🔬 mesoscale physics

Non-Abelian quantum Hall states at strong Landau level mixing: finding anyons with first-principles AI

This paper presents the first complete first-principles solution of the non-Abelian quantum Hall problem at $\nu = 3/2$ using a momentum-resolved neural-network variational wave function, demonstrating that strong Landau level mixing favors the anti-Pfaffian state and confirming the emergence of Ising-type non-Abelian anyons with quantum dimension $\sqrt{2}$ .

Original authors: Pierre-Antoine Graham, Ahmed Abouelkomsan, Liang Fu

Published 2026-10-15

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Pierre-Antoine Graham, Ahmed Abouelkomsan, Liang Fu

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 strange world of quantum physics, particles do not always behave like the solid objects we see around us. Sometimes, they act more like ripples in a pond, and when many of them interact under extreme conditions, they can form a new kind of matter with rules all its own. One of the most fascinating places where this happens is in the fractional quantum Hall effect, a state of matter that appears when electrons are trapped in a flat layer and subjected to a powerful magnetic field. Under these conditions, the electrons organize themselves into a highly ordered, fluid-like state. Within this fluid, the electrons can act as if they are carrying a fraction of their usual electric charge, and they can form exotic particles called anyons. These anyons are special because swapping their positions does not just change the system slightly; it fundamentally alters the quantum state of the entire group in a way that is impossible to undo with simple measurements. This unique behavior makes them a leading candidate for building the next generation of quantum computers, which could solve problems that are currently impossible for today's machines. However, figuring out exactly which type of anyon exists in these systems has been a decades-long puzzle, complicated by the fact that the electrons are constantly jostling each other in ways that are difficult to calculate.

For a long time, scientists have struggled to predict the exact nature of these quantum fluids because the standard methods of calculation break down when the electrons interact too strongly. In many experimental setups, the electrons are so crowded and energetic that they jump between different energy levels, a phenomenon known as Landau level mixing. Traditional computer models usually ignore these jumps or treat them as a small, manageable error, but in reality, this mixing is often the dominant force shaping the system. Without accounting for it, theories have produced conflicting predictions, leaving researchers unsure whether the electrons form one specific type of topological order or another. The question has remained open for over thirty years, with different experiments pointing toward different answers and theoretical models failing to agree on the ground truth.

A team of researchers at the Massachusetts Institute of Technology has now tackled this problem using a completely different approach: artificial intelligence. Instead of trying to force the physics into a simplified equation, they trained a neural network—a type of computer program inspired by the human brain—to learn the behavior of the electrons directly from the fundamental laws of nature. They started with the basic equations that describe how electrons repel each other and move in a magnetic field, without telling the computer what kind of state it should find. The neural network was allowed to explore the vast landscape of possible electron arrangements on its own, searching for the configuration that minimized the system's energy. This method is powerful because it does not rely on human guesses or pre-existing theories about what the answer should look like; it simply lets the data speak.

The researchers focused on a specific scenario where the electrons are fully polarized in their spin and occupy a filling factor of 3/2. This means that the lowest energy level is completely full, and the next level is half full. In this setting, the team simulated a system with twenty-one electrons and found that the neural network successfully discovered a stable, liquid-like state. When they analyzed the properties of this state, they found a specific signature of a non-Abelian topological order. This signature appeared in the way the electrons were entangled with one another, revealing the presence of anyons with a specific mathematical property known as a quantum dimension of the square root of two. This value is the hallmark of a particular type of order called the Ising type, which is exactly what is needed for fault-tolerant quantum computing. The simulation showed that this state is robust and stable, even when the electrons are interacting very strongly.

To determine exactly which version of this topological order the electrons had chosen, the researchers compared their AI-discovered state against several competing theories. There were three main candidates for the ground state: the Moore-Read Pfaffian, its particle-hole conjugate known as the anti-Pfaffian, and a third possibility called the particle-hole Pfaffian. By calculating how closely the AI's solution matched each of these theoretical models, the team found a clear winner. The state discovered by the neural network overlapped most strongly with the anti-Pfaffian. This result is significant because it suggests that when Landau level mixing is strong, the system naturally favors the anti-Pfaffian order over the others. The study explicitly ruled out the particle-hole Pfaffian as a viable candidate for the ground state in this regime, as the overlap with that model remained very small regardless of the strength of the interactions.

The power of this approach lies in its ability to handle the full complexity of the problem without simplifying the physics. Previous studies had to limit their calculations to just one or two energy levels or treat the mixing between levels as a minor correction. This new method included all the electrons and all the energy levels in the simulation, capturing the full effect of the strong interactions. The researchers verified their findings by checking the energy of the system and comparing it to older, more limited calculations, finding that their AI model consistently produced lower, more accurate energies. They also observed that as the strength of the mixing increased, the electrons began to redistribute themselves across higher energy levels, a detail that older models missed entirely.

This work provides a direct, first-principles solution to a problem that has resisted theoretical description for decades. It demonstrates that artificial intelligence can be used to uncover the hidden rules of quantum matter, finding patterns that are too complex for traditional mathematics to reveal. The findings are particularly relevant to recent experiments in materials like monolayer tungsten diselenide and zinc oxide, where similar strong mixing effects are observed. In these materials, the heavy mass of the electrons makes the mixing between energy levels unavoidable, creating a perfect environment for the type of non-Abelian order the researchers identified. By showing that the anti-Pfaffian state is the likely ground state under these conditions, the study offers a concrete target for experimentalists to verify and helps clarify the path toward building practical topological quantum computers. The research confirms that when the electrons are allowed to interact fully and without artificial constraints, they settle into a specific, highly ordered state that holds the key to a new era of quantum technology.

