Fractional Dissipation, Magnetization Dynamics, and Long-Range Interactions in Fractals

This thesis began with an interest in quantum fractals, leading to an investigation of fractional derivatives as a tool to describe fractality in uniform spaces. We found that these operators can effectively model anomalous diffusion. Seeking justification for their use, we showed that fractional derivatives can emerge from microscopic models of open quantum systems. In particular, we studied a generalized Caldeira-Leggett model and discovered a time glass, alongside an alternative derivation of fractional dissipation. We applied our findings to magnetization dynamics, resulting in a fractional generalization of the Landau-Lifshitz-Gilbert equation. The order of the fractional derivative was shown to be experimentally accessible via steady-state spin dynamics. Finally, we explored spin systems on complex geometries, including a fractal Ising model. Using graph-theoretic numerical techniques, we computed ground-state magnetizations for higher generations of the Sierpiński triangle, with both short- and long-range interactions, and observed qualitative agreement with preliminary experimental data. The first part of this thesis focuses on the role of fractional derivatives in dissipative dynamics. Starting from the Caldeira-Leggett framework, we showed that a power-law spectral function naturally leads to friction terms described by fractional derivatives. Depending on the chosen definition, different aspects such as causality, boundary conditions, and memory effects can be modeled. This is relevant in both classical and quantum regimes and can be grounded in the microscopic system-plus-bath interaction. By applying these methods to Brownian motion in a sub-Ohmic bath, we derived a fractional Langevin equation in which standard friction is replaced by a Caputo derivative. Manipulating the bath’s temperature gradient allowed us to isolate the effect of white noise. This led to the identification of the “time glass” phase: a regime in which a fluid periodically freezes and melts, exhibiting spontaneous time-translation symmetry breaking. By varying the derivative order, we could describe four distinct phases—liquid, glass, time glass, and marginal glass—within a unified framework. In an alternative approach, we showed that an Ohmic bath with a fractional coupling term also leads to fractional dissipation, resulting in a Liouville derivative in the equation of motion. This captures a wide range of anomalous diffusion behaviors, from ballistic to saturated, depending on the derivative order. This framework was applied to spin dynamics, where coupling a macrospin to a bath led to a fractional Landau-Lifshitz-Gilbert equation. This predicts altered frequency-to-linewidth ratios measurable via ferromagnetic resonance. Additionally, high-frequency bath modes universally induce spin inertia, an effect recently confirmed experimentally. The second part of the thesis shifts focus to spin systems on fractal networks. We studied Rydberg atoms arranged in a Sierpiński triangle, modeled as a transverse-field Ising model. For large systems, we developed the SIM-GRAPH method, which uses graph symmetries to reduce Hilbert space dimensionality. This enabled us to compute ground states for up to 81 spins within seconds on a standard laptop. Our results show that fractal ground states only emerge with van der Waals interactions. The magnetization patterns agree qualitatively with data from a quantum simulator at the University of Amsterdam.