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Nonlinear thermal and thermoelectric transport from quantum geometry

This paper investigates nonlinear thermal and thermoelectric responses as powerful probes of quantum geometry, revealing a network of connections analogous to standard transport relations that offer new insights into topological systems like Weyl-Kondo semimetals and Bernal bilayer graphene.

Original authors: Yuan Fang, Shouvik Sur, Yonglong Xie, Qimiao Si

Published 2026-05-11

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yuan Fang, Shouvik Sur, Yonglong Xie, Qimiao Si

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to understand the shape of a hidden landscape. In the world of quantum materials, electrons don't just move like cars on a flat road; they move across a complex, warped terrain shaped by the material's atomic structure. This "shape" is called quantum geometry.

For a long time, scientists have had a few tools to peek at this landscape, but they only gave them a flat, 2D snapshot. This paper introduces a new set of tools that let us see the landscape in 3D, specifically by looking at how heat and electricity behave when pushed hard (nonlinearly).

Here is a breakdown of the paper's main ideas using everyday analogies:

1. The Landscape: Quantum Geometry

Think of the electrons in a material as hikers on a mountain.

Berry Curvature: This is like a twist in the path. If you walk in a circle, the twist makes you end up facing a different direction than when you started. It's a "topological" feature.
Quantum Metric: This is like the stretchiness or the actual distance between points on the map. It tells you how "tight" or "loose" the fabric of the electron's world is.

2. The Old Tools: Linear Responses

Previously, scientists mostly studied what happens when you give the electrons a gentle nudge (a small electric field or a tiny temperature difference).

The Wiedemann-Franz Law: This is a famous rule that says, "If you can conduct electricity well, you can also conduct heat well." It's like saying, "If a highway is good for cars, it's also good for trucks."
The Mott Relation: This connects how well a material conducts electricity to how well it generates voltage from heat (thermoelectric effect).

3. The New Discovery: Nonlinear Responses

The authors asked: "What happens if we push the electrons hard? What if we turn up the electric field or the heat gradient significantly?"

When you push hard, the electrons don't just move faster; they start reacting to the shape of the landscape in new ways. The paper discovers that even in this "hard push" scenario, there are still strict rules connecting electricity and heat, but they are more complex than the old rules.

They found two main scenarios, depending on the symmetry of the material (like how the material is built):

Scenario A: The "Twisted" Path (Time-Reversal Symmetric)

Imagine a material where the "twist" (Berry curvature) is the main feature, but the material looks the same if you run time backward.

The Discovery: The authors found a new "web" of rules. Just as the old rules linked electricity and heat, these new rules link the nonlinear versions of them.
The Analogy: Think of a river. In a gentle flow, the water moves straight. But if you flood the river (nonlinear), the water starts swirling in specific patterns based on the riverbed's shape. The paper shows that if you measure how much the water swirls (nonlinear Hall effect), you can predict exactly how much heat will be carried by those swirls, using a new version of the old rules.

Scenario B: The "Stretched" Fabric (Time-Reversal Broken)

Imagine a material where the "twist" cancels itself out, but the "stretchiness" (Quantum Metric) is the dominant feature. This happens in certain magnetic materials.

The Discovery: Here, the rules are different again. The "stretchiness" of the quantum fabric drives the nonlinear currents.
The Analogy: Imagine a trampoline. If you bounce gently, it behaves normally. But if you jump hard, the way the fabric stretches and snaps back creates a specific pattern of motion. The paper shows that the way heat moves in this "stretching" scenario is mathematically locked to how electricity moves, creating a new set of predictable relationships.

4. The Real-World Test: Bilayer Graphene

To prove these ideas aren't just math on paper, the authors looked at Bernal bilayer graphene (two layers of graphene stacked like a sandwich).

Why this material? It's like a perfectly tunable lab. You can change the "chemical potential" (essentially the number of electrons) by applying a gate voltage, like turning a dial.
The Result: They showed that by tuning this dial, you can isolate the "twist" effects from the "stretch" effects.
- In one setting, the "twist" dominates, and you can see the new nonlinear rules for twisted paths.
- In another setting, the "stretch" dominates, allowing scientists to measure the "quantum metric dipole" directly for the first time.

5. Why This Matters (According to the Paper)

The paper claims that these new relationships act as a Rosetta Stone for quantum materials.

Verification: If you measure the nonlinear electrical response, you can use these new rules to predict the nonlinear thermal response without even measuring the heat. If the prediction matches the measurement, you know you truly understand the quantum geometry of the material.
New Probes: This gives scientists a way to "see" the quantum metric (the stretchiness), which was previously very hard to measure directly.

In summary: The paper says that when you push quantum materials hard, electricity and heat still dance together in a predictable way. By understanding the steps of this new dance, we can finally map out the hidden, warped geometry of the quantum world with much greater precision.

Technical Summary: Nonlinear thermal and thermoelectric transport from quantum geometry

Problem Statement
Quantum geometry, characterized by the Berry curvature (antisymmetric) and quantum metric (symmetric) components of the quantum geometric tensor, is fundamental to understanding novel quantum phases such as correlated topological states and superconductivity. While the nonlinear Hall effect has emerged as a powerful probe for the Berry curvature dipole in time-reversal ( $T$ ) invariant systems with broken inversion symmetry ( $P$ ), and for the quantum metric dipole in systems with broken $T$ but preserved $PT$, these electrical measurements provide only limited "slices" of the full quantum geometry tensor. A significant gap exists in understanding how quantum geometry governs intrinsic nonlinear thermal and thermoelectric transport. Specifically, it is unclear whether fundamental relationships, analogous to the standard Wiedemann-Franz and Mott relations in linear response, exist between nonlinear electrical, thermal, and thermoelectric coefficients.

