Conventional phase field modeling of fracture uses the degraded strain energy density (SED) at the crack tip as a material damage index to drive crack growth. To avoid non-physical evolution in the crack phase-field, various SED splitting schemes have been adopted, leading to the development of "anisotropic"-SED-based formulations that better capture realistic crack nucleation and propagation under mixed-mode loading. In this work, we propose a stress-intensity-factor-driven (SIF-driven) phase field method as an alternative to achieve the same goal. Using the crack phase-field distribution as a marker of the material configurational change and leveraging the phase-field landscape and its gradient, the nonlocal SIF-powered fracture energy release rate near the crack tip is computed within the framework of linear elastic fracture mechanics (LEFM) and hyperplastic materials. This non-local energy release rate is then incorporated into a variational phase field modeling framework as the driving force for material configurational changes, i.e., the crack phase-field evolution. The proposed formulation is validated through multiple numerical examples, demonstrating its capability to capture mode I, mode II, and mixed-mode fracture behaviors without mesh dependency. The key contributions of this work include: (1) accurate representation of crack-tip stress asymptotic field, (2) precise prediction of crack growth and material failure without the need for additional splitting techniques, (3) introducing a physics-based stress-intensity-factor-governed crack driving force to replace the SED-based approach, thereby effectively bridging the gap between phase-field formulation for fracture and well-established LEFM theory, and (4) providing a numerically efficient and straightforward implementation that closely resembles that of conventional phase field methods. This work establishes a robust connection between the phase field method and full-fledged fracture mechanics, providing a practical, physics-consistent tool for cleavage fracture analysis in engineering applications.