Reply to: Comment on "Electric conductivity of graphene: Kubo model versus a nonlocal quantum field theory model"

In this reply, the authors refute the concerns raised by Bordag et al. regarding their previous work on graphene's electric conductivity by demonstrating that the criticisms stem from misinterpretations and misapplications of their model, while reaffirming the validity, physical consistency, and correctness of their original findings derived from both the Kubo formula and quantum field theory.

The Gist

Imagine two groups of scientists are having a heated debate over a very specific recipe for calculating how electricity flows through a super-thin material called graphene.

One group (the authors of the original paper, let's call them Team Graphene) published a recipe. The other group (the "Commenters," let's call them Team Skeptic) looked at the recipe and said, "Wait a minute! This recipe is broken. It predicts that electricity will flow even when you haven't turned on the power switch. That's impossible!"

This paper is Team Graphene's reply. They are saying, "You misunderstood the recipe. We didn't make a mistake; you just read the instructions wrong. Our recipe is actually perfect, and your version is the one that causes the impossible 'ghost electricity'."

Here is the breakdown of their argument using simple analogies:

1. The "Ghost Current" Problem

The Accusation: Team Skeptic claims that Team Graphene's math suggests that if you have a piece of graphene, electricity will start flowing on its own, even if there is no battery or electric field connected to it. It's like saying a car will drive itself down the highway with the engine off.

The Defense: Team Graphene says, "No, that's not what we said."

• The Analogy: Imagine you are trying to measure how fast a car goes.
  • Team Skeptic's method: They measure the car's speed relative to the ground, but they forget to subtract the speed of the ground itself (which might be moving). This makes it look like the car is moving even when it's parked.
  • Team Graphene's method: They use a "Luttinger Formula." This is like a smart GPS that automatically subtracts the background noise. They subtract the "static" part of the math so that if the electric field is zero, the current is definitely zero.
• The Result: Team Graphene insists their math correctly predicts that no current flows without an electric field. The "ghost current" only appears if you use the wrong math (the one Team Skeptic is using).

2. The "Friction" Debate (Losses)

The Accusation: Team Skeptic argues that Team Graphene included a "friction" factor (called $\Gamma$) in their math. They say, "Graphene is a perfect, ideal material in theory; adding friction ruins the purity of the model."

The Defense: Team Graphene says, "Real life isn't perfect, and neither is our model."

• The Analogy: Think of a skateboarder. In a perfect video game, the skateboarder glides forever on a frictionless surface. But in the real world, there is air resistance and rough pavement. If you want to predict how the skateboarder actually stops, you must include friction.
• The Point: Graphene in the real world has impurities and interactions that act like friction. Team Graphene included this "friction" to make the model match reality. Team Skeptic wants to ignore friction to keep the math "pure," but that leads to predictions that don't match what we see in experiments.

3. The "Magnetic vs. Electric" Mix-up

The Accusation: Team Skeptic claims that Team Graphene's model breaks the fundamental laws of physics (specifically "gauge invariance," which is a fancy way of saying the laws shouldn't change just because you look at them from a different angle).

The Defense: Team Graphene says, "You are applying our rules to a situation we never intended."

• The Analogy: Imagine a rulebook for a soccer game that says, "You can't use your hands." Team Skeptic says, "But what if you are playing basketball? Your rulebook is broken!"
• The Point: Team Graphene's model is specifically designed for electric fields (like the battery in a circuit). Team Skeptic tried to apply those rules to a constant magnetic field (like a magnet sitting still next to the graphene). Team Graphene explains that their model handles magnetic fields differently. When you separate the two, the "broken rule" accusation disappears. The model works perfectly for electricity.

4. The "Double Pole" Confusion

The Accusation: Team Skeptic points out a weird mathematical spike (a "double pole") in the data and says it proves the model is wrong.

The Defense: Team Graphene says, "That spike isn't a mistake; it's a feature of a different part of the system."

• The Analogy: Imagine you are listening to a song. You hear a weird high-pitched squeal. Team Skeptic says, "The song is ruined!" Team Graphene says, "No, that squeal is coming from the magnetic speaker, not the electric one. If you are only listening to the electric part, that squeal isn't there."
• The Point: The weird mathematical spike comes from magnetic effects, not the electric conductivity they are studying. It doesn't break the physics of the electric current.

The Bottom Line

Team Graphene is standing firm. They are saying:

1. We are right: Our model correctly predicts that electricity needs a push (an electric field) to flow.
2. You are wrong: The "impossible" results you found are because you used a simplified version of the math that ignores friction and mixes up magnetic and electric effects.
3. Real world matters: Including "losses" (friction) is standard practice and necessary to match real experiments.

They conclude that their original paper is solid, the math is correct, and the "ghost currents" are just a mirage created by looking at the math the wrong way.

Technical Summary

Based on the provided text, here is a detailed technical summary of the reply paper by Rodriguez-Lopez, Wang, and Antezza.

Paper Title: Reply to Comment on "Electric conductivity of graphene: Kubo model versus a nonlocal quantum field theory model"

1. Problem Statement

The authors are responding to a "Comment" by Bordag et al. [1] which challenges the validity of their previous work [2]. The Comment raised several critical concerns regarding:

• Theoretical Consistency: Alleged misinterpretation of the relationship between the polarization tensor ($\Pi_{\mu\nu}$) and the electric current ($J_\mu$), specifically claiming the authors incorrectly modified the relation to satisfy the Luttinger formula.
• Causality and Formalism: Accusations that the derivation of the Luttinger formula from the Kubo formalism relied on incorrect "nonrelativistic" causality and one-sided Fourier transforms, rather than proper Feynman Green functions.
• Physical Validity: Claims that the inclusion of dissipation (losses, parameter $\Gamma$) violates the relativistic Dirac model and that the authors' model predicts unphysical results, such as electric currents existing in the absence of an electric field (specifically a "double pole" in frequency $\omega$).
• Tensor Structure: Arguments that the subtraction term in the Luttinger formula violates the tensorial structure and gauge invariance of the conductivity.
• Experimental Relevance: Claims that the authors' model contradicts established literature regarding the real nature of conductivity in pure graphene at zero temperature and fails to explain thermal Casimir force effects.

