Dynamical Phase Boundary in Long-Range Quantum Ising Chains

We systematically characterize a previously unidentified dynamical phase boundary alpha*(h/J) in the one-dimensional transverse-field Ising model with power-law decaying interactions ~1/r^alpha. Three main results are established: (i) the dominant oscillation frequency of the mean magnetization satisfies omega = Delta/2pi universally across all interaction ranges alpha tested; (ii) the finite-size scaling of the energy gap changes qualitatively at alpha*(h/J), separating exponential (long-range) from power-law (short-range effective) scaling; (iii) the boundary follows alpha*(h/J) = (c*J/h)^3 + b with c ~ e/2 and b ~ 5/3, and is characterized by the effective crossover condition Jeff(alpha*) ~ (h/J)/sqrt(3) + 1/2, suggesting the crossover is controlled by a balance between collective interaction strength and transverse field energy scale. Results are obtained via exact diagonalization (QuTiP, N=4-12) and symbolic regression (PySR).