A neural network, trained without pre-programmed expectations, has revealed a degenerate ground-state manifold whose entanglement reveals anyons with a quantum dimension of $\sqrt{2}$ , a key signature of Ising-type non-Abelian order. Researchers at the Massachusetts Institute of Technology developed the momentum-resolved neural-network wave function to analyze the behavior of electrons at filling factor ν = 3/2, a specific condition for observing these quantum phenomena.

The work presents the first complete solution to the non-Abelian quantum Hall problem including all electrons and Landau levels, and further demonstrates that Landau level mixing favors the anti-Pfaffian state over other candidates. These results showcase the power of artificial intelligence in characterizing anyons from first principles, relevant to recent experiments in the field.

AI Identifies Anyons at ν = 3/2 Filling Factor

The AI’s success stems from a momentum-resolved neural-network wave function, optimized directly against the microscopic Hamiltonian, allowing access to the degenerate ground states where these anyons reside. The filling factor ν = 3/2 represents a state where the lowest Landau level is completely filled and the first excited Landau level is half-filled, a configuration known to be conducive to the formation of these unusual particles. The structure factor, a measure of order within the system, was analyzed to confirm the presence of these anyons and their unique properties.

This resolves a long-standing debate within the field regarding the precise nature of the anyonic state at this filling factor. “Non-Abelian quantum Hall states at strong Landau level mixing: finding anyons with first-principles AI,” encapsulates the core achievement of this work, as the AI’s approach offers a new pathway to characterizing these elusive particles. The team, comprised of Pierre-Antoine Graham, Ahmed Abouelkomsan and Liang Fu, all affiliated at the Massachusetts Institute of Technology, used a neural-network variational method to optimize the network parameters.

By directly computing the overlap with trial wave functions, we further show that Landau level mixing favors the anti-Pfaffian over other candidate states. This work demonstrates the power of neural-network wave functions in identifying and characterizing non-Abelian anyons from first principles, even in regimes of strong Landau level mixing, conditions increasingly relevant to recent experimental observations.

Landau Level Mixing Favors Anti-Pfaffian State

This finding, achieved through a neural network approach, moves beyond theoretical predictions by directly addressing the complexities of Landau level mixing, a critical factor often omitted in prior studies. The analysis demonstrates that, as the strength of this mixing increases, the anti-Pfaffian state emerges as the most likely topological order, surpassing the Pfaffian and particle-hole Pfaffian candidates.

This is not merely a theoretical prediction; observations in wide and double GaAs quantum wells have already revealed states at ν = 1/2 exhibiting energy gaps which reached about 6 K, the largest reported for any even-denominator state. Further observations in GaAs hole systems, ZnO, AlAs, bilayer graphene, and monolayer WSe2 also demonstrate fractional quantum Hall states appearing with significant Landau level mixing, establishing a broad experimental context for these findings.

The neural network’s ability to accurately model these conditions is a key advancement, as it accounts for the depletion of both the lowest and first Landau levels, transferring weight to higher energy levels. As the degree of Landau level mixing decreases, the overlaps between the Pfaffian and anti-Pfaffian states converge, aligning with expectations of restored particle-hole symmetry. The overlap with the particle-hole Pfaffian remains minimal across all levels of mixing, effectively ruling it out as a viable ground state candidate.

This work distinguishes itself from previous neural-network methods by eschewing pre-programmed physical assumptions about the expected ground state. By starting solely from the microscopic Hamiltonian of interacting electrons in a magnetic field, the approach inherently captures topological order, Landau level mixing and the competition between candidate states within a unified framework. The inclusion of virtual inter-Landau-level excitations, through “dressing” the trial states, further emphasises the importance of accurately modeling Landau level mixing, a feature fully integrated into the neural-network’s variational wave function.

Fractional Quantum Hall States Across Multiple Platforms

These diverse systems share a common characteristic: significant Landau level mixing, a phenomenon where electrons occupy multiple energy levels simultaneously, complicating accurate simulations. Previous numerical calculations often truncated the computational space or treated this mixing as a minor effect, limiting their ability to definitively identify topological order. This comprehensive treatment is important because Landau level mixing increasingly influences the overlap between the neural-network generated states and candidate wave functions as the degree of mixing, represented by the parameter κ, increases.

Remarkably, at larger values of κ, the overlap with the anti-Pfaffian state not only becomes comparable to that of the lowest Landau level-projected Coulomb ground state but, in some instances, exceeds it. This suggests that Landau level mixing actively favors the anti-Pfaffian topological order over other potential states.

