Phase Transitions in Quantum Spin Systems with Long-Range Interactions: A Nonlinear Variational Inequality Approach

This paper presents a novel mathematical framework for modeling phase transitions in quantum spin systems with long-range interactions using nonlinear variational inequalities (NVIs). Traditional methods such as mean-field theory and Landau-Ginzburg models often fall short in capturing spatial inhomogeneities, critical phenomena, and constraints inherent in realistic spin systems. Our approach addresses these limitations by reformulating the energy-based governing equations as an NVI problem defined over a suitable convex set of physically admissible spin configurations. We incorporate both Ising and Heisenberg-type quantum lattice models with power-law decaying interactions to accurately represent nonlocal effects. This research provides rigorous mathematical preliminaries, including operator-theoretic foundations in Hilbert and Banach spaces, and prove existence and uniqueness results under monotonicity and coercivity assumptions. Analytical insights include bifurcation analysis, critical point theory, and regularity of solutions, offering a robust theoretical foundation for understanding phase transitions. A finite element–Galerkin discretization is developed to simulate the variational system numerically. Simulation results demonstrate the emergence of ordered phases, spin textures, and domain formation, consistent with known physical behavior. Comparisons with classical models reveal the advantages of the NVI framework in capturing phase boundaries and metastable states with higher fidelity. The proposed framework also opens pathways for extending the analysis to quantum entanglement, topological order, and non-equilibrium phenomena. This research thus contributes both to the mathematical theory of variational inequalities and to the physical modeling of quantum spin systems, providing tools for deeper exploration of complex condensed matter phenomena.