Holographic nature of critical quantum states of proteins

The Anderson metal-insulator transition is a fundamental phenomenon in condensed matter physics, describing the transition from a conducting (metallic) to a non-conducting (insulating) state driven by disorder in a material. At the critical point of the Anderson transition, wave functions exhibit multifractal behavior, and energy levels display a universal distribution, indicating non-trivial correlations in the eigenstates. Recent studies have shown that proteins, traditionally considered insulators, exhibit much higher conductivity than previously assumed. In this paper, we investigate several proteins known for their efficient electron transport properties. We compare their energy level statistics, eigenfunction correlation, and electron return probability to those expected in metallic, insulating, or critical states. Remarkably, these proteins exhibit properties of critically disordered metals in their natural state without any parameter adjustment. Their composition and geometry are self-organized into the critical state of the Anderson transition, and their fractal properties are universal and unique among critical systems. Our findings suggest that proteins' wave functions may fulfill "holographic" area laws, since their correlation fractal dimension is $d_2\approx 2$ .