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

Scientists have discovered a new, very stable quantum state in special layers of graphene. This state, called the 1/3 fractional quantum anomalous Hall (FQAH) state, behaves much like similar states found in high magnetic fields. However, it works here without any magnetic field at all.

The discovery addresses a long-standing mystery in condensed matter physics. In conventional systems, the fractional quantum Hall effect—a phenomenon where electrons form complex, collective states—requires massive external magnetic fields. Researchers have recently sought to replicate these exotic states at zero magnetic field using "moiré superlattices." These are artificial patterns created by stacking atomic layers at a slight twist. While some fractional states had been spotted in these materials, the 1/3 state had remained missing.

The authors report that they have finally observed this 1/3 state in rhombohedral graphene aligned to hexagonal boron nitride (R5G/hBN). This discovery changes how we view the relationship between magnetism and topology (the study of shapes that remain unchanged under continuous deformation).

The search for the missing 1/3 state

Until now, the search for the FQAH effect has involved incomplete sequences. In existing FQAH systems, researchers have identified several states belonging to the "Jain sequence." This is a mathematical progression describing how electrons organize in high magnetic fields. However, the 1/3 state was absent in the leading candidate materials.

This absence created a theoretical bottleneck. Without the 1/3 state, scientists could not definitively link the FQAH effect to the high-field quantum Hall effect. Furthermore, previous attempts to measure these states using standard transport—measuring how voltage and current flow—were often blocked. High resistance in the surrounding environment frequently obscured the signals. This left the bulk properties of these quantum phases a mystery.

Probing the bulk with penetration capacitance

To bypass the limits of traditional transport, the researchers used penetration capacitance (also known as quantum capacitance) measurements. Transport measurements look at how charges move through a sample. In contrast, penetration capacitance probes the "compressibility" of the electron liquid. Think of it like testing a sponge. Instead of watching water flow through it, you measure how much pressure is required to squeeze more water into the pores.

The methodology follows a precise sequence:

1. Thermodynamic Mapping: The team uses an AC excitation on a top gate to measure penetration capacitance ($C_P$). This identifies "incompressible" phases. These are states where electron density resists change, signifying a stable energy gap.
2. Topological Verification: To prove these states are topological, the authors perform "Landau fan" measurements. By applying varying magnetic fields, they track how state density shifts. They use the Středa formula to calculate the "Chern number." This is a topological integer that acts as a fingerprint for the state's identity.
3. Dual-Mode Validation: They combine this with transport measurements on specialized devices. This confirms that thermodynamic gaps correspond to dissipationless edge currents (currents that flow without losing energy).

The results, mapped in, reveal a landscape of phases.

These range from a Wigner solid (an electronic crystal) to the new FQAH states.

Symmetry and a record-breaking gap

The most significant result reported by the authors is the 1/3 fractional Chern insulator. Through their Landau fan analysis, the study finds a Chern number of $0.329 \pm 0.001$ at the 1/3 filling factor.

This confirms its topological nature.

Crucially, the researchers report that the 1/3 state possesses a thermodynamic gap ($\Delta_\mu$) of approximately 0.6 meV. This is roughly 7 K. The authors note this is the largest reported gap for any graphene fractional Chern insulator. This large gap is vital. It provides the thermal stability needed for future tasks like anyon braiding (moving quasiparticles to perform logic operations).

Beyond the gap size, the study finds a "surprising level of particle-hole symmetry" about the half-filling of the moiré band .

This symmetry means the physics for "under-filled" states is nearly identical to "over-filled" states. This behavior closely mimics the conventional fractional quantum Hall effect seen in high magnetic fields.

Distinguishing the gapless from the gapped

The paper also examines the "extended quantum anomalous Hall" (EQAH) regime. Previously, transport measurements suggested this region showed quantized Hall resistance. Many assumed it was a continuous topological phase.

However, the authors' compressibility measurements reveal a contradiction. While transport shows no change when moving from the integer state ($\nu = 1$) into the EQAH region, the bulk thermodynamics differ .

The study finds the $\nu = 1$ state is a gapped, incompressible insulator. Conversely, the EQAH regime is a gapless, highly compressible state.

This means that in the EQAH region, the edges remain conductive. Yet, the bulk of the material is "soft" and compressible. The authors suggest that strong correlations localize charge carriers in the bulk. This prevents them from scattering, even without a formal energy gap.

Verdict: A roadmap for topological engineering

These findings represent a step toward controlling topological matter without massive magnets. By proving the 1/3 state exists in rhombohedral graphene, the authors have validated a key theoretical prediction. They have also provided a high-stability platform for studying fractional physics.

Is this ready for production? Not yet. The authors note that these gaps are still smaller than those in gallium arsenide or $\text{MoTe}_2$. Also, competing "trivial" states like the charge density wave (CDW) exist. Precise control over the displacement field is required to stay within the target phase. However, for researchers aiming to engineer quasiparticles for quantum computing, this paper provides a promising blueprint.

Figures from the paper