A numerical investigation on percolation, quantum transport and correlated states

Abstract

This work presents a numerical investigation of disorder-induced phenomena and strong correlation effects in lattice models of condensed matter physics. The work is divided into two main parts. The first focuses on classical and quantum percolation in flat-band systems as well on quantum state transfer (QST) protocols, while the second addresses strongly correlated physics of heavy-fermions within the Periodic Anderson Model (PAM). We begin by studying classical percolation on two- and three-dimensional Lieb lattices using Monte Carlo simulations and finite-size scaling. We determine precise site and bond percolation thresholds, establishing a coordination-dependent hierarchy consistent with mean-field predictions. The extracted critical exponents confirm universal behavior based on lattice dimensionality. Extending to the quantum regime, we analyze quantum percolation (QP) via level spacing statistics and random matrix theory (RMT). Thresholds and critical exponents are computed for the Lieb lattice and its extensions, showing that QP retains a hierarchy similar to the classical case. The localization length exponent ν is found to match known universality classes, with no emergent behavior in flat-band systems due to symmetry-breaking disorder. Next, we explore QST in two-dimensional lattices, focusing on the role of particle statistics, interactions, and quantum chaos. By employing an annealing-based coupling optimization scheme, we achieve high-fidelity transfer for both spinless fermions and hard-core bosons. Our results demonstrate that QST performance is sensitive to the interplay between integrability, interactions, and optimization, offering insights into controllable quantum dynamics in many-body systems. In the final part, we perform determinant Quantum Monte Carlo simulations of the three-dimensional PAM. By analyzing thermodynamic, magnetic, and transport observables across different hybridization strengths, we identify a quantum critical point separating antiferromagnetic and Kondo-singlet phases. We map out the corresponding finite-temperature phase diagram, establishing energy scales that characterize coherence, magnetic ordering, and quantum fluctuations.