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 Delta(q). In the context of Anderson transitions, the multi-fractality of critical wave functions is described by operators O-q with scaling dimensions Delta(q) in a fieldtheory 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 Delta(q) (and its generalized form) must be quadratic in its arguments in any dimension d >= 2.