Anomalous thermoelectric Hall response of interacting 2D Dirac fermions

This paper demonstrates that the anomalous thermoelectric Hall response of interacting 2D massive Dirac fermions remains non-zero at zero temperature even after subtracting magnetization currents, a surprising result attributed to the inevitable manifestation of locality violation in the infrared physics of quantized field theories.

The Gist

The Big Picture: A Traffic Jam in a Quantum City

Imagine a tiny, flat city made of electrons (the "2D Dirac fermions"). In this city, the electrons move like light, and the city has a special property: it breaks the rules of symmetry, meaning traffic flows differently depending on which way you look. This is called Time-Reversal Symmetry Breaking.

Scientists are interested in a specific phenomenon here: The Thermoelectric Hall Effect.

• The Setup: Imagine you heat one side of this city and cool the other.
• The Result: You expect the electrons to flow from hot to cold. But because of the city's special "twisted" geometry (Berry curvature), the electrons don't just go straight; they get pushed sideways, creating a current perpendicular to the temperature difference. This is the Nernst Effect.

The paper asks a simple question: What happens if these electrons start talking to each other? (i.e., if they interact).

The Problem: The "Ghost" Traffic

In physics, when you calculate how much current flows, you have to be careful not to count "ghost traffic."

1. Real Traffic (Transport): Electrons actually moving from A to B, carrying heat and charge. This is what we want to measure.
2. Ghost Traffic (Circulating Currents): Electrons spinning in little loops right where they are. They aren't going anywhere useful, but they create a magnetic field and look like current if you aren't careful.

In the past, scientists had a magic eraser (a formula by Niu, Qin, and Shi) to wipe out the "Ghost Traffic." They calculated the total current, subtracted the "magnetization" (the measure of those spinning loops), and what was left was the true transport current.

The Golden Rule: At absolute zero temperature ($T=0$), if you remove the ghost traffic, the real transport current should vanish. Why? Because at absolute zero, there is no thermal energy to drive the electrons. If the current doesn't vanish, the math is broken.

The Experiment: Adding "Chatter" to the Electrons

The authors of this paper decided to test this "Magic Eraser" on a system where electrons interact (they push and pull on each other, like people in a crowded room).

They used a model of Massive Dirac Fermions. Think of these as electrons that have a little bit of "weight" (mass) and live in a 2D plane. They added a "contact interaction," which is like saying, "If two electrons get too close, they instantly repel each other."

They did the math to first order (the first step of complexity) and found something shocking.

The Surprise: The Magic Eraser Failed!

When they subtracted the "Ghost Traffic" (magnetization) from the total current in the interacting system, the result did not go to zero at absolute zero.

Instead of a clean, empty road at absolute zero, they still saw a "phantom" current flowing.

The Analogy:
Imagine you are trying to measure how much water flows through a pipe when you turn off the tap.

• Non-interacting case (Old Physics): You turn off the tap, and the pipe is perfectly dry. The math works.
• Interacting case (This Paper): You turn off the tap, but water is still dripping from the pipe. You check your bucket (the magnetization subtraction), and you think you removed all the water, but the pipe is still wet.

Why Did This Happen?

The authors realized that the "Magic Eraser" relies on a concept called Locality.

• Locality: This means that an electron only cares about what is happening right next to it.
• The Violation: In the quantum world, especially when you have interactions, things aren't truly local. An electron here can be influenced by an electron far away, even if they are separated by a tiny distance.

The paper suggests that the "Magic Eraser" formula assumes everything is perfectly local. But in the real quantum world, at the tiniest scales, locality breaks down. This tiny, unavoidable violation of "local rules" leaks through the math and manifests as a leftover current at absolute zero.

The Takeaway

1. The Expectation: We thought that if we account for electron interactions and subtract the "spinning loops" (magnetization), the thermoelectric current would behave perfectly and vanish at absolute zero.
2. The Reality: It doesn't. Even with the best subtraction method, a tiny, anomalous current remains.
3. The Cause: The universe isn't perfectly "local" at the smallest scales. The rules we use to clean up our math (which assume locality) fail when interactions are strong, leading to a "glitch" in the physics at absolute zero.

In short: The authors found a new, weird behavior in quantum materials where the standard tools for cleaning up calculations fail because the quantum world is too interconnected to be treated as a collection of isolated, local events. This suggests that our understanding of how heat and electricity flow in complex materials needs a major update.

Technical Summary

1. Problem Statement

The paper addresses a fundamental issue in the theoretical description of anomalous thermoelectric Hall transport (specifically the Nernst effect) in systems with broken time-reversal symmetry, such as massive Dirac fermions.

• The Core Issue: In standard linear response theory (Kubo formula), the calculation of the thermoelectric Hall coefficient ($L_{12}^{xy}$) includes spurious contributions from equilibrium circulating currents. These currents do not represent genuine transport and lead to unphysical divergences (infinite response) as temperature $T \to 0$.
• The Standard Fix: For non-interacting systems, these spurious terms are removed by subtracting the particle magnetization ($M_N$), a procedure formalized by the Kubo-Středa relation. This subtraction ensures the transport coefficient vanishes at $T=0$, consistent with physical expectations.
• The Open Question: It is unclear if this subtraction procedure remains valid when electron-electron interactions are introduced. Previous methods rely on locality assumptions that may be violated in interacting quantum field theories, particularly at short length scales (ultraviolet regime). The authors investigate whether the cancellation between the Kubo kernel and magnetization holds to first order in perturbation theory for interacting 2D massive Dirac fermions.

