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Mathematical Physics Seminar

Webinar: Ilya A. Gruzberg - Multifractals and conformal invariance at Anderson transitions.

Location:  Zoom
Date & time: Wednesday, 24 April 2024 at 10:45AM - 11:45AM


Ilya A. Gruzberg – The Ohio State University

Wednesday, April 24th, 10:45AM EDT 


Multifractals and conformal invariance at Anderson transitions.

Multifractal measures arise in such diverse subjects as dynamical chaos, weather and climate, turbulence, fractal growth, critical clusters in statistical mechanics, disordered magnets and other random critical points, Anderson transitions, mathematical finance, random energy landscapes, Gaussian multiplicative chaos, and rigorous approaches to conformal field theory. A multifractal measure is characterized by a continuous spectrum of multifractal exponents Δ_q that describe the scaling of the moments of the measure with the system size. In the context of Anderson transitions, the multifractality of critical wave functions is described by operators 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 must be quadratic in its arguments in any dimension d ≥ 2 (parabolicity of the spectrum). We confront this finding with available numerical and analytical data for various Anderson transitions that unambiguously show clear deviations of the multifractal spectra Δ_q from parabolicity, and discuss possible reasons for the discrepancy.