Fractional multiscale materials are common in classical and biological systems, being in fact quite typical in disparate natural systems ranging from anomalous diffusion of pollutants in ground water systems to faster than expected infection rates of catheters in hospital settings. Until now multiscale quantum problems of this nature 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 I demonstrate in the first part of this talk, treating classical and quantum phase transitions in a fractional Ising model for 0<q≤2 (q=2 is the usual Ising model). We show that fractional derivatives not only enable continuous tuning of critical exponents such as ν, δ, and η, but also define the Hausdorff dimension H_D of the system tied geometrically to the anomalous dimension η. 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, with BKT transitions in the q=1 and q=2 borderline cases, respectively. 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.
In the second part of this talk, I will present the propagation of quantum information in a one-dimensional fractional transverse-field Ising model, where Riesz fractional derivatives generate interactions beyond the scope of standard power laws. Using matrix product states (MPS) and time-dependent variational principle (TDVP) methods adapted for nonlocal couplings, we systematically vary the fractional order q and show that the dynamical critical exponent takes the form z = q/2. This finding directly links fractional interactions to a Lévy flight framework, since the mean-square displacement of a classical Lévy flight scales as t^(2/q), mirroring the t^(1/z) dependence of correlation fronts in our spin chain. As a result, the usual short-range limit is recovered for q≤2, whereas q>2 gives rise to a unique frustration-driven regime that remains genuinely nonlocal and displays sublinear growth of entanglement and correlations. These observations illustrate how fractional derivatives unify short-range, power-law, and frustrated long-range interactions within a single framework, offering a window into exotic phases and nonlocal critical phenomena.
References:
1. Bruce J. West, “Colloquium: Fractional calculus view of complexity: A tutorial.” Reviews of Modern Physics 86, 1169 (2014), https://doi.org/10.1103/RevModPhys.86.1169
2. Mark J. Ablowitz, Joel B. Been, and Lincoln D. Carr, “Fractional Integrable Nonlinear Soliton Equations,” Phys. Rev. Lett., v. 128, p.184101 (2022)
3. Joshua M. Lewis and Lincoln D. Carr, “Exploring Multiscale Quantum Media: High-Precision Efficient Numerical Solution of the Fractional Schrödinger equation, Eigenfunctions with Physical Potentials, and Fractionally-Enhanced Quantum Tunneling,” J. Phys. A, v. 58, p. 175303 (2025)
4. Joshua M. Lewis and Lincoln D. Carr, “Classical and Quantum Phase Transitions in Multiscale Media: Universality and Critical Exponents in the Fractional Ising Model,” Phys. Rev. Lett., under review (2025), https://arxiv.org/abs/2501.14134
5. Joshua M. Lewis, Zhexuan Gong, and Lincoln D. Carr, “Fractional Ising Model and Lévy Light Cones: Nonlocal Causality Constraints Beyond Power-Law Decays,” Quantum Science and Technology, under review (2025), https://arxiv.org/abs/2505.05645