🔬 mesoscale physics

Non-universal localization transition in the quantum Hall effect probed through broken-symmetry states of graphene

By systematically measuring localization lengths in broken-symmetry quantum Hall states of graphene, this study reveals significant non-universality and deviations from standard scaling, which are successfully explained by a model involving the co-existence of localized states from two successive sub-Landau levels.

Original authors: Aifei Zhang, Torsten Röper, Manjari Garg, Kenji Watanabe, Takashi Taniguchi, Carles Altimiras, Patrice Roche, Erwann Bocquillon, Olivier Maillet, François D. Parmentier

Published 2026-09-11

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CC BY 4.0

Original authors: Aifei Zhang, Torsten Röper, Manjari Garg, Kenji Watanabe, Takashi Taniguchi, Carles Altimiras, Patrice Roche, Erwann Bocquillon, 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 coldest corners of the universe, where thermal jitters are silenced, electrons in certain materials can behave in a way that defies ordinary intuition. When a strong magnetic field is applied to a thin sheet of conducting material, the electrons are forced into a rigid, organized state known as the quantum Hall effect. In this state, the material's interior becomes an insulator, blocking the flow of electricity, while the edges become perfect highways where current flows without any resistance. This phenomenon is so precise that it serves as a global standard for electrical resistance. However, the transition between these insulating and conducting states is not a simple switch; it is a complex journey where electrons shift from being trapped in place to moving freely. For decades, physicists have believed that this shift follows a universal rule, a mathematical pattern that should look the same regardless of the specific material or the type of disorder present within it. This idea of universality suggests that nature has a single, elegant script for how electrons localize, or get stuck, in these extreme conditions.

A team of researchers in France, Germany, and Japan has now challenged this long-held belief by looking closely at a special type of material: graphene. Graphene is a single layer of carbon atoms arranged in a honeycomb lattice, and when placed in a magnetic field, it reveals a rich landscape of different quantum states. The researchers used a device shaped like a donut, known as a Corbino geometry, to measure how electricity flows through the center of the graphene sheet, away from the edges. By carefully adjusting the number of electrons and the temperature, they mapped out how the electrons became trapped in the disordered parts of the material. They found that the size of these trapped regions, known as the localization length, did not follow the single, universal rule that had been predicted. Instead, the size of the trapped regions changed dramatically depending on the specific type of quantum state the electrons were in. In some states, the electrons were confined to tiny pockets just twenty nanometers across, while in others, they remained spread out over distances as large as one micrometer. This variation, which can be a factor of ten or more, suggests that the simple, universal picture of electron localization is incomplete.

The study focused on the different ways the electrons in graphene can organize themselves. In a strong magnetic field, the energy levels of the electrons split into distinct groups. Some of these groups are separated by large energy gaps, while others are split by smaller gaps due to the breaking of symmetries in the electron's spin or valley properties. The researchers measured the localization length for each of these different states. They discovered that when the energy gap separating the states was large, the electrons localized very tightly, fitting the predictions of the old universal theory. However, when the energy gap was small, the electrons behaved differently. In these cases, the trapped states from one energy level began to overlap with the trapped states from the next level. This overlap meant that the electrons were not confined to a single, isolated pocket but could hop between the overlapping regions of two different energy levels. This mixing made the electrons appear much less localized than expected, leading to a much larger localization length and a different mathematical pattern for how they transitioned between states.

To understand this, imagine the energy levels as a series of floors in a building, and the disorder in the material as a layer of fog that blurs the edges of each floor. In the case of large gaps, the fog on one floor does not reach the next floor, so an electron trapped on one floor stays there. But when the gap is small, the fog from the floor below rises high enough to touch the floor above. An electron trapped in this overlapping fog can drift between the two floors, effectively making its "home" much larger. The researchers found that this overlap explains why the universal scaling law failed for the smaller gaps. The data showed that for the states with the smallest gaps, the electrons were not following the standard rule of a single, isolated energy level. Instead, they were navigating a complex landscape where two levels were intertwined. This finding suggests that the previous observations of non-universal behavior in other experiments were likely caused by this same overlap of energy levels, rather than by some unknown flaw in the theory itself.

The implications of this work extend beyond just graphene. It provides a clearer picture of how disorder affects quantum states in general. The researchers showed that the apparent failure of universal scaling in many past experiments could be resolved by recognizing that the energy levels were not isolated but were overlapping. This means that the "localization length" measured in such experiments is not always the size of a single trapped state, but rather an effective size that includes the influence of neighboring states. The study confirms that while the universal scaling law holds true for well-separated energy levels, it breaks down when those levels get too close together. By systematically measuring these effects in graphene, the team has provided a simple and intuitive explanation for a phenomenon that has puzzled physicists for years. They have shown that the key to understanding these transitions lies not in a single, rigid rule, but in the delicate interplay between the size of the energy gaps and the amount of disorder in the material. This work does not discard the old theory but refines it, showing exactly where and why it applies, and where the complexity of overlapping states takes over.

