🔬 mesoscale physics

Quantized heat flow in moiré chern bands of bilayer graphene

Using Johnson-noise thermometry in a bilayer graphene-hexagonal boron nitride moiré superlattice, researchers demonstrate that the thermal conductance of various topological states is universally quantized in units determined solely by their Chern number, establishing thermal transport as a robust probe for moiré topological matter.

Original authors: Santanu Samai, Debangan Sarkar, Abhijit Halder, T. Taniguchi, K. Watanabe, Subroto Mukerjee, Saurabh Kumar Srivastav, Anindya Das

Published 2026-07-22

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Santanu Samai, Debangan Sarkar, Abhijit Halder, T. Taniguchi, K. Watanabe, Subroto Mukerjee, Saurabh Kumar Srivastav, Anindya Das

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 electricity doesn't just flow like water in a pipe, but dances to the rhythm of invisible, repeating patterns. This is the realm of condensed matter physics, where scientists study how electrons behave in special materials. Usually, when you push electrons through a wire, they bump into atoms and generate heat, making the wire warm. But in certain exotic materials, under strong magnetic fields, electrons can organize themselves into "topological" states. Think of these states as a super-highway where electrons are forced to travel in one direction only, like cars on a one-way street that never get stuck in traffic.

For decades, scientists have known that these highways conduct electricity perfectly. But a big mystery remained: do they also conduct heat perfectly? In the quantum world, heat is carried by the same electrons, but measuring it is like trying to weigh a ghost while it's running a marathon. If the heat flow follows the same strict rules as the electricity, it would prove that the "topology" (the shape of the electron's path) is the only thing that matters, regardless of how the highway was built. This question is crucial because if heat behaves differently, it might mean there are hidden, invisible particles or strange interactions we haven't discovered yet.

Now, enter a team of researchers who decided to settle this mystery using a material that looks like a cosmic moiré pattern. They took two layers of graphene (a super-thin sheet of carbon atoms) and sandwiched them with hexagonal boron nitride (hBN), aligning them so precisely that the atoms created a giant, repeating grid of ripples. This "moiré superlattice" acts like a giant playground for electrons, creating a fractal energy landscape known as the "Hofstadter butterfly." In this landscape, electrons get trapped in specific lanes defined by their "Chern number," a fancy integer that counts how many times the electron's path twists.

The researchers, led by Santanu Samai and Anindya Das, set out to measure the heat flow through these lanes. They built a tiny device with a "floating reservoir"—a small island in the middle of the electron highway. They heated up this island by pushing a current through it and then measured how much heat leaked out to the cold edges. They used a clever trick called "Johnson-noise thermometry," which listens to the tiny electrical whispers (noise) generated by the heat to figure out the temperature, rather than using a traditional thermometer that would be too clumsy for such a small scale.

What they found was a beautiful confirmation of nature's symmetry. They measured the heat flow for different types of electron highways: the standard "Quantum Hall" lanes, the "Chern Insulator" lanes created by the moiré pattern, and even the "Symmetry-Broken" lanes where electrons interacted with each other to form new patterns. In every single case, the heat flow was perfectly quantized. It came out in exact steps, determined solely by the Chern number (the twist count of the path).

Specifically, they found that the thermal conductance ( $G_Q$ ) was exactly equal to $2t\kappa_0T$ , where $t$ is the Chern number, $\kappa_0$ is a fundamental constant, and $T$ is the temperature. For example, a state with a Chern number of -2 carried exactly $4.12\kappa_0T$ of heat, and a state with -4 carried $8.08\kappa_0T$ . This was a big deal because a previous experiment on a similar material had suggested that heat flow might be "blocked" or reduced by a phenomenon called "heat Coulomb blockade." However, in this new experiment, thanks to the very thin insulating layer (less than 7–8 nm) between the floating island and the gate, that blocking effect vanished. The heat flowed freely, just as the theory predicted.

The team also looked at states where the electrons were interacting strongly, creating "Symmetry-Broken Chern Insulators" with fractional numbers. Even here, the heat flow followed the rules, matching the integer part of the twist count. This tells us that whether the electron highway is built by a simple magnetic field, a complex crystal pattern, or by electrons pushing and pulling on each other, the way they carry heat is universal. It depends only on the topological shape of the path, not on the microscopic details of the road.

In short, this paper shows that in the quantum world of moiré superlattices, heat is just as disciplined as electricity. The electrons carry thermal energy in perfect, quantized packets, proving that the "bulk-edge correspondence"—the idea that the inside of the material dictates the behavior of its edges—holds true even for heat. This discovery doesn't just solve a puzzle; it gives scientists a new, reliable tool to explore even stranger phases of matter, like "fractional Chern insulators," where the rules might get even more exotic. By listening to the heat, we are learning to read the secret topological code of the universe.