Challenges in Accurately Modeling Strong Landau Level Mixing

The redistribution of electrons among Landau levels, a consequence of strong Landau level mixing, presents a significant challenge to accurately modeling non-Abelian quantum Hall states. Beyond simply shifting electron occupation, this mixing fundamentally alters the nature of the ground state, as observed in the current work. This unexpected depletion in the first Landau level suggests a more complex interplay between electron scattering than previously understood.

The study demonstrates that even as electrons scatter from the lowest Landau level, they do not simply populate the next available level. Instead, the distribution extends to even higher energy states. This redistribution is not a finite-size effect, as the occupations for the two system sizes examined nearly coincide, indicating a robust phenomenon intrinsic to the strong mixing regime. The implications extend to the stability of non-Abelian states, as the altered electron distribution directly impacts the topological order.

This allows for a comprehensive study of the system’s behavior across a wide range of mixing strengths, from weak to ultra-strong. The researchers state that the network can still identify the underlying topological order. The neural network’s success in modeling strong Landau level mixing is particularly noteworthy given the limitations of previous approaches. Traditional methods struggle to accurately treat the fractional quantum Hall problem under these conditions, making this work a significant step forward in the field.

An upcoming preprint will detail related results, further expanding on the findings presented. The ability to accurately model these complex interactions is essential for realizing the potential of non-Abelian anyons in fault-tolerant topological quantum computing, offering a pathway to more robust and reliable quantum information processing.

Coulomb and Cyclotron Energies Define Mixing Parameter κ

The ratio defining the strength of Landau level mixing, denoted as κ and calculated from Coulomb and cyclotron energies, proves critical in determining the observed topological order within fractional quantum Hall states. Specifically, the calculations reveal that when κ increases, the ability to distinguish between different anyonic states, particles theorized to be essential for fault-tolerant quantum computing, becomes increasingly reliant on accurately accounting for this mixing parameter.

Researchers found that in systems where the Coulomb energy is comparable to or larger than the cyclotron energy, traditional perturbative methods for analyzing Landau level mixing become inadequate, necessitating a more comprehensive approach. A momentum-resolved neural-network variational wave function was developed to address this challenge, allowing for direct access to the topologically degenerate ground states without pre-programmed expectations about the outcome.

This method optimizes the wave function based solely on the microscopic Hamiltonian, an important step in accurately modeling the complex interactions between electrons. The resulting spectra demonstrated significantly lower energies compared to those obtained through exact diagonalization, a benchmark calculation, confirming the network’s ability to find more stable and accurate ground states. Quantifying the overlap with trial wave functions further illuminates the preference for certain topological orders.

The overlap was computed by directly comparing the neural-network generated wave functions with trial states, providing a measure of how well the network’s solution aligns with known theoretical models. Analysis revealed that the overlap with the anti-Pfaffian state remains consistently higher than with other candidate states, including the Pfaffian and particle-hole Pfaffian, even as the Landau level mixing parameter, κ, increases up to a value of 5.

This was demonstrated using both bare and “dressed” trial states, the latter incorporating virtual excitations to account for the influence of higher Landau levels. This approach allows for a detailed examination of how Landau level mixing influences the stability of different topological phases, providing key insights for realizing non-Abelian anyons in experimental settings. The ability to accurately model these interactions is essential for the development of robust and scalable quantum computing architectures.

Topological Order Determined from First-Principles Simulations

This discovery, achieved without pre-programmed expectations about the outcome, establishes a new method for identifying these elusive particles essential for fault-tolerant quantum computing. The work extends beyond simply finding anyons; it provides a means to differentiate between competing theoretical models of their behavior under realistic conditions. This capability addresses a long-standing challenge in the field, as previous theoretical studies often lacked accurate treatment of strong Landau level mixing, hindering definitive identification of topological order.

The methodology employed goes beyond traditional variational wave function approaches by incorporating all electrons and all Landau levels within the simulation. This contrasts with other neural network methods that incorporate prior physical assumptions into the wave function, offering a truly first-principles approach. By directly computing the overlap with trial wave functions, the entanglement properties of these ground states were analyzed, revealing the quantum dimensions of the anyons and confirming the Ising-type non-Abelian order.

👉 More information
🗞 Non-Abelian quantum Hall states at strong Landau level mixing: finding anyons with first-principles AI
✍️ Pierre-Antoine Graham, Ahmed Abouelkomsan and Liang Fu (Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA)
🧠 ArXiv: https://arxiv.org/pdf/2610.18967

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