2. Methodology

The authors perform a rigorous perturbative calculation to first order in the electron-electron interaction strength for a continuum model of 2D massive Dirac fermions.

• Model: A single-flavor 2D massive Dirac Hamiltonian with a contact-like interaction (modeling overscreened Coulomb repulsion), $V(r-r') = V_c \delta(r-r')$. The interaction is treated with a momentum-dependent form factor $V_q$ to regularize ultraviolet divergences before taking the contact limit.
• Formalism:
  • Kubo Kernel ($K_{12}^{xy}$): Calculated using the current-current correlation function between particle current ($\hat{j}^N$) and heat current ($\hat{j}^Q$).
  • Magnetization ($M_N^z$): Calculated via the Středa formula, relating it to the static density-current response.
  • Diagrammatics: The authors compute Feynman diagrams to first order, including:
    1. Exchange diagrams (density-density interaction).
    2. Self-energy corrections (Fock insertions on propagators).
    3. Vertex corrections arising specifically from the interaction-dependent part of the heat-current operator (a crucial term often overlooked in charge transport but essential for thermoelectricity).
• Definitions: The true transport coefficient is defined as $(L_{12}^{xy})_{tr} = K_{12}^{xy} + M_N^z$. The authors explicitly verify the continuity equations and scaling laws required for the validity of the magnetization subtraction.

3. Key Contributions

1. First-Order Interaction Calculation: The paper provides the first complete calculation of the thermoelectric Hall response for interacting Dirac fermions to first order in perturbation theory, explicitly separating the Kubo kernel and magnetization contributions.
2. Identification of Vertex Corrections: The authors highlight the critical role of the interaction-dependent heat-current vertex. Unlike charge transport, where the current operator is often just the velocity, the heat current operator in interacting systems contains explicit interaction terms. These terms generate specific diagrams (Fig. 1e, 1f) that are essential for the calculation.
3. Analytical Formula for Magnetization: They derive a remarkably simple, compact analytical formula for the magnetization correction in the presence of interactions (Eq. 28), which depends on the self-energy components ($\Sigma_0, \Sigma_z$).

4. Key Results

The central finding of the paper is negative: the standard cancellation mechanism fails in the presence of interactions.

• Failure of Cancellation: In the non-interacting limit, the magnetization term $M_N^z$ exactly cancels the finite $T \to 0$ limit of the Kubo kernel $K_{12}^{xy}$, resulting in a vanishing transport coefficient. However, with interactions included to first order, $(L_{12}^{xy})_{tr}$ does not vanish at $T=0$.
• The Source of the Residual Term: The non-vanishing result arises because the specific combination of terms in the interacting magnetization correction does not perfectly match the interacting Kubo kernel. Specifically, in the final expression for the transport coefficient (Eq. 29), a factor of $1/2$ in one of the interaction terms prevents the exact cancellation that occurs in the non-interacting case.
• Locality Violation: The authors attribute this failure to a violation of locality on the smallest length scales. While the interaction is modeled as a contact term, the underlying quantum field theory requires regularization (non-locality) to handle ultraviolet divergences. This short-scale non-locality manifests in the infrared physics (the $T \to 0$ limit), breaking the scaling laws required for the magnetization subtraction to work perfectly.
• Dependence on Fermi Surface: The residual term depends on the existence of a Fermi surface. If the chemical potential lies in the gap (no Fermi surface), the result vanishes as expected. The anomaly is specific to the metallic phase.

5. Significance and Implications

• Theoretical Challenge: The result challenges the universality of the Kubo-Středa formalism for interacting systems. It suggests that for interacting topological materials, the "true" transport coefficient may not be simply the sum of the Kubo kernel and magnetization, or that the definition of the current operator requires more subtle renormalization in the presence of interactions.
• Experimental Relevance: The finding implies that in real materials (like topological magnets or Dirac semimetals) where electron-electron interactions are significant, the anomalous Nernst effect might exhibit a non-zero residual signal at zero temperature, contrary to standard non-interacting predictions.
• Ward Identities and Scaling: The paper discusses whether this failure stems from a violation of Ward identities or the specific scaling laws (Eqs. 11-12) required for the subtraction. The authors conclude that the violation likely stems from the inevitable non-locality introduced by regularization in the UV, which affects the IR transport properties.
• Future Directions: The work highlights the need for higher-order perturbation theory or non-perturbative methods (e.g., summing infinite leading diagrams) to determine if this is a genuine physical feature of interacting Hall thermoelectricity or an artifact of the first-order truncation.

In summary, the paper demonstrates that electron-electron interactions break the delicate cancellation between Kubo and magnetization terms in the thermoelectric Hall response of 2D Dirac fermions, leading to a potentially unphysical (or physically novel) non-vanishing response at zero temperature, attributed to the breakdown of locality in the quantum field theoretic description.