Technical Summary: Quantized Heat Flow in Moiré Chern Bands of Bilayer Graphene

Problem Statement
While the topological nature of electronic states in moiré superlattices—specifically the fractal Hofstadter spectrum arising from the interplay of periodic potentials and magnetic fields—has been well-established through electrical transport, the universality of their heat transport remains unexplored. Previous electrical measurements have identified Quantum Hall (QH), Chern Insulator (CI), and interaction-driven Symmetry-Broken Chern Insulator (SBCI) states via quantized electrical conductance. However, it is unknown whether the thermal conductance of these states follows the same universal quantization rules dictated by their topological invariants (Chern numbers). Furthermore, recent experiments on monolayer graphene-hBN superlattices observed a reduction in thermal conductance attributed to the heat Coulomb blockade (HCB) effect, leaving the intrinsic quantized thermal conductance ( $G_Q = t\kappa_0T$ ) unverified in moiré Chern bands.

Methodology
The authors investigated bilayer graphene (BLG) encapsulated in hexagonal boron nitride (hBN), forming a moiré superlattice with a wavelength of $\sim14$ nm. The device features a graphite back gate and a central metallic floating reservoir (FR) fabricated via 1D edge contacts, with a separation of less than 7–8 nm between the reservoir and the gate.

The study employed Johnson-noise thermometry to measure thermal conductance. The experimental setup involved:

Electrical Transport: Identifying topological states (QH, CI, SBCI) by mapping the quasi-longitudinal resistance ( $R_{xx}$ ) and Hall resistance ( $R_{xy}$ ) as a function of carrier density ( $n/n_0$ ) and magnetic flux ( $\phi/\phi_0$ ). States were classified using the Diophantine equation $n/n_0 = t(\phi/\phi_0) + s$ , where $t$ is the Chern number.
Thermal Measurement: A source current ( $I_{SD}$ ) was injected into the FR, where it split into outgoing chiral edge channels. The power dissipated ( $J_Q$ ) heated the electrons in the FR to a steady-state temperature ( $T_M$ ).
Noise Thermometry: The electron temperature $T_M$ was determined by measuring the excess thermal noise ( $S_I$ ) along the outgoing edge channels.
Extraction: Thermal conductance ( $G_Q$ ) was extracted by plotting the dissipated power $J_Q$ against $T_M^2 - T_0^2$ (where $T_0$ is the bath temperature). The slope of this linear relationship yields $G_Q$ .

Key Contributions and Results
The paper reports the first experimental demonstration of quantized thermal conductance in moiré Chern bands, free from the heat Coulomb blockade effect that plagued previous monolayer studies.

Suppression of Heat Coulomb Blockade (HCB): Unlike previous monolayer results, the authors observed no reduction in thermal conductance due to HCB. This is attributed to the large capacitance between the floating reservoir and the graphite back gate, enabled by the thin hBN dielectric in the bilayer device.
Quantization in QH States: For conventional QH states (e.g., $t = \pm2, \pm4$ ), the measured thermal conductance was $G_Q \approx 2|t|\kappa_0T$ . The factor of 2 arises from the device geometry where the floating reservoir splits the current into two outgoing channels.
Quantization in Chern Insulators: For single-particle Chern insulator states (e.g., $(t, s) = (-2, 2), (-3, -2), (-4, -2)$ ), the thermal conductance was strictly quantized according to the Chern number $t$ , yielding $G_Q \approx 2|t|\kappa_0T$ .
Quantization in Interaction-Driven States: The study extended these findings to interaction-driven Symmetry-Broken Chern Insulators (SBCI), such as the state $(t, s) = (-4, 7/6)$ . These states also exhibited quantized thermal conductance consistent with their integer Chern number $t$ .
Universality: Across all measured states—whether originating from conventional QH physics, single-particle Chern bands, or strong electron-electron interactions—the thermal conductance was determined solely by the Chern number $t$ , independent of the microscopic origin of the topological state.

Significance
The authors claim that this work establishes thermal conductance as a stringent and universal probe of moiré topological matter. By directly revealing universal topological heat transport in Hofstadter bands, the study confirms the bulk-edge correspondence for heat transport beyond conventional Quantum Hall systems. The ability to measure intrinsic quantized thermal conductance without HCB interference provides a new pathway to investigate more exotic phases, including fractional Chern insulators and non-Abelian states, in moiré quantum materials. The results strengthen the fundamental connection between thermal response and topological invariants in two-dimensional materials.

More like this