Termination of typical wave-function multifractal spectra at the Anderson metal-insulator transition: Field theory description using the functional renormalization group

We revisit the problem of wave-function statistics at the Anderson metal-insulator transition (MIT) of noninteracting electrons in d>2 spatial dimensions. At the transition, the complex spatial structure of the critical wave functions is reflected in the nonlinear behavior of the multifractal spectrum of generalized inverse participation ratios (IPRs). Beyond the crossover from narrow to broad IPR statistics, which always occurs for sufficiently large moments of the wave-function amplitude, the spectrum obtained from a typical wave function associated with a particular disorder realization differs markedly from that obtained from the disorder-averaged IPRs. This phenomenon is known as the termination of the multifractal spectrum. We provide a field theoretical derivation for the termination of the typical multifractal spectrum by combining the nonlinear sigma model framework, conventionally used to access the MIT in d=2+ dimensions, with a functional renormalization-group (FRG) technique. The FRG method deployed here was originally pioneered to study the properties of the two-dimensional (2D) random-phase XY model. The same method was used to demonstrate the termination of the multifractal spectrum in the very special problem of 2D Dirac fermions subject to a random Abelian vector potential. Our result shows that the typical multifractal wave-function spectrum and its termination can be obtained at a generic Anderson localization transition in d>2, within the standard field theoretical framework of the nonlinear sigma model, when combined with the FRG.