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The Flexoelectric Twist: Persistence of Heavy Fermion Picture in Small-Angle Twisted Bilayer Graphene

By combining scanning tunneling microscopy with phenomenological modeling, this study reveals that in small-angle twisted bilayer graphene, moiré lattice reconstruction and atomic dilation induce a flexoelectric response that breaks particle-hole symmetry, thereby stabilizing compact heavy fermion states at AA sites that coexist with itinerant electrons and confirming the persistence of the heavy fermion picture in this regime.

Original authors: Qirong Yao, Ka Cheuk Fan, Tianyu Qiao, Hao Shi, Xi Dai, Berthold Jäck

Published 2026-10-30

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CC BY 4.0

Original authors: Qirong Yao, Ka Cheuk Fan, Tianyu Qiao, Hao Shi, Xi Dai, Berthold Jäck

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 a world where the rules of electricity change simply by twisting a sheet of material. In the realm of quantum physics, scientists have discovered that stacking two layers of graphene—a material made of a single layer of carbon atoms arranged in a honeycomb pattern—and twisting them slightly creates a new kind of landscape for electrons. When these layers are twisted to a specific, very precise angle, the electrons slow down dramatically, forming what physicists call "flat bands." In this slowed-down state, electrons interact with each other so strongly that they create exotic phases of matter, including superconductivity, where electricity flows with zero resistance. For years, researchers have focused on a specific "magic angle" of about 1.05 degrees, where these effects are most famous. However, the question remained: what happens if you twist the layers even more, to angles smaller than one degree? Does the magic disappear, or does a different kind of physics emerge?

A team of researchers at the Hong Kong University of Science and Technology has now answered this question by looking directly at these tiny, twisted structures. They built devices using graphene layers twisted at angles between 0.3 and 0.7 degrees and used a powerful microscope called a scanning tunneling microscope to watch how electrons behave. This microscope acts like a sensitive probe that can feel the surface of the material atom by atom and measure the energy of the electrons sitting there. What they found was surprising. Instead of the electrons behaving in a balanced way, they discovered a strong imbalance. The electrons on one side of the energy scale were behaving very differently from those on the other side, a phenomenon known as breaking particle-hole symmetry. This imbalance was not random; it was caused by the physical stretching of the carbon atoms themselves.

When the graphene layers are twisted to these small angles, the material does not stay perfectly flat. The atoms relax and shift, causing the carbon lattice to stretch or dilate specifically at the points where the two layers align perfectly. The researchers measured this stretching to be about 2.1 percent larger at these alignment points compared to the rest of the material. This physical stretching creates an internal electric field, a response known as flexoelectricity, which pushes the energy levels of the electrons down. This shift is crucial because it changes the entire electronic structure of the material. The electrons at these stretched points become heavy and sluggish, getting stuck in place, while the electrons in the surrounding areas remain free to move around.

This separation creates a unique situation that physicists call a "heavy fermion" state. In this state, the stuck, heavy electrons act like a fixed background, while the free, moving electrons flow around them. This is similar to how a swimmer moves through a crowd of stationary people; the swimmer is the free electron, and the crowd represents the heavy, localized electrons. The researchers confirmed that this heavy fermion picture, which was previously thought to only exist at the specific magic angle of 1.05 degrees, actually persists and even thrives at these smaller angles. They showed that the heavy electrons are tightly packed at the alignment points, while the free electrons form a broad, interconnected network that fills the space between them.

The study also revealed that the behavior of these electrons changes depending on the exact angle of the twist. As the angle gets smaller, the area where the heavy electrons live shrinks, but the space for the free electrons grows. This means the free electrons become more dominant, effectively screening or shielding the interactions between the heavy electrons. This finding is significant because it suggests that small-angle twisted graphene is not just a variation of the magic-angle system, but a distinct and highly tunable platform for studying complex quantum interactions. By simply adjusting the twist angle or applying an electric field, scientists can now control the balance between the heavy, stuck electrons and the free, moving ones.

This work opens a new door for exploring the "Kondo lattice" problem, a long-standing puzzle in physics that describes how magnetic atoms interact with a sea of moving electrons. In traditional materials, it is very difficult to tune these interactions to see how they change. But in these small-angle graphene devices, the researchers have found a way to tune the system with high precision. The ability to observe these heavy fermion states without needing to hit a single, perfect angle means that scientists can now study how these quantum states evolve and interact in a much more flexible environment. The discovery confirms that the physics of these twisted materials is richer and more robust than previously thought, offering a new laboratory for understanding how matter behaves when electrons are forced to slow down and interact.

