Emergent Quantum Droplets in Logarithmic Klein-Gordon Models of Bose-Einstein Condensates

We study a relativistic scalar field model for self-bound Bose-Einstein condensates (BECs) by analyzing a nonlinear Klein-Gordon equation with cubic and logarithmic interactions. This framework captures essential features of quantum droplets, such as self-trapping and finite energy configurations, which emerge from the interplay between attractive and repulsive terms. By performing the non-relativistic limit, we derive a generalized Gross-Pitaevskii equation with a logarithmic correction, consistent with recent models used to describe ultra-cold atomic gasses beyond mean-field theory. We construct the corresponding Lagrangian density, identify conserved quantities via Noether's theorem, and compute the energy-momentum tensor. Numerical solutions of the BEC parameters are shown, establishing the foundations for a field theoretical description of relativistic condensates with a logarithmic interaction. This model provides a unified approach to investigate relativistic effects in quantum droplets and enriches the theoretical landscape of Bose-Einstein condensates with non-standard interactions. The resulting dynamics exhibit stable oscillatory regimes consistent with self-bound condensate configurations.