Technical Summary: Non-Abelian Quantum Hall States at Strong Landau Level Mixing

Problem Statement
The nature of non-Abelian fractional quantum Hall (FQH) states at even-denominator filling factors, particularly $\nu = 5/2$ in GaAs and $\nu = 3/2$ in materials with strong Landau level (LL) mixing (e.g., ZnO, WSe $_2$ ), remains an open question. While the Moore–Read Pfaffian state was proposed as the leading candidate, the competition between the Pfaffian, its particle-hole conjugate (the anti-Pfaffian), and the particle-hole symmetric Pfaffian (PH-Pfaffian) has proven difficult to resolve. A critical limitation in previous theoretical studies is the inability to treat strong Landau level mixing non-perturbatively. Existing numerical approaches either truncate the Hilbert space to a single or few Landau levels or treat mixing as a small perturbation. This is insufficient for regimes where the Coulomb energy is comparable to or exceeds the cyclotron energy, a condition prevalent in recent experiments on wide quantum wells and transition metal dichalcogenides.

Methodology
The authors present a first-principles solution to the non-Abelian FQH problem for spin-polarized electrons at $\nu = 3/2$ , incorporating all electrons and all Landau levels without truncation. The core of their approach is a momentum-resolved neural-network variational wave function (NN-VMC) based on the MagNet architecture.

Variational Ansatz: The wave function is constructed as a sum of Slater determinants of many-body orbitals, $\Psi_\theta = \sum_n \det(\phi^n_j)$ . Each orbital $\phi^n_j$ factorizes into a periodic amplitude $F^n_j$ and a quasiperiodic factor $\chi^n_j$ that encodes magnetic boundary conditions.
Neural Network Architecture: The amplitudes and winding maps (which control the zeros of the orbitals) are parameterized by a self-attention neural network. The network inputs are high-dimensional features of electron coordinates mapped to a torus. The architecture includes $L$ layers combining self-attention and multilayer perceptrons (MLPs).
Momentum Projection: To access topologically degenerate ground states, the authors exploit magnetic translation symmetry. They construct center-of-mass (COM) momentum eigenstates $\Phi_k$ by summing translated copies of the variational wave function over the magnetic unit cell, weighted by appropriate phases.
Optimization: The network parameters are optimized solely by minimizing the variational energy $E_k(\theta) = \langle \Phi_k | H | \Phi_k \rangle / \langle \Phi_k | \Phi_k \rangle$ using Monte Carlo sampling. No prior knowledge of the expected topological order is injected into the ansatz.
Characterization:
- Topological Order: Identified via ground-state degeneracy on the torus and the pattern of momentum eigenvalues.
- Anyon Content: Probed using Minimally Entangled States (MESs). The authors construct linear superpositions of degenerate ground states to minimize entanglement entropy across a bipartition. The differences in Rényi entanglement entropy between MESs yield the quantum dimensions of the anyons.
- State Identification: Overlaps are computed between the NN-VMC ground-state manifold and trial manifolds for the Pfaffian, anti-Pfaffian, and PH-Pfaffian. To account for LL mixing, trial states are "dressed" using a Schrieffer–Wolff transformation to include virtual inter-Landau-level excitations.

Key Results

Ground State Degeneracy and Anyons: For $\nu = 3/2$ at $\kappa = 1.0$ (where $\kappa$ is the LL mixing parameter), the method yields a twofold degenerate ground state for an odd number of electrons ( $N=21$ ) and a sixfold degeneracy for an even number ( $N=18$ ) in the active Landau level. This matches the expected degeneracy for a Moore–Read-type state.
Quantum Dimensions: Analysis of the entanglement entropy of the six MESs for $N=18$ reveals a structure consistent with Ising-type non-Abelian order. Four MESs correspond to Abelian sectors (quantum dimension $d=1$ ), while two MESs exhibit an entropy increase of $\log 2$ , corresponding to a non-Abelian anyon with quantum dimension $d_\sigma = \sqrt{2}$ .
Effect of Landau Level Mixing:
- The NN-VMC energies are significantly lower than those obtained from exact diagonalization (ED) restricted to the lowest two Landau levels, demonstrating that higher LLs substantially lower the energy.
- As $\kappa$ increases, the occupation of higher Landau levels grows, and the lowest two levels are depleted.
- At very strong mixing ( $\kappa = 20$ ), the system transitions from a liquid state to a Wigner crystal, evidenced by pronounced peaks in the pair correlation function.
Topological Order Selection: By comparing overlaps with dressed trial states, the authors find that the anti-Pfaffian state consistently has a higher overlap with the NN-VMC ground state than the Pfaffian or PH-Pfaffian across all values of $\kappa$ studied. The overlap with the PH-Pfaffian remains small, ruling it out as the ground state candidate in this regime. The preference for the anti-Pfaffian strengthens with increasing $\kappa$ .

Significance and Claims
The paper claims to provide the first complete, non-perturbative solution to the non-Abelian FQH problem that includes all electrons and all Landau levels. The primary significance lies in demonstrating that neural-network wave functions, optimized purely on the microscopic Hamiltonian without physics priors, can successfully identify non-Abelian anyons and their quantum dimensions.

The authors emphasize that their approach resolves the competition between candidate topological orders in regimes of strong Landau level mixing, a regime relevant to recent experiments in WSe $_2$ and ZnO where the effective mass is large. The results indicate that Landau level mixing favors the anti-Pfaffian topological order over the Pfaffian. The work establishes neural-network variational methods as a powerful tool for characterizing topological phases from first principles, particularly in parameter regimes where traditional perturbative or truncated Hilbert space methods fail.

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