Exploring Itinerant Ferromagnetism: From Puzzle Games to Correlated Quantum Phenomena

Itinerant ferromagnetism in materials, characterized by the complex interplay of electron correlations, represents a fundamental quantum phenomenon in condensed matter physics. Despite the prevalence of ferromagnetic metals, deciphering the microscopic origins and magnetic phase transitions of such materials remains a formidable challenge. We use non-perturbative approaches, including graph theory, analytical analysis, classical Monte Carlo and sign problem-free quantum Monte Carlo simulations, to examine the ground state, thermodynamic properties, and magnetic phase transitions of itinerant electrons in single-orbital, multi-orbital, as well as flat band systems. Our discussion aims to provide concrete starting points towards understanding and identifying novel itinerant ferromagnetic states and supporting the experimental quest for discovering new materials.