Technical Summary: The Flexoelectric Twist: Persistence of Heavy Fermion Picture in Small-Angle Twisted Bilayer Graphene

Problem Statement
Twisted bilayer graphene (TBG) near the first "magic angle" ( $\theta \approx 1.05^\circ$ ) is known to host nearly particle-hole (PH) symmetric flat bands, describable by a topological heavy fermion (HF) model where localized $f$ -orbitals at AA stacking sites coexist with itinerant $c$ -electrons at AB/BA sites. However, at smaller twist angles ( $\theta < 1^\circ$ ), strong lattice relaxation significantly alters the moiré landscape, reducing the AA stacking area and expanding AB/BA domains separated by domain walls. While continuum models have predicted the existence of other magic angles in this regime, the impact of electro-mechanical coupling—specifically lattice reconstruction and strain-induced effects—on the low-energy electronic structure remains unresolved. It is unclear whether the heavy fermion picture persists in this small-angle regime or if the electronic landscape is fundamentally altered by these structural changes.

Methodology
The authors employed cryogenic scanning tunneling microscopy and spectroscopy (STM/STS) at $T = 4.5$ K to investigate TBG devices with twist angles ranging from $\theta \approx 0.3^\circ$ to $1.1^\circ$ .

Experimental Characterization: The study involved analyzing topographic maps to determine moiré lattice wavelengths and fitting a uniaxial heterostrain model to extract twist angles and strain. Atomic-resolution STM was used to measure lattice constants at AA and AB sites via 2D fast-Fourier transform (FFT) analysis. Spectroscopic measurements ( $dI/dV$ ) were performed to map the local density of states (LDOS) with spatial resolution.
Theoretical Modeling: To interpret the experimental data, the authors developed a phenomenological continuum model. This model incorporated two distinct effects: (1) relative lattice reconstruction (minimizing elastic and interlayer adhesion energy) and (2) a common atomic lattice dilation at AA sites relative to AB/BA regions. The latter introduces a scalar strain field and flexoelectric coupling. The model utilized orbital projectors based on the zeroth Landau level (ZLL) doublet to map the calculated spectral functions onto the heavy fermion basis ( $H_{ff}$ , $H_{cc}$ , and coupling $V_{fc}$ ).

Key Results

Structural Observations: STM topographies of small-angle TBG ( $\theta \approx 0.37^\circ$ ) revealed a significant expansion of AB/BA domains and a relative dilation of the atomic lattice at AA sites compared to AB sites, quantified as a $\sim 2.1\%$ increase in the lattice constant ( $\Delta \eta$ ). In contrast, samples near the first magic angle showed identical lattice constants at AA and AB/BA sites.
Electronic Asymmetry: STS measurements on small-angle TBG showed a pronounced $dI/dV$ peak localized at the AA site, centered at a negative bias voltage ( $V_B \approx -80$ mV). This spectrum is highly asymmetric about the charge neutrality point (CNP), contrasting with the approximately PH-symmetric spectra observed near the first magic angle. The peak position shifts to more negative voltages as the twist angle decreases.
Model Validation: Theoretical calculations incorporating the experimentally observed lattice dilation successfully reproduced the PH symmetry breaking and the energy shift of the AA-localized resonance. The scalar deformation potential associated with the dilation lowers the orbital energy in the AA region.
Persistence of the Heavy Fermion Picture: The study demonstrates that the low-energy electronic structure of small-angle TBG can be mapped to the heavy fermion model. The system comprises compact, localized $f$ $f$ -states at AA sites embedded within a broad, itinerant continuum of $c$ $c$ -states at AB/BA sites.
- The $f$ -orbitals remain highly localized, with their bandwidth suppressed further than at the first magic angle due to increased separation between AA centers.
- Unlike the first magic angle where $f$ -states are spectrally isolated, the flexoelectric coupling in small-angle TBG pushes the $f$ -states down in energy, causing them to energetically coexist with the itinerant $c$ -electrons.
- The density of states (DOS) of the itinerant $c$ -background at the energy of the $f$ -orbitals scales approximately exponentially with decreasing twist angle.

Significance and Claims
The paper claims that the heavy fermion model, previously established for the first magic angle, naturally extends to the small-angle regime ( $\theta < 1^\circ$ ) without the need for fine-tuning to a specific "magic" condition. The primary contribution is the identification of flexoelectricity (arising from atomic lattice dilation) as the mechanism that breaks particle-hole symmetry and enables the coexistence of localized $f$ -states and itinerant $c$ -states.

The authors posit that small-angle TBG establishes a new class of moiré materials for exploring the interacting Kondo lattice problem. The system offers high tunability: the twist angle controls the density of itinerant $c$ -electrons (and thus the screening of Coulomb interactions), while electrostatic back gates control the $f$ -orbital occupation. This tunability provides a platform to study the competition between Kondo screening and long-range magnetic order (RKKY interaction) in a regime where the itinerant background is not spectrally isolated from the localized moments, a condition distinct from both the first magic angle TBG and bulk heavy fermion compounds.

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