Methodology
The authors develop a semiclassical framework treating the electric field ( $\mathbf{E}$ ) and temperature gradient ( $\nabla T$ ) on equal footing as driving forces. The approach utilizes the Boltzmann equation in the relaxation time approximation within the weak-field regime, where $|\mathbf{E}|$ and $|\nabla T|$ are small compared to the Fermi energy and inter-band gaps.

Key methodological steps include:

Expansion of Distribution and Dynamics: The nonequilibrium distribution function is expanded in powers of $\mathbf{E}$ and $\nabla T$ . The group velocity and Berry curvature density are expanded to include corrections induced by the quantum metric dipole, treating the temperature gradient as a statistical force analogous to the electric field.
Current Formulation: Anomalous charge and heat currents are derived using edge-current formulas involving the Berry curvature density and the non-equilibrium distribution function.
Symmetry Analysis: The authors systematically analyze two distinct symmetry settings:
- $T$ -symmetric, $P$ -broken systems: Where the Berry curvature is odd in momentum ( $k$ ), leading to vanishing linear anomalous responses but finite second-order responses driven by the Berry curvature dipole.
- $PT$-symmetric systems: Where $T$ and $P$ are broken but $PT$ is preserved. Here, Kramers degeneracy cancels the net Berry curvature, leaving the quantum metric dipole as the sole contributor to intrinsic anomalous nonlinear responses.
Sommerfeld Expansion: To establish inter-relations between coefficients, the authors perform a Sommerfeld expansion at low temperatures to extract finite transport coefficients and their dependencies on the chemical potential ( $\mu$ ).

Key Contributions and Results
The paper uncovers a "web of connections" between nonlinear thermal, thermoelectric, and electrical transport coefficients that generalize the standard Wiedemann-Franz and Mott relations. The specific findings are:

$T$ -Symmetric Case (Berry Curvature Dipole):
- In systems with $T$ symmetry and broken $P$ , the quantum metric contributions to second-order terms vanish. The transport is controlled entirely by the Berry curvature dipole.
- The authors identify six independent second-order transport coefficients.
- They derive three generalized Wiedemann-Franz relations and two generalized Mott relations that constrain these coefficients. For example, the nonlinear thermal Hall coefficient is related to the nonlinear charge Hall coefficient via the Lorenz number ( $L$ ), and the nonlinear Nernst coefficient is related to the derivative of the nonlinear charge Hall coefficient with respect to $\mu$ .
- Crucially, the authors demonstrate that a naive generalization of Onsager reciprocity to the second order (e.g., $L_{1,22} \propto L_{2,11}$ ) is invalid in these systems.
$PT$-Symmetric Case (Quantum Metric Dipole):
- In systems with broken $T$ and $P$ but preserved $PT$, the Berry curvature contributions vanish due to Kramers degeneracy. The response is driven by the quantum metric dipole.
- A distinct set of relations emerges, including three Wiedemann-Franz type relations and one Mott type relation.
- Notably, the quantum metric dipole sustains an intrinsic anomalous charge-Hall effect at the nonlinear order, which is not present at the linear order in these specific symmetry settings.
Application to Bernal Bilayer Graphene:
- The authors apply their formalism to gate-tunable Bernal bilayer graphene.
- They show that by tuning the displacement field ( $u_D$ ) and strain ( $\delta$ ), one can isolate specific quantum geometric contributions.
- In the $T$ -symmetric, $P$ -broken phase (induced by strain), the system exhibits a large Berry curvature dipole, allowing for the experimental verification of the derived Mott-type relation between nonlinear Nernst and Ettingshausen coefficients.
- In the $PT$-broken phase (induced by displacement field with preserved $C_{3z}$ ), the Berry curvature dipole is constrained to zero by symmetry, while the quantum metric dipole remains finite. This allows for a direct probe of the quantum metric dipole via the slope of the nonlinear Hall conductivity versus electric field.

Significance and Claims
The paper claims to provide a systematic theoretical framework that extends the understanding of quantum geometry beyond electrical transport to thermal and thermoelectric domains. The primary significance lies in:

Generalized Relations: Establishing that the Wiedemann-Franz and Mott relations, which are central to characterizing Fermi liquid Drude transport, have distinct, symmetry-dependent counterparts in the nonlinear regime. These relations encode the nature of the underlying quantum geometry.
Experimental Probes: Proposing that these generalized relations offer new, experimentally accessible methods to probe quantum geometry. Specifically, the concurrence of enhanced density of states and high quantum geometric components in materials like Bernal bilayer graphene makes them ideal testbeds.
Topological Material Characterization: Suggesting that these nonlinear thermal and thermoelectric responses can elucidate the physics of complex topological systems, such as Weyl-Kondo semimetals (e.g., Ce $_3$ Bi $_4$ Pd $_3$ ), where spontaneous nonlinear Hall effects have been observed. The derived relations (e.g., Eq. 10) suggest that measuring nonlinear Nernst responses can provide insight into the Weyl-Kondo physics.
Onsager Reciprocity: Highlighting that the standard Onsager reciprocity relations do not simply generalize to the second order, implying a need for new forms of reciprocity in nonlinear nonequilibrium thermodynamics.

The authors conclude that their work lays the groundwork for further studies of nonlinear thermal effects, including the fully nonequilibrium regime and the role of interactions, while providing concrete, testable predictions for gate-tunable 2D materials and topological antiferromagnets.