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

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>ν</a:mi><a:mo>=</a:mo><a:mn>1</a:mn></a:math>, 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 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 <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>ν</c:mi><c:mo>=</c:mo><c:mn>1</c:mn></c:math> 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.