🔬 mesoscale physics

Quantized heat flow in the Hofstadter butterfly

This study demonstrates that heat transport in the Hofstadter butterfly, realized in graphene/hexagonal boron nitride moiré superlattices, is universally quantized according to topological invariants across quantum Hall states, Chern insulators, and interaction-driven symmetry-broken phases, thereby confirming the deep connection between thermal transport and topology.

Original authors: Aifei Zhang, Gibril Aissani, Quan Dong, Yong Jin, Kenji Watanabe, Takashi Taniguchi, Carles Altimiras, Patrice Roche, Jean-Marc Berroir, Emmanuel Baudin, Gwendal Fève, Gerbold Ménard, Olivier Maillet

Published 2026-09-11

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CC BY 4.0

Original authors: Aifei Zhang, Gibril Aissani, Quan Dong, Yong Jin, Kenji Watanabe, Takashi Taniguchi, Carles Altimiras, Patrice Roche, Jean-Marc Berroir, Emmanuel Baudin, Gwendal Fève, Gerbold Ménard, Olivier Maillet, François D. Parmentier

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

In the quiet, cold world of quantum physics, electrons moving through a flat, two-dimensional sheet do not behave like water flowing in a river. Instead, when a powerful magnetic field is applied, these tiny particles organize themselves into a rigid, crystalline structure of energy levels. This arrangement creates a state of matter where electricity can flow without any resistance, a phenomenon known as the quantum Hall effect. In these conditions, the flow of electric charge is not just smooth; it is quantized, meaning it moves in exact, indivisible steps, much like counting individual coins rather than pouring a continuous stream of water. For decades, scientists have known that this electrical flow is tied to a deep, mathematical property of the material called topology, which describes how the shape of the electron's energy landscape is connected.

Recently, researchers discovered that if you place these electrons on a specific type of patterned surface, the energy landscape becomes far more complex and beautiful. Instead of simple steps, the energy levels split into a fractal pattern that looks like a butterfly with infinite wings, a shape known as the Hofstadter butterfly. This pattern creates new types of insulating states where electricity is blocked in the middle of the material but flows perfectly along the edges. While scientists have long suspected that the flow of heat in these exotic states should also follow the same strict, quantized rules as electricity, no one had ever directly measured it. The question remained whether heat, which is carried by the same electrons, would obey the same topological laws or behave differently in these complex, patterned environments.

A team of physicists has now answered this question by building a delicate experiment to measure how heat moves through these fractal states. They created a device using a single layer of carbon atoms, known as graphene, sandwiched between layers of a different material called hexagonal boron nitride. Because the atomic spacing of the two materials is slightly different, they form a large, repeating pattern called a moiré superlattice. When the researchers applied a strong magnetic field to this setup, the electrons inside formed the Hofstadter butterfly pattern. To test the heat flow, they placed a tiny, metallic island in the center of the graphene sheet and heated it with a small electric current. This island was connected to the rest of the material by two narrow paths where the electrons travel along the edges.

The researchers then measured how much the temperature of this central island rose as they increased the heating power. They did this for many different states within the Hofstadter butterfly pattern, including the standard quantum Hall states and the more complex, fractal states that appear when the electrons interact strongly with one another. What they found was strikingly consistent. In every single case, the heat flowing away from the island was not a random amount but was locked into exact, discrete values. The amount of heat flow depended entirely on the topological number that defined the state of the electrons, just as the electrical flow does. Whether the state was a simple, well-known type or a complex, newly discovered one where the electrons broke their own symmetry, the heat flow remained perfectly quantized.

This result confirms that the rules governing heat transport are universal for these topological states. The experiment showed that the heat flow is determined solely by the number of channels available for the electrons to travel, a number set by the material's topology. Even when the electrons formed complex, interacting states that were previously difficult to understand, the heat still flowed in the same precise, quantized steps as electricity. The researchers also observed a subtle effect where the tiny island's ability to cool down was slightly hindered by its own electric charge, a phenomenon known as heat Coulomb blockade. This effect matched their theoretical predictions perfectly, further proving that their understanding of the system was correct.

The significance of this work lies in its demonstration that topology is the master key for both electricity and heat in these materials. Before this study, it was possible that the complex interactions between electrons in the fractal states might disrupt the flow of heat, making it different from the flow of charge. By showing that heat flow remains quantized and tied to the same topological numbers as electricity, the researchers have firmly established that these exotic states are indeed true replicas of the quantum Hall effect, just with a more intricate internal structure. This finding strengthens the link between the abstract mathematics of topology and the physical reality of how energy moves through matter, suggesting that the laws of nature governing these tiny particles are far more robust and universal than previously thought.

