Condensed Matter

Fuzzy sphere models conjecturally realize 3d CFTs in small systems of spinful fermions, but why they work so well is still not fully understood. Their Hamiltonians are built from electron density operators projected to the lowest Landau level. In this talk I will discuss the algebra generated by these density modes, which forms a spin-enriched spherical analogue of the Girvin–MacDonald–Platzman (GMP) algebra and is closely related to the matrix algebra of fuzzy spherical harmonics from noncommutative field theory. I will then describe two natural thermodynamic limits of the fuzzy-sphere geometry. In a local planar limit, high–angular-momentum modes reproduce the planar GMP algebra. In the commutative limit, which appears to govern the low-energy subspace of critical fuzzy sphere Hamiltonians, the low–angular-momentum modes become semiclassical. This talk is based on joint work with Zhenghan Wang (https://arxiv.org/abs/2602.15025).