Conformal invariance and multifractality at Anderson transitions in arbitrary dimensions

Multifractals arise in various systems across nature whose scaling behavior is characterized by a continuous spectrum of multifractal exponents Δ_q. In the context of Anderson transitions, the multifractality of critical wavefunctions is described by operators O_q with scaling dimensions Δ_q in a field theory description of the transitions. The operators O_q satisfy the so-called Abelian fusion, expressed as a simple operator product expansion. Assuming conformal invariance and Abelian fusion, we use the conformal bootstrap framework to derive a constraint that implies that the multifractal spectrum Δ_q (and its generalized form) must be quadratic in its arguments in any dimension d ≥ 2.