🔬 mesoscale physics

1/3 Fractional and Gapless Integer Quantum Anomalous Hall States in Rhombohedral Graphene

This study reports the first observation of the fundamental 1/3 fractional quantum anomalous Hall state and a gapless extended quantum anomalous Hall phase in rhombohedral pentagonal graphene/hBN moiré superlattices, revealing a particle-hole symmetric phase diagram and enabling thermodynamic characterization of topological transitions at zero magnetic field.

Original authors: Jackson P. Butler, Tonghang Han, Andrew DiFabbio, Zach Hadjri, Emily Aitken, Kenji Watanabe, Takashi Taniguchi, Long Ju, Raymond C. Ashoori

Published 2026-06-05

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CC BY 4.0

Original authors: Jackson P. Butler, Tonghang Han, Andrew DiFabbio, Zach Hadjri, Emily Aitken, Kenji Watanabe, Takashi Taniguchi, Long Ju, Raymond C. Ashoori

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 city built on a perfectly repeating grid of tiny, hexagonal streets. In this city, electrons are the citizens trying to move around. Usually, when you push electrons through a material, they bump into things, creating resistance (like traffic jams). But sometimes, under very specific conditions, these electrons can organize themselves into a perfect, frictionless dance where they flow without any resistance at all. This is the "Quantum Anomalous Hall" effect.

For a long time, scientists had found two types of these perfect dances:

The "Whole" Dance: Where every street is filled with exactly one electron per block (a perfect integer).
The "Fractional" Dance: Where the electrons organize into groups that act like they are only a part of an electron (like 2/5 or 2/3 of a citizen).

However, there was one missing piece in the puzzle: the most famous and fundamental version of the fractional dance, the 1/3 state. It's like finding a dance where everyone moves in perfect thirds, but nobody had ever seen it happen in these special graphene materials without a giant magnetic field.

The Discovery: Finding the Missing 1/3

This paper reports that the researchers finally found this missing 1/3 dance in a special material called "Rhombohedral Graphene" (stacked layers of carbon atoms) aligned with a hexagonal boron nitride substrate.

Think of the material as a trampoline with a pattern of bumps (the "moiré superlattice"). The researchers could push the electrons away from the bumps to a "distant" area where they could move more freely. By adjusting the "push" (called a displacement field), they managed to coax the electrons into this elusive 1/3 formation.

Why is this a big deal?

The "Gold Standard": The 1/3 state is the "gold standard" of these quantum dances. Finding it here proves that the rules governing these materials are very similar to the famous rules of the "Fractional Quantum Hall" effect, even though no giant magnets were used.
Symmetry: Before this, the dance floor looked lopsided. Now that the 1/3 state is found, the whole pattern looks perfectly balanced (symmetrical) around the halfway point, just like the classic theories predicted.

The Two Different "States" of the 1/3 Dance

The researchers discovered something fascinating: the 1/3 state isn't just one thing; it can change costumes depending on how hard they push the electrons.

The "Fancy Dress" (Fractional Chern Insulator): When they push hard enough, the electrons form a topological state. This is a robust, protected state where the electrons are locked in a specific pattern that is hard to break. It has a "thermodynamic gap," which is like a deep moat protecting the castle. The researchers measured this gap and found it to be the largest and most stable of all the fractional states they saw.
The "Plain Clothes" (Charge Density Wave): If they relax the push, the electrons stop doing the fancy topological dance and just form a simple, repeating pattern (like a grid of people standing still). This is a "trivial" state, meaning it doesn't have the special topological protection.

The paper shows they can switch the electrons back and forth between these two costumes just by turning a knob (adjusting the displacement field).

The Mystery of the "Extended" State

The paper also looked at what happens when the material is almost full (1 electron per block) but not quite.

At exactly 1 (Full): The material is a perfect insulator in the middle (like a solid block of ice) but conducts electricity perfectly on the edges. This is the "Integer" state.
Just below 1 (Slightly less full): Previous experiments showed that the electricity still flowed perfectly on the edges, even though the middle wasn't full. Scientists called this the "Extended" state.

The big question was: Is the middle of this "Extended" state solid (gapped) or liquid (gapless)?

Using a special "squeezing" measurement (compressibility), the researchers found the answer:

At 1: The middle is solid (gapped).
Below 1: The middle becomes a highly squishy, compressible liquid (gapless).

The Analogy: Imagine a highway. At the "Integer" point, the highway is a solid wall of traffic (no movement in the middle, only on the shoulders). As soon as you remove a few cars (doping), the middle of the highway turns into a soft, squishy marshmallow that can be compressed easily, yet the cars on the shoulders (the edges) keep driving perfectly without crashing. This is a rare and surprising combination: a "gapless" middle that still supports a "perfect" edge.

Summary

In simple terms, this paper:

Found the missing 1/3 dance in a graphene material, proving these materials follow the same deep rules as the classic magnetic-field experiments.
Measured the energy cost (the gap) to break these dances, finding the 1/3 state is the most robust.
Solved a mystery about the "Extended" state, showing it is a strange hybrid where the middle is squishy and gapless, but the edges remain perfectly conductive.

This work helps scientists understand how electrons can organize themselves into these complex, frictionless patterns, which is a crucial step toward understanding the fundamental laws of quantum matter.

