Redes livres de escala de osciladores harmônicos acoplados: aplicação para teoria quântica de campos

Abstract

We investigate the transport of excitations in discretized quantum fields, employing the Complex Networks formalism to model spatial geometry. The central objective is to analyze how the propagation of a bosonic field, described as a network of coupled harmonic oscillators, differs from classical diffusion when space is represented by non-trivial topologies. To this end, we implemented the Continuous-Time Quantum Walk (CTQW) on Barabási–Albert networks and compared the results with classical diffusion (CTRW) on the same structure. The analysis focuses on the temporal return probability, revealing a distinct difference between the classical and quantum regimes. While the classical walker explores the entire network, the quantum walker tends to remain partially localized in regions of high connectivity. To this end, we identified the spectral origin of this topological localization by characterizing the Density of States (DOS) and the Inverse Participation Ratio (IPR), demonstrating that modes associated with high eigenvalues generate quantum trapping. As a supplementary analysis, we examined the consistency of these phenomena by varying the initial network connectivity (𝑚) and the system size (𝑁). We observed a continuous transition, where sparsely connected networks exhibit strong localization and inefficient transport, while denser networks approach a regime of partial delocalization, featuring enhanced spreading behavior and reduced trapping. The results suggest that the topology of discrete space acts as an effective potential for bosonic excitations, such that sparse networks generate long-range quantum trapping regimes even for large systems (𝑁 = 2000), whereas higher connectivities induce progressive spreading and more efficient transport. Asymptotic analysis reveals that the average return probability tends toward a finite value in irregular topologies, establishing a topological confinement mechanism analogous to effective mass generation. In view of this, quantum dynamics proves to be intensely dependent on discrete geometry, imposing restrictions on coherence and field propagation in non-continuous spacetime models.