Quantum geometric contribution to the diffusion constant

This paper establishes that the diffusion constant and DC conductivity in systems with linear Dirac dispersion can be rigorously separated into ordinary band velocity and quantum geometric contributions, revealing that the diffusion constant of three-dimensional Dirac fermions at charge neutrality is entirely quantum geometric due to an accidental cancellation of the band velocity term, a phenomenon not observed in two dimensions.

The Gist

The Big Picture: How Particles Move in a Weird World

Imagine you are trying to understand how a crowd of people moves through a city. In a normal city (a standard metal), people have clear streets to walk on, and they move based on how fast they are running and how crowded the streets are. This is the "ordinary" way we think about electricity flowing through wires.

But this paper is about a weird, quantum city (a Dirac semimetal) where the streets don't exist in the usual way. Here, the "people" are electrons, but they behave like massless particles moving at the speed of light, and the "streets" are actually made of invisible, twisting shapes in a mathematical space called "momentum space."

The author asks a simple question: When these electrons diffuse (spread out) in this weird city, what is actually pushing them?

The Two Forces: The "Runner" vs. The "Dancer"

The paper discovers that the movement of these electrons is driven by two different forces, which the author separates like two different types of motion:

1. The "Runner" (Ordinary Band Velocity):

   • Analogy: Imagine a runner sprinting down a straight track. Their speed depends entirely on how fast their legs are moving (their velocity). If you double their speed, they get to the finish line twice as fast.
   • In Physics: This is the standard way we think about electricity. Electrons move because they have momentum and velocity. In normal metals, this is the only thing that matters.

2. The "Dancer" (Quantum Geometry):

   • Analogy: Imagine a dancer spinning on a stage. Even if they aren't moving across the stage, the way they twist, turn, and overlap with the space around them creates a unique motion. It's not about how fast they run; it's about the shape of their movement and how their "shadow" overlaps with the shadows of others.
   • In Physics: This is the "Quantum Geometry." It comes from the fact that electron waves have a specific shape and phase. When they scatter off impurities (like bumps in the road), the way their wave shapes overlap determines how they move. This is the "metric" or "ruler" of the quantum world.

The Surprise: The 3D Magic Trick

The most exciting part of the paper is what happens when you change the dimension of the world.

• In a 2D World (Flatland):
  Imagine a flat sheet of paper. Here, the electrons are a mix. About 25% of their movement is due to the "Runner" (ordinary speed), and 75% is due to the "Dancer" (quantum geometry). It's a team effort, but the geometry does most of the heavy lifting.

• In a 3D World (Our World):
  Now, imagine adding a third dimension (up and down). The author performs a calculation and finds a miraculous cancellation.

  • The "Runner" force (ordinary speed) tries to push the electrons one way.
  • The "Dancer" force (geometry) tries to push them another way.
  • In 3D, these two forces cancel each other out perfectly. The "Runner" contribution drops to exactly zero.

The Result: In a 3D Dirac semimetal, the electrons don't move because they are "running" fast. They move entirely because of their quantum dance steps. The diffusion is 100% driven by the shape of the quantum waves, not their speed.

Why is this "Accidental"?

The author is careful to say this isn't a deep, fundamental law of the universe like gravity. He calls it an "accidental cancellation."

Think of it like a seesaw. If you put a heavy rock on one side and a heavy rock on the other, they balance. But if you change the weight of the rocks slightly (change the disorder or the model), the balance breaks.

• In 3D, the math just happens to line up so perfectly that the "speed" part vanishes.
• If you were in 4D, or if the material wasn't perfectly pure, this magic trick would fail, and the "Runner" would come back.

The Takeaway: A New Way to See Electricity

Usually, we think of electricity as a classical process: particles bumping into things and wandering around randomly (like a drunk person walking home).

This paper tells us that in these special 3D materials, that intuition is wrong. The "drunk person" isn't walking because of their legs; they are gliding because the floor itself is shaped in a way that forces them to slide.

In summary:

1. Ordinary metals: Electrons move because they are fast (Velocity).
2. 2D Quantum materials: Electrons move because they are fast AND because of their quantum shape (Geometry).
3. 3D Quantum materials: Electrons move only because of their quantum shape. The "speed" part vanishes completely due to a lucky mathematical accident.

This helps scientists understand that in the quantum world, the shape of the wave is just as important as the speed of the particle, and in some cases, the shape is the only thing that matters.

Technical Summary

1. Problem Statement

The paper addresses the microscopic origins of DC electrical conductivity and the diffusion constant in metallic systems lacking a well-defined Fermi surface, specifically Dirac and Weyl semimetals with linear dispersion relations.

• Context: In conventional metals with large Fermi surfaces, transport properties are dominated by the "ordinary" band velocity ($v_F$), with quantum geometric effects (like Berry curvature and the quantum metric) appearing only as small corrections.
• Gap: In systems where the Fermi surface shrinks to a point (charge neutrality in Dirac semimetals) or is absent (flat bands), the standard description breaks down. While previous work established that quantum geometry dominates in flat bands, its specific role in the diffusion constant of linear Dirac systems (as opposed to flat bands) remained less clear.
• Goal: To rigorously separate the contributions of ordinary band velocity and quantum geometry to the diffusion constant ($D$) and DC conductivity ($\sigma$) in $d$-dimensional Dirac fermions, and to determine if one mechanism dominates the other.

2. Methodology

The author employs a diagrammatic approach within the Self-Consistent Born Approximation (SCBA) for systems with Gaussian-distributed scalar disorder.

