Emergence of fractional quantum mechanics in condensed matter physics

Condensed matter provides a fertile ground for the emergence of exotic phenomena. For instance, polymers typically lack long range order and crystalline symmetry, leading to difficulties in establishing a general theory to describe the properties of these materials. They exhibit memory, some extent of self-similarity and non-locality effects, which are better described by fractional calculus. In this letter we demonstrate the emergence of an effective Hamiltonian possessing fractional dispersion taking as the starting point a tight-binding model in which the hopping parameters from next-nearest neighbours cannot be neglected. Our model was applied to describe the Pauli paramagnetism in a system with fractional dispersion, the magnetic susceptibility of vanadyl acetate and the electronic transport as a function of temperature in conductive polymers, displaying good agreement between theory and experimental data. © 2021 Elsevier B.V.