Technical Summary: Non-universal Localization Transition in the Quantum Hall Effect Probed through Broken-Symmetry States of Graphene

Problem Statement
The Quantum Hall Effect (QHE) is characterized by the emergence of an insulating bulk and quantized edge channels, stabilized by disorder which creates localized bulk states. The transition between these localized states and extended states (where the Fermi level crosses the center of a Landau level) is theoretically predicted to follow universal scaling laws derived from percolation theory. Specifically, the localization length $\xi$ is expected to diverge as a power law of the distance from the critical filling factor $\nu_c$ : $\xi \propto |\nu - \nu_c|^{-\gamma}$ , with a universal exponent $\gamma \approx 2.35$ . However, experimental observations have frequently deviated from this universality. Reported scaling exponents vary, and minimum localization lengths are sometimes observed to be orders of magnitude larger than theoretical predictions. These discrepancies are particularly pronounced in graphene, where previous measurements in spin/valley degenerate Landau levels (LLs) reported observations corresponding to $\gamma$ markedly higher than 2.35, and non-universality persists even in extremely clean devices.

Methodology
To investigate the origin of these non-universal behaviors, the authors utilized monolayer graphene (MLG) flakes encapsulated in hexagonal boron nitride (hBN) with a graphite back-gate. The devices were fabricated in a Corbino geometry, featuring a central contact and edge contacts, to exclusively probe bulk transport while eliminating edge channel contributions.

The study systematically measured the bulk conductance ( $G_{bulk}$ ) as a function of the filling factor ( $\nu$ ) and temperature ( $T$ ) over a range from 50 mK to 55 K at a magnetic field of $B = 14$ T. The high magnetic field lifts the fourfold spin and valley degeneracy of graphene's Landau levels, creating distinct energy gaps: large cyclotron gaps (at $\nu = \pm 2, \pm 6$ ) and smaller symmetry-broken sub-gaps (spin-polarized at $\nu = \pm 4, \pm 8$ and valley-polarized at $\nu = 3, 5, 7$ ).

The temperature dependence of $G_{bulk}$ was analyzed by fitting the data to a model combining two conduction mechanisms:

Efros-Shklovskii Variable Range Hopping (ES-VRH): Dominant at low temperatures, describing transport through localized states.
Arrhenius Thermal Activation: Dominant at higher temperatures.

From the ES-VRH fits, the authors extracted the characteristic hopping temperature $T_0$ , which is inversely proportional to the localization length $\xi$ via the relation $\xi = \frac{Ce^2}{4\pi\epsilon k_B T_0}$ . This allowed for the direct extraction of $\xi$ across different filling factors and gap types. The authors then tested the universality of the scaling by rescaling the conductance data using the predicted power law and examining the collapse of data onto a single exponential curve.

Key Results
The study reveals a strong dependence of the localization length and scaling behavior on the nature of the energy gap:

Localization Length ( $\xi$ ): There are significant quantitative differences in the minimum localization length ( $\xi_{min}$ ) depending on the gap type.
- Cyclotron Gaps: $\xi$ reduces drastically from sample-size scales (micrometers) at the plateau edges to approximately 100 nm (or even 20 nm when fitting only the low-temperature VRH regime) at the plateau center.
- Spin-Polarized Gaps: $\xi_{min} \approx 500$ nm.
- Valley-Polarized Gaps: $\xi_{min} \approx 1$ $\mu$ m.
- N=0 Landau Level: Despite being symmetry-broken, the partially filled $N=0$ LL exhibits a small $\xi_{min} \approx 100$ nm, attributed to its unique wavefunction properties and reduced disorder broadening.
Scaling Exponents ( $\gamma$ ):
- Transitions involving cyclotron gaps (e.g., $\nu=2 \leftrightarrow 3$ ) exhibit data collapse consistent with the universal exponent $\gamma \approx 2.35$ .
- Transitions involving symmetry-broken gaps (spin and valley) fail to collapse with the universal exponent. Instead, they yield significantly smaller exponents, closer to half the universal value ( $\gamma \approx 1.18$ ).
Scaling Failure: The universal scaling law, which predicts a collapse of rescaled conductance data, holds well for cyclotron gaps but fails for symmetry-broken gaps.

Proposed Mechanism and Significance
The authors propose a physical picture based on the overlap of localized states from successive sub-Landau levels to explain the observed non-universality.

Non-Overlapping Regime (Cyclotron Gaps): For large gaps ( $\Delta \gg V_0$ , where $V_0$ is the disorder amplitude), the disorder-broadened sub-LLs are energetically separated. The Fermi level jumps from the top of one sub-LL to the bottom of the next without intersecting localized states of both simultaneously. This preserves the conditions for universal scaling.
Overlapping Regime (Symmetry-Broken Gaps): For smaller gaps ( $\Delta \sim V_0$ ), such as valley-polarized states, the disorder broadening causes significant overlap between the localized states of adjacent sub-LLs ( $N$ and $N+1$ ). As the filling factor changes, the Fermi level intersects localized states from both sub-LLs simultaneously. This co-existence leads to a competition in hopping contributions, preventing $\xi$ from reaching the theoretical minimum and flattening the dependence of $\xi$ on $\nu$ . This results in a reduced scaling exponent and a larger effective minimum localization length.

Significance
The paper claims that this simple picture of overlapping localized states from successive sub-LLs fully captures the observed non-universality, including the discrepancies in scaling exponents and minimum localization lengths. It suggests that the "non-universal" behavior is not a failure of the underlying theory but a consequence of the specific energy scales (gap size vs. disorder broadening) in the system. The authors emphasize that the extracted localization length in overlapping regimes should be interpreted as an effective hopping length rather than the intrinsic localization length of an isolated sub-LL. This framework reconciles previous experimental discrepancies, particularly those in spin/valley degenerate systems (where previous reports showed higher exponents), by highlighting the critical role of the ratio between the energy gap and disorder broadening.

Technical Summary: Non-universal Localization Transition in the Quantum Hall Effect Probed through Broken-Symmetry States of Graphene

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