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 multifractality of critical wave functions is described by operators $O_q$ with scaling dimensions $\Delta_q$ 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 $\Delta_q$ (and its generalized form) must be quadratic in its arguments in any dimension $d \geq 2$.