We propose a phase-field fracture model, termed the Ω^2-model, that naturally incorporates singular strain and displacement jump. The new model faithfully recovers the fundamental feature of displacement discontinuity in classical fracture mechanics, while preserving the spatially diffuse nature of the phase-field variable. This enables natural tracking of crack initiation and propagation on a fixed spatial mesh, effectively eliminating mesh dependency and related issues. By exploiting the variational structure and the particular form of the Ω^2-model’s energy functional, we prove that the extreme value of energy requires the d-dimensional volume of the damage activation region to vanish, reducing it to a sharp (d−1)-dimensional crack surface. Here the damage is a independent field variable different from phase-field. Furthermore, we prove in multi-dimension that, as the phase-field length scale l→0 , the Ω^2-model converges to the target cohesive zone model (CZM) in terms of both the traction-separation law and the energy functional. The proofs avoid advanced techniques such as Γ-convergence, remaining accessible to researchers with standard engineering or physics backgrounds. One of the numerical examples demonstrates that in the brittle fracture limit (as the Irwin’s length l_ch→0), the crack-tip stress fields are in excellent agreement with classical linear elastic fracture mechanics (LEFM), including the r^(-1/2) singularity and the value of stress intensity factor.