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Regularizing 3D Conformal Field Theories via Anyons on the Fuzzy Sphere

Cristian Voinea1, Ruihua Fan2, Nicolas Regnault3,4,5, and Zlatko Papić1

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Phys. Rev. X 15, 031007 – Published 7 July, 2025

DOI: https://doi.org/10.1103/bf4k-phl9

Abstract

The recently introduced “fuzzy-sphere” method has enabled accurate numerical regularizations of certain three-dimensional (3D) conformal field theories (CFTs). The regularization is provided by the noncommutative geometry of the lowest Landau level filled by electrons, such that the charge is trivially gapped due to the Pauli exclusion principle at filling factor ν=1, while the electron spins encode the desired CFT. Successful applications of the fuzzy sphere to paradigmatic CFTs, such as the 3D Ising model, raise an important question: How finely tuned does the underlying electron system need to be? Here, we show that the 3D Ising CFT can also be realized at fractional electron fillings. In such cases, the CFT spectrum is intertwined with the charge-neutral spectrum of the underlying fractional quantum Hall state—a feature that is trivially absent in the previously studied ν=1 case. Remarkably, we show that the mixing between the CFT spectrum and the fractional quantum Hall spectrum is strongly suppressed within the numerically accessible system sizes. Moreover, we demonstrate that the CFT critical point is unaffected by the exchange statistics of the particles and by the nature of topological order in the charge sector. Our results set the stage for the fuzzy-sphere exploration of conformal critical points between topologically ordered states.

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Popular Summary

Simulating three-dimensional conformal field theories (3D CFTs), like the 3D Ising model, is important for understanding critical points in physics but is usually very difficult. A recent approach uses a “fuzzy sphere,” where particles live on the surface of a sphere with a magnetic monopole in the center. In earlier work, the particles filled the lowest energy level completely. In our study, we show that the same critical behavior still appears even when the level is only partially filled. In this case, the system forms fractional quantum Hall (FQH) states, which involve strange particles called anyons and exhibit topological order—yet the 3D Ising CFT still shows up at the phase transition.

To explore this, we create a model with two layers of particles, where each particle has an internal property like spin. By adjusting interactions and a field that connects the layers, we cause a transition between an ordered phase and a disordered one. We use numerical simulations and theory to check for signs of the 3D Ising CFT. Key features—like how physical quantities scale and how entangled the system is—match what we expect from the theory, even though the system also includes more complicated FQH physics.

This shows that the fuzzy-sphere method is a powerful tool for studying CFTs, even in the presence of more complex quantum behavior. It opens the door to simulating other kinds of phase transitions and could help us discover new types of interacting quantum theories.

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Phys. Rev. X 15, 031007– Published 7 July, 2025
  • Received 5 December 2024
  • Revised 16 April 2025
  • Accepted 20 May 2025
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