Technical Summary: 1/3 Fractional and Gapless Integer Quantum Anomalous Hall States in Rhombohedral Graphene

Problem Statement
The fractional quantum anomalous Hall (FQAH) effect has recently been observed in moiré superlattices, specifically twisted bilayer MoTe $_2$ (tMoTe $_2$ ) and rhombohedral $n$ -layer graphene aligned to hexagonal boron nitride (R $_n$ G/hBN). While several fractional states within the Jain sequence have been identified in these systems, the 1/3 filling factor state—the most fundamental and robust state in conventional high-field fractional quantum Hall (FQH) systems described by the Laughlin wavefunction—had remained missing in zero-field FQAH systems. Its absence raised questions regarding the mechanism of FQAH in R $_n$ G/hBN, particularly whether it relates to the conventional Landau level picture. Furthermore, the bulk electronic properties of the extended quantum anomalous Hall (EQAH) and re-entrant integer quantum anomalous Hall (RIQAH) phases, which exhibit quantized Hall resistance over ranges of filling factors, were poorly understood due to a lack of thermodynamic characterization beyond transport measurements.

Methodology
The authors investigated R $_5$ G/hBN devices in the "moiré-distant" regime, where electrons are displaced away from the graphene/hBN interface via a tunable displacement field ( $D$ ). The study employed a combination of:

Penetration Capacitance Measurements: Used to probe the bulk thermodynamic density of states and compressibility. This technique identifies incompressible phases (gapped states) and quantifies thermodynamic gaps ( $\Delta\mu$ ) by measuring the step in chemical potential as electron density is swept.
Transport Measurements: Conducted on two devices (D1 and D2) to measure longitudinal resistance ( $R_{xx}$ ) and Hall resistance ( $R_{xy}$ ).
Magnetic Field Dependence: Landau fan measurements were performed to extract Chern numbers ( $C$ ) using the Středa formula ( $C = \phi_0 \partial n / \partial B$ ), distinguishing topological states from trivial correlated insulators.

Key Contributions and Results

Discovery of the 1/3 FQAH State: The primary result is the observation of a robust FQAH state at the moiré filling factor $\nu = 1/3$ in R $_5$ G/hBN. This state was previously unobserved in either tMoTe $_2$ or R $_n$ G/hBN.
- Thermodynamic Gap: The 1/3 state exhibits the largest thermodynamic gap among the fractional states in the system, measured at $\Delta\mu \approx 0.6$ meV ( $\sim 7$ K).
- Topological Transition: By tuning the displacement field, the authors observed a phase transition from the 1/3 fractional Chern insulator (FCI) to a trivial charge density wave (CDW) state at lower fields.
- Verification: Transport measurements on a second device (D2) confirmed the state's topological nature, showing quantized Hall resistance ( $R_{xy} \approx 3h/e^2$ ) and vanishing longitudinal resistance. Landau fan analysis yielded a Chern number $C \approx 1/3$ .
Particle-Hole Symmetry: With the inclusion of the 1/3 state, the FQAH states in R $_5$ G/hBN exhibit a striking particle-hole symmetry about the half-filling of the moiré band. The sequence of observed fractions (1/3, 2/5, 3/5, 2/3) and their gap sizes closely resemble the behavior of FQH states in the lowest Landau level (LLL), despite the fundamentally different physical origins of the two systems.
Characterization of the Extended Quantum Anomalous Hall (EQAH) State: The study provides the first direct thermodynamic characterization of the EQAH region ( $\nu < 1$ ).
- Gapless Nature: While transport measurements show no change in resistance between the integer quantum anomalous Hall (IQAH) state at $\nu=1$ and the EQAH region, compressibility measurements reveal a distinct transition.
- Phase Transition: At $\nu=1$ , the system is an incompressible IQAH state with a large thermodynamic gap. As density decreases below $\nu=1$ , the system transitions to a highly compressible state with strong negative compressibility.
- Implication: This indicates that the EQAH state lacks a bulk thermodynamic gap. The retention of quantized Hall resistance and vanishing longitudinal resistance in this gapless, compressible bulk suggests a picture where strong correlations localize bulk charge carriers (potentially forming a compressible crystalline state), thereby preventing back-scattering of the ballistic edge modes.
Symmetry-Broken Chern Insulators (SBCI): The authors identified additional states at non-zero magnetic fields, including a $C=1$ SBCI state emerging from $\nu=1/3$ and a correlated insulator at $\nu=1/2$ . Notably, the $\nu=1/2$ state shows a thermodynamic gap, distinguishing it as a zero-field SBCI rather than a gapless EQAH state.

Significance
The paper claims that the direct thermodynamic characterization of the R $_5$ G/hBN phase diagram, particularly the discovery of the 1/3 FQAH state, significantly advances the understanding of FQAH mechanisms. The observation that these zero-field states mimic the hierarchy and symmetry of conventional LLL FQH states suggests a deep connection between the two regimes, despite the lack of an external magnetic field and the different microscopic origins (e.g., moiré potential vs. Landau quantization).

Furthermore, the identification of a gapless EQAH state that retains topological edge transport challenges the conventional view that quantized Hall resistance requires a bulk gap. This finding paves the way for engineering anyon braiding and non-Abelian quasiparticles at zero magnetic field, although the authors note that the underlying mechanism for the stability of the 1/3 state in R $_5$ G/hBN (which theory suggests should be unstable to CDW states with minimal band mixing) requires further theoretical investigation.

The Discovery: Finding the Missing 1/3

The Two Different "States" of the 1/3 Dance

The Mystery of the "Extended" State

Summary

Technical Summary: 1/3 Fractional and Gapless Integer Quantum Anomalous Hall States in Rhombohedral Graphene

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