Classical and Quantum Phase Transitions in Multiscale Media: Universality and Critical Exponents in the Fractional Ising Model

Multiscale quantum phenomena can be effectively addressed using fractional derivatives. In this study, we investigate classical and quantum phase transitions within a fractional Ising model, also known as a Lévy crystal, characterized by a fractional order $0 < q \leq 2$, where $q = 2$ corresponds to the conventional nearest-neighbor spin-spin Ising coupling. Notably, we demonstrate that long-range interactions induced by fractional derivatives enable continuous tuning of both the critical transition values and the associated critical exponents, including the anomalous dimension $\eta$, as well as $\alpha$, $\beta$, $\gamma$, $\delta$, and $\nu$. These critical exponents govern fundamental aspects of the phase transition, such as specific heat, magnetization, susceptibility, critical isotherm, and correlation length. Furthermore, we reveal that the fractional order $q$ directly tunes the Hausdorff dimension of the scale-invariant behavior during phase transitions, with the Hausdorff dimension being equivalent to the fractional order. This tuning mechanism allows us to overcome the conventional Mermin-Wagner-Hohenberg dimensional restrictions on both classical and quantum phase transitions in one dimension, specifically for $q < 1$ in the classical case and $q < 2$ in the quantum case. In the critical borderline case of $q = 1$, a Berezinskii-Kosterlitz-Thouless transition emerges. Our analysis confirms the validity of Rushbrooke, Josephson, and other scaling relations, while also resolving hyperscaling relations by utilizing the fractional order to account for the dangerous irrelevant variables that arise from the fractional interaction. This many-body application of fractional calculus not only broadens the theoretical framework for understanding quantum materials but also opens new avenues for exploration using Noisy Intermediate-Scale Quantum (NISQ) devices.