Technical Summary: Quantized Heat Flow in the Hofstadter Butterfly

Problem and Motivation
When electrons on a two-dimensional lattice are subjected to a strong magnetic field, they acquire a fractal energy spectrum known as the Hofstadter butterfly. This spectrum hosts a cascade of ground states characterized by non-zero topological invariants, including Quantum Hall (QH) states, Chern insulators (CIs), and symmetry-broken Chern insulators (SBCIs) arising from strong electronic interactions. While these states are understood as topological replicas of QH states with quantized electrical conductance, their thermal properties remain largely unexplored. Specifically, it has not been demonstrated whether the heat flow in these Hofstadter states is quantized according to their topological invariants, as is the case in standard integer and fractional QH effects. Establishing this quantization is crucial for validating the universality of the Streda formula for heat transport and for confirming the topological nature of these states, particularly regarding the presence or absence of heat-carrying neutral modes.

Methodology
The authors investigated heat transport in a graphene/hexagonal boron nitride (hBN) moiré superlattice, a system where the moiré potential acts as a periodic lattice allowing the observation of the Hofstadter butterfly at magnetic fields between 5 and 40 T. The experimental setup involved a central metallic island (1 µm × 3 µm) embedded within the graphene flake, connected to colder electrodes via ballistic edge channels.

The measurement geometry adapted techniques previously developed for QH systems:

Heating: A constant Joule power ( $J_Q$ ) was applied to the central island via a DC current ( $I_0$ ), raising its electron temperature to $T_c$ .
Cooling: The island exchanged heat with cold electrodes kept at a base temperature $T_0$ (20 mK) through two distinct edge paths.
Detection: The temperature rise $\Delta T_c = T_c - T_0$ was inferred from thermal noise measurements, while electrical conductance was simultaneously measured to confirm the topological state.
Chirality Control: The magnetic field direction was adjusted relative to the topological invariant ( $t$ ) to ensure the chiral edge states directed current flow correctly, allowing the island to split the current evenly between the two outgoing paths.

The study probed various states defined by the Diophantine equation $n/n_0 = t \times \phi/\phi_0 + s$ , covering integer QH states ( $s=0$ ), Chern insulators ( $s=\pm 4$ ), and symmetry-broken Chern insulators ( $s$ fractional).

Key Results
The study yielded three primary findings:

Quantized Electrical Conductance and Current Splitting: The researchers confirmed that all investigated states (QH, CI, and SBCI) exhibited quantized two-point electrical conductance $G = t \times G_0$ . Furthermore, transconductance measurements verified that the metallic island correctly split the chiral edge currents evenly between the two sides of the device, with transconductances equal to $2t \times G_0$ when the chirality was correct. This established the presence of $t$ ballistic channels.
Quantized Heat Flow with Heat Coulomb Blockade: By measuring the relationship between the dissipated Joule power and the resulting temperature difference ( $T_c^2 - T_0^2$ ), the authors observed that the heat flow was quantized. The data followed a quantized heat flow given by $(2N-1)J_e^Q$ , where $J_e^Q = \frac{\pi^2 k_B^2}{3h}T$ is the universal quantum limit of heat flow and $N$ corresponds to the number of ballistic channels ( $N = |t|$ ). This specific quantization arises because the large charging energy of the metallic island induces a Heat Coulomb Blockade (HCB) effect, which suppresses up to one ballistic channel, reducing the expected $2N$ channels to $2N-1$ .
Universality Across Topological States: Crucially, the quantization of heat flow was observed for all states regardless of their specific topological classification (QH vs. CI vs. SBCI) or the value of the parameter $s$ . The data for states with the same $t$ collapsed onto the same theoretical line, demonstrating that the heat transport is set uniquely by the topological invariant (Chern number).

The analysis accounted for the Heat Coulomb Blockade (HCB) effect, where the large charging energy of the metallic island suppresses one heat channel at low temperatures. The experimental data matched theoretical predictions including HCB corrections, particularly for states with $N=1$ , while states with $N \ge 2$ showed fully developed quantization consistent with the $(2N-1)$ channel suppression model.

Significance and Claims
The paper claims to firmly establish the universality of heat transport quantization in the Hofstadter butterfly. By demonstrating that heat flow is quantized and set by the topological invariant for QH states, Chern insulators, and symmetry-broken Chern insulators, the work confirms that these states share the same fundamental topological transport properties as standard QH states.

The authors emphasize that quantized heat flow is a robust signature of topology, as it implies a quantized electrical conductance, whereas the reverse is not always true. The results suggest that the analogy between Chern insulators and QH states holds even for thermal transport, including subtle effects like the Heat Coulomb Blockade. The paper concludes by noting that these findings invite further investigation into zero-field fractional Chern insulators, which may host non-Abelian anyonic phases, though no specific future applications or proposals are detailed beyond this scientific direction.

Technical Summary: Quantized Heat Flow in the Hofstadter Butterfly

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