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 ${\mathrm{\Delta }}_q$. In the context of Anderson transitions, the multifractality of critical wave functions is described by operators $O_q$ with scaling dimensions ${\mathrm{\Delta }}_q$ 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 ${\mathrm{\Delta }}_q$ (and its generalized form) must be quadratic in its arguments in any dimension $d\ge 2$.