on the electric-field ramp (e.g., the adiabatic parameter scaling) in the statement of the linear-response theorem, rather than only in the discussion section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work and for recommending minor revision. The referee's description accurately reflects the manuscript's content and contributions. No specific major comments are provided in the report, so we have no individual points to address. We will proceed with minor editorial polishing as appropriate for the revision.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives an explicit formula for longitudinal conductivity from the Kubo formula applied to tight-binding models with conical intersections at the Fermi level, and separately proves the validity of linear response from the time-dependent Schrödinger equation under weak, slowly varying electric fields. The result is stated to depend only on the number and local shape of the cones, and is verified to recover known quantized values for graphene and the critical Haldane model. No load-bearing self-citation, self-definitional steps, or fitted parameters renamed as predictions appear in the abstract or description; the central claim follows from the stated first-principles setup rather than reducing to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; ledger entries are therefore minimal and provisional.

axioms (2)
  • domain assumption Kubo formula gives the longitudinal conductivity for these models under adiabatic switching of a homogeneous electric field
    Invoked to obtain the explicit expression
  • domain assumption Linear response holds for weak, slowly varying electric fields starting from the time-dependent Schrödinger equation
    Used to justify validity of the Kubo approach

pith-pipeline@v0.9.0 · 5675 in / 1264 out tokens · 48178 ms · 2026-05-23T01:54:28.883112+00:00 · methodology

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Reference graph

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