High-quality reconstruction of controllable 3D head avatars from 2D videos is highly desirable for virtual human applications in movies, games, and telepresence. Neural implicit fields provide a powerful representation to model 3D head avatars with personalized shape, expressions, and facial parts, e.g., hair and mouth interior, that go beyond the linear 3D morphable model (3DMM). However, existing methods do not model faces with fine-scale facial features, or local control of facial parts that extrapolate asymmetric expressions from monocular videos. Further, most condition only on 3DMM parameters with poor(er) locality, and resolve local features with a global neural field. We build on part-based implicit shape models that decompose a global deformation field into local ones. Our novel formulation models multiple implicit deformation fields with local semantic rig-like control via 3DMM-based parameters, and representative facial landmarks. Further, we propose a local control loss and attention mask mechanism that promote sparsity of each learned deformation field. Our formulation renders sharper locally controllable nonlinear deformations than previous implicit monocular approaches, especially mouth interior, asymmetric expressions, and facial details. Project page:https://imaging.cs.cmu.edu/localdeformationfields/