To uncover the mechanism behind dynamical quantum phase transitions (DQPTs), we employ initial-state engineering by tuning the effective inverse temperature $\beta$ of the coherent Gibbs state in the one-dimensional XY model. We find that DQPTs arising from interphase quenches are topologically protected, originating from the topological mismatch between the initial state and the postquench Hamiltonian, whereas those triggered by intraphase quenches are topologically independent. Despite this distinction, both types of DQPTs are governed by a unified geometric condition: A DQPT occurs when the Bloch vector of a given momentum mode becomes orthogonal to the postquench Hamiltonian axis, causing the mode-resolved Loschmidt amplitude to vanish at critical times. We show that the effective inverse temperature $\beta$ of the coherent Gibbs state acts as a geometric knob that continuously rotates the initial Bloch vector. For interphase quenches, the topological mismatch guarantees this orthogonality for any $\beta$, making topological protection a natural consequence of this geometric picture. For intraphase quenches, the orthogonality condition reveals a particularly insightful limit: at $\beta =0$, DQPTs can occur even without any quench; as $\beta$ increases, the initial Bloch vector rotates smoothly, and this rotation must be compensated for by a corresponding rotation of the Hamiltonian axis, precisely achieved by the external quench. This interplay gives rise to two critical modes that emerge from opposite Brillouin-zone boundaries, migrate inward with increasing $\beta$, and eventually coalesce and annihilate at a finite critical inverse temperature. These results establish a unified geometric description of DQPTs and elucidate how intraphase critical modes can arise and persist in the absence of any topological protection.

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• Dynamical phase transitions
• Phase transitions
• XY model
• 1-dimensional spin chains
• 1-dimensional systems
• Dynamic critical phenomena
• Exact diagonalization