Accurate global localization is an essential ingredient for autonomous mobile robots (AMRs) operating in enclosed or partially enclosed repetitive environments (e.g., office corridors, industrial warehouses, transportation centers). In such environments, the Global Navigation Satellite System (GNSS) signals are unreliable or severely degraded. The highly ambiguous structures in such challenging scenarios would also lead the ordinary geometric feature-based LiDAR/visual localization methods to fail. The ambient magnetic field (MF) has exhibited high distinctiveness at different location, which makes it a viable alternative for infrastructure-free AMR localization. However, few of the previous research has been focused on the orientation-dependency and similar-sequential-route limitations of MF-based localization. Thus, this paper proposes a novel probabilistic global localization system with 2-D LiDAR and rotation-invariant magnetic field for AMRs operating in challenging repetitive and ambiguous environments. The proposed localization system mainly consists of: 1) Two-step Initialization: laser distance and MF sequence based matching, and 2) MF-based Pose Tracking: recursive multi-dimensional MF sequence based matching. Extensive experimental results demonstrate the advantageous localization performances of the proposed localization system over the existing methods.