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

TL;DR

This paper introduces a fractional Ising model using fractional derivatives to tune critical exponents and explore phase transitions in classical and quantum multiscale media, revealing new universal behaviors and geometric properties.

Contribution

It demonstrates how fractional derivatives can control critical exponents and fractal geometry, enabling phase transitions in lower-dimensional systems and providing a novel framework for multiscale quantum problems.

Findings

Fractional derivatives tune critical exponents continuously.

Classical fractional systems have Hausdorff dimension equal to fractional order q.

Quantum fractional systems show a gradual deviation in Hausdorff dimension from q.

Abstract

Until now multiscale quantum problems have appeared to be out of reach at the many-body level relevant to strongly correlated materials and current quantum information devices. In fact, they can be modeled with $q$-th order fractional derivatives, as we demonstrate in this work, treating classical and quantum phase transitions in a fractional Ising model for $0 < q \leq 2$ ($q = 2$ is the usual Ising model). We show that fractional derivatives not only enable continuous tuning of critical exponents such as $\nu$, $\delta$, and $\eta$, but also define the Hausdorff dimension $H_D$ of the system tied geometrically to the anomalous dimension $\eta$. We discover that for classical systems, $H_D$ is precisely equal to the fractional order $q$. In contrast, for quantum systems, $H_D$ deviates from this direct equivalence, scaling more gradually, driven by additional degrees of freedom introduced by quantum fluctuations. These results reveal how fractional derivatives fundamentally modify the fractal geometry of many-body interactions, directly influencing the universal symmetries of the system and overcoming traditional dimensional restrictions on phase transitions. Specifically, we find that for $q < 1$ in the classical regime and $q < 2$ in the quantum regime, fractional interactions allow phase transitions in one dimension. This work establishes fractional derivatives as a powerful tool for engineering critical behavior, offering new insights into the geometry of multiscale systems and opening avenues for exploring tunable quantum materials on NISQ devices.