2. Methodology

The authors employ a rigorous point-by-point rebuttal strategy, combining mathematical re-derivation, physical argumentation, and literature cross-referencing:

• Mathematical Clarification: They re-examine the definitions of the polarization tensor ($\Pi_{\mu\nu}$) and conductivity ($\sigma_{\mu\nu}$) to demonstrate that their original equations were correct and that the Comment misapplied them.
• Causality Analysis: They verify the use of standard two-sided Fourier transforms and Matsubara time-ordered Green functions in their original work, refuting claims of non-relativistic approximations.
• Limit Analysis: They analyze the behavior of their conductivity models in specific limits (local limit $q \to 0$, zero temperature $T \to 0$, and zero frequency $\omega \to 0$) to compare with cited literature (Falkovsky, Katsnelson).
• Physical Mechanism Differentiation: They distinguish between different physical contributions to the conductivity peaks:
  • Drude Peak: Arising from finite losses ($\Gamma > 0$).
  • Plasma Peak: Arising in the lossless limit ($\Gamma \to 0$).
  • Anomalous Third Peak: Arising from magnetically induced currents in the non-relativistic (NR) model, which the Luttinger formula naturally excludes.
• Gauge Invariance Check: They clarify the scope of their work, noting that the Luttinger formula derived in [2] is a spatial tensor relating spatial vectors ($J$ and $E$), not a covariant space-time tensor, rendering the Comment's gauge invariance critique inapplicable.

3. Key Contributions

• Validation of the Luttinger Formula: The authors confirm that the Luttinger formula, derived via the Kubo formalism, is the correct expression for electric conductivity in graphene. It ensures that no electric current flows in the absence of an electric field, a fundamental physical requirement.
• Clarification of the "Double Pole" Issue: They demonstrate that the "double pole" at zero frequency claimed by the Commenters arises only in the non-local (NR) model when incorrectly applying Faraday's law to static magnetic fields. The Kubo-derived Luttinger model naturally avoids this unphysical behavior.
• Role of Dissipation ($\Gamma$): The authors defend the inclusion of a phenomenological loss parameter $\Gamma$. They argue that losses are ubiquitous in real materials and that setting $\Gamma = 0$ leads to non-dissipative, permanent currents that do not reflect physical reality. They show that $\Gamma > 0$ is consistent with the Dirac model and widely accepted in the literature.
• Resolution of Contradictions with Literature: They prove that their results are consistent with established works (e.g., Falkovsky [16] and Katsnelson [17]) when the same limits (local, zero temperature) are applied correctly. The imaginary part of the conductivity vanishes in the appropriate limits, resolving the alleged contradiction.
• Correction of Minor Errors: The authors acknowledge and correct minor typographical errors in their original paper [2] regarding constants ($\alpha_c$ vs $e^2/\hbar$) and variable definitions.

4. Results

• Current-Field Relationship: The authors rigorously show that for their model ($\sigma^K$), if the electric field $E_\nu = 0$, the induced current $J_\mu = 0$. This holds true even in the presence of static magnetic fields, provided the correct vector potential formulation is used.
• Behavior of Peaks:
  • The Drude peak (with $\Gamma > 0$) leads to dissipative currents that decay over time.
  • The Plasma peak (with $\Gamma \to 0$) represents a non-dissipative current induced by an electric field, but it does not imply a current exists without a field.
  • The Anomalous peak in the Comment's model is identified as a magnetic contribution that incorrectly appears as an electric conductivity term due to a flawed application of the relation between vector potential and electric field.
• Casimir Force: The authors refute the Comment's claim that their model is required to explain the "big thermal effect" in the Casimir force. They argue this effect is a general property of any 2D Drude metal with $\lim_{\omega \to 0} \sigma_L(\omega) > 0$ and does not validate the specific (and flawed) conductivity model proposed by the Commenters.
• Permittivity: They confirm that the electric permittivity does not exhibit a double pole in $\omega$, contrary to the Comment's assertion.

5. Significance

• Theoretical Robustness: The reply solidifies the validity of using the Kubo formalism combined with the Luttinger formula for describing graphene's non-local conductivity. It establishes that the subtraction of the $\omega \to 0$ limit is a necessary and physically correct step to ensure causality and the absence of spontaneous currents.
• Clarification of Gauge Invariance: By distinguishing between spatial tensors and covariant space-time tensors, the authors resolve the confusion regarding gauge invariance, showing that the Commenters criticized an extension of the model that was never proposed in the original work.
• Experimental Consistency: The work reinforces that the inclusion of dissipation ($\Gamma$) is not only physically necessary but also consistent with experimental measurements of graphene's conductivity and relaxation times.
• Impact on Casimir Physics: By clarifying the origin of thermal effects in the Casimir force, the authors prevent the misattribution of experimental results to incorrect theoretical models, ensuring that future theoretical developments in graphene electrodynamics are built on a sound foundation.

In conclusion, the authors maintain that all results in their original paper [2] are fully valid, correct, and consistent with established physics, and that the concerns raised in the Comment stem from misinterpretations and the application of models outside their domain of validity.