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Quantum geometric contribution to the diffusion constant

A. A. Burkov

TL;DR

The paper investigates how quantum geometry, through the quantum metric, contributes to diffusion and DC conductivity in metals and Dirac/Weyl semimetals with linear dispersion. Employing a diffusion-propagator approach and the self-consistent Born approximation, it decomposes the diffusion constant into longitudinal (band-velocity) and transverse (geometric) parts, tying this separation to a rank-two tensor decomposition on momentum space. At 2D Dirac neutrality, about one-quarter of the diffusion originates from band velocity while three-quarters arise from quantum geometry; in 3D, the band-velocity contribution cancels, rendering diffusion entirely geometric with $D_3 = \frac{\gamma^2}{8\pi v_F}$ and $\sigma_3 = \frac{e^2}{4\pi h \ell}$. The results illuminate the central role of quantum geometry in transport for Dirac/Weyl systems without a conventional Fermi surface and establish universal relationships guiding future investigations, including extensions to lower-symmetry settings.

Abstract

We discuss the quantum geometric contribution to the diffusion constant and the DC conductivity in metals and semimetals with linear Dirac dispersion. We demonstrate that, for systems with perfectly linear dispersion, there exists a clear and rigorous separation of the quantum geometric from the ordinary band velocity contributions to the diffusion constant, which turns out to be directly related to the separation of a rank two tensor into transverse and longitudinal parts. We also demonstrate that the diffusion constant of three-dimensional Dirac fermions at charge neutrality is entirely quantum geometric in origin, which is not the case for two-dimensional Dirac fermions. This is the result of an accidental perfect cancellation of the band velocity contribution in three dimensions.

Quantum geometric contribution to the diffusion constant

TL;DR

The paper investigates how quantum geometry, through the quantum metric, contributes to diffusion and DC conductivity in metals and Dirac/Weyl semimetals with linear dispersion. Employing a diffusion-propagator approach and the self-consistent Born approximation, it decomposes the diffusion constant into longitudinal (band-velocity) and transverse (geometric) parts, tying this separation to a rank-two tensor decomposition on momentum space. At 2D Dirac neutrality, about one-quarter of the diffusion originates from band velocity while three-quarters arise from quantum geometry; in 3D, the band-velocity contribution cancels, rendering diffusion entirely geometric with and . The results illuminate the central role of quantum geometry in transport for Dirac/Weyl systems without a conventional Fermi surface and establish universal relationships guiding future investigations, including extensions to lower-symmetry settings.

Abstract

We discuss the quantum geometric contribution to the diffusion constant and the DC conductivity in metals and semimetals with linear Dirac dispersion. We demonstrate that, for systems with perfectly linear dispersion, there exists a clear and rigorous separation of the quantum geometric from the ordinary band velocity contributions to the diffusion constant, which turns out to be directly related to the separation of a rank two tensor into transverse and longitudinal parts. We also demonstrate that the diffusion constant of three-dimensional Dirac fermions at charge neutrality is entirely quantum geometric in origin, which is not the case for two-dimensional Dirac fermions. This is the result of an accidental perfect cancellation of the band velocity contribution in three dimensions.

Paper Structure

This paper contains 5 sections, 58 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Diffusion propagator for massless Dirac fermions
  3. d-dimensional metal
  4. Dirac semimetal
  5. Discussion and conclusions

Figures (1)

  • Figure 1: Diagrammatic representation of (a) The SCBA Green's function. Thin line represents the bare Green's function, thick line is the SCBA impurity-averaged Green's function and the dashed line represents the disorder potential correlator $\langle V({{\bf{r}}}) V({{\bf{r}}}') \rangle = \gamma^2 \delta({{\bf{r}}} - {{\bf{r}}}')$. (b) The equation for the diffusion propagator ${\cal D}$.