• Model: Massless Dirac (Weyl) fermions in $d \geq 2$ dimensions described by the Hamiltonian $H = v_F \boldsymbol{\sigma} \cdot \mathbf{k}$.
• Approach: Instead of calculating conductivity directly via the Kubo formula, the paper calculates the diffusion propagator ($D(q, \Omega)$). This is advantageous because the diffusion constant reveals the separation of contributions more clearly than the conductivity tensor.
• Key Steps:
  1. SCBA Green's Functions: Derive the disorder-averaged retarded and advanced Green's functions, incorporating the self-energy $\Sigma$.
  2. Diffusion Propagator: Construct the diffusion propagator by summing ladder diagrams (impurity averaging). The propagator is expressed as $D(q, \Omega) = [1 - I(q, \Omega)]^{-1}$.
  3. Expansion: Expand the function $I(q, 0)$ to second order in the wavevector $q$. The coefficient of the $q^2$ term yields the diffusion constant.
  4. Decomposition: The expansion separates into two distinct physical origins:
     • Longitudinal terms: Proportional to $\hat{k}_a \hat{k}_b$, arising from the momentum dependence of the band dispersion (band velocity).
     • Transverse terms: Proportional to the quantum metric tensor $g_{ab}(\mathbf{k}) \propto (\delta_{ab} - \hat{k}_a \hat{k}_b)$, arising from the overlap of Bloch wavefunctions (quantum geometry).

3. Key Contributions

• Rigorous Separation: The paper demonstrates a clean separation of the diffusion constant into "ordinary" (band velocity) and "quantum geometric" contributions. This separation maps directly to the decomposition of a rank-2 tensor into longitudinal and transverse parts in rotationally invariant systems.
• Dimensional Dependence: The analysis reveals that the ratio of band velocity to geometric contributions is a universal function of dimensionality $d$.
• Accidental Cancellation in 3D: The most significant finding is that for 3D Dirac/Weyl fermions at charge neutrality, the ordinary band velocity contribution to the diffusion constant vanishes exactly (within SCBA and Gaussian disorder). Consequently, the diffusion constant is entirely of quantum geometric origin.
• Physical Interpretation: The paper clarifies that in 3D, the "random walk" intuition of diffusion (driven by particle velocity) fails. Instead, diffusion is driven by the overlap of wavefunctions at different momenta (quantum geometry).

4. Detailed Results

A. Doped Case (Large Fermi Energy)

For a heavily doped system ($\epsilon_F \tau \gg 1$), the standard result is recovered:
$$D_d = \frac{v_F^2 \tau}{d}$$
Here, the diffusion is dominated by the band velocity, and quantum geometric corrections are negligible (higher order in $1/\epsilon_F \tau$).

B. Charge Neutrality (Dirac Semimetals)

At charge neutrality ($\epsilon_F \to 0$), both conduction and valence bands contribute equally. The diffusion constant $D_d$ is derived as:
$$D_d = D_d^{\parallel} + D_d^{\perp}$$
Where $D_d^{\parallel}$ is the longitudinal (band velocity) part and $D_d^{\perp}$ is the transverse (geometric) part. The ratio is universal:
$$\frac{D_d^{\parallel}}{D_d^{\perp}} = \frac{3-d}{3(d-1)}$$

Specific Dimensions:

• 2D Dirac Semimetals ($d=2$):

  • The ratio is $D_2^{\parallel} / D_2^{\perp} = 1/3$.
  • Result: 25% of the diffusion constant comes from band velocity, and 75% comes from the quantum metric.
  • The DC conductivity is $\sigma_2 = e^2 / (\pi h)$, a universal value independent of disorder strength.

• 3D Dirac/Weyl Semimetals ($d=3$):

  • The ratio is $D_3^{\parallel} / D_3^{\perp} = 0$.
  • Result: The band velocity contribution vanishes identically due to an accidental cancellation of terms in the integral (specifically, the positive contribution from the inverse particle-hole propagator cancels the negative contribution from the spectral function).
  • Conclusion: The diffusion constant is 100% quantum geometric.
  • The diffusion constant is $D_3 = \frac{\gamma^2}{8\pi v_F}$, and the conductivity is $\sigma_3 = \frac{e^2}{4\pi h \ell}$.

5. Significance and Implications

• Paradigm Shift in Transport: The results challenge the classical intuition that diffusion is solely a result of particles moving with random velocities. In 3D Dirac semimetals, the mechanism is purely quantum mechanical, relying on the geometric structure of the Hilbert space (wavefunction overlap).
• Universality: The finding that 3D Dirac fermions share the property of "geometry-dominated transport" with flat-band systems (despite having zero effective mass rather than infinite mass) suggests a deep connection between systems without a Fermi surface.
• Conceptual Clarity: By using the diffusion constant rather than the Kubo formula, the author exposes a hidden structure in transport theory that is obscured in direct conductivity calculations.
• Limitations: The author notes that the exact cancellation in 3D is "accidental" (dependent on the specific SCBA and Gaussian disorder model) and may not be robust against all types of disorder or non-perturbative effects (e.g., rare region effects). However, the conceptual demonstration of the dominance of quantum geometry remains a robust insight.

In summary, Burkov provides a rigorous theoretical framework showing that in 3D Dirac semimetals, the ability of charge carriers to diffuse is governed entirely by the quantum geometry of their wavefunctions, marking a departure from standard metallic transport theory.