3d Conformal Field Theories via Fuzzy Sphere Algebra

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. We analyze the Lie algebra generated by these density modes and its large-$s$ limits. Depending on how the limit is taken, the algebra approaches either the Girvin-MacDonald-Platzman algebra in a local planar limit or a semiclassical algebra for low-angular-momentum modes in a global commutative limit. With an additional restriction to a low-excitation sector above the paramagnetic state, the density modes become approximate harmonic oscillators. We also test whether the conformal algebra $so(3,2)$ can be realized directly by density modes. Such a representation exists only in the minimal two-electron system; its natural coproduct extension does not match the physical thermodynamic limit of a single growing fuzzy sphere.