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Longitudinal conductivity at integer quantum Hall transitions

Giovanna Marcelli, Lorenzo Pigozzi, Marcello Porta

TL;DR

This work derives a closed-form expression for the longitudinal conductivity at integer quantum Hall transitions in a broad class of two-dimensional periodic lattice models exhibiting conical band intersections at the Fermi level. Using a Kubo-type linear response with a complex-time deformation, the authors show that $\sigma_{jj}$ is dictated solely by local cone geometry, encoded in the matrices $S_l$, and the number of cones $n$, via $\sigma_{jj} = \frac{1}{16}\sum_{l=1}^n \frac{s_{l,1j}^2 + s_{l,2j}^2}{|\det S_l|}$. In isotropic-cone cases, this recovers the familiar quantized values, $\sigma_{jj} = n/16$, including graphene and the critical Haldane model, highlighting a non-topological transport property controlled by critical-band geometry. The results extend prior analyses from interacting graphene and specific models to a wide, non-interacting lattice class with generic conical crossings, and open avenues for extensions to spin transport and interacting systems.

Abstract

We compute the longitudinal conductivity for a wide class of two-dimensional tight-binding models, whose Hamiltonian displays conical intersections of the Bloch bands at the Fermi level. Our setting allows to consider generic transitions between quantum Hall phases. 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 obtained for graphene and for the critical Haldane model.

Longitudinal conductivity at integer quantum Hall transitions

TL;DR

This work derives a closed-form expression for the longitudinal conductivity at integer quantum Hall transitions in a broad class of two-dimensional periodic lattice models exhibiting conical band intersections at the Fermi level. Using a Kubo-type linear response with a complex-time deformation, the authors show that is dictated solely by local cone geometry, encoded in the matrices , and the number of cones , via . In isotropic-cone cases, this recovers the familiar quantized values, , including graphene and the critical Haldane model, highlighting a non-topological transport property controlled by critical-band geometry. The results extend prior analyses from interacting graphene and specific models to a wide, non-interacting lattice class with generic conical crossings, and open avenues for extensions to spin transport and interacting systems.

Abstract

We compute the longitudinal conductivity for a wide class of two-dimensional tight-binding models, whose Hamiltonian displays conical intersections of the Bloch bands at the Fermi level. Our setting allows to consider generic transitions between quantum Hall phases. 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 obtained for graphene and for the critical Haldane model.

Paper Structure

This paper contains 13 sections, 10 theorems, 138 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Model and main result
  4. Periodic operators
  5. The model
  6. Linear response
  7. Main result
  8. Proof of Theorem \ref{['thm:main']}
  9. $\sigma_{jl}$ as right derivative of $f_{jl}$
  10. Complex time deformation
  11. Singular and regular parts of $\Tilde{f}_{jj}$
  12. Final rewriting of $\sigma_{jj}$
  13. Auxiliary results

Key Result

Theorem 2.4

Under Assumption assum:H, the longitudinal conductivity $\sigma_{jj}$ is given by, for $j=1,2$: where the scalar $s_{l,ij}$ is the $ij$-th element of the matrix $S_l$, appearing in the energy dispersion relation eqn:conical of the two eigenvalues $\Lambda_{\pm}$, near the Fermi energy $\mu$ around the Fermi point $\omega_l$.

Figures (4)

  • Figure 1: Path used in the complex-time deformation.
  • Figure 2: Graphs of the smooth cut-off functions $\chi$, $1-\chi$.
  • Figure 3: Graphical representation of the energy cutoff. The colored regions represent the region of the spectrum selected by $\chi_{\alpha_1} P_\pm$. In all the terms that we estimate a gap appears, which allows us to write the inequality \ref{['eqn:ineqwithone>']}.
  • Figure 4: Complex path used to prove Lemma \ref{['lem:norm-derivative-projector-estimate']}. We denote the minimum of the spectrum of $H$ as $\mathrm{Sp}_\text{m}$.

Theorems & Definitions (22)

  • Remark 2.2
  • Definition 2.3
  • Theorem 2.4: Main result
  • Lemma 2.5
  • Proposition 3.1
  • Lemma 3.2
  • proof : Proof of Proposition \ref{['prop:right-derivative']}
  • Proposition 3.3
  • proof
  • Remark 3.4
  • ...and 12 more