Real-time Scattering in φ^4 Theory using Matrix Product States

We investigate the critical behavior and real-time scattering dynamics of the interacting φ⁴ quantum field theory in (1+1) dimensions using uniform matrix product states and the time-dependent variational principle. A finite-entanglement scaling analysis at λ= 0.8 bounds the critical mass-squared to μ_c² in [-0.3190,-0.3185] and provides a quantitative map of the symmetric, near-critical, weakly broken, and deeply broken regimes. Using these ground states as asymptotic vacua, we simulate two-particle collisions in a sandwich geometry and extract the elastic scattering probability P_{11→ 11}(E) and Wigner time delay Δt(E) following the prescription of Jha et al. [Phys. Rev. Research 7, 023266 (2025)]. We find strongly inelastic scattering in the symmetric phase $P_{11→ 11} simeq 0.63$, $Δt simeq -180$ for $μ² = 0.2$, almost perfectly elastic collisions in the spontaneously broken phase $P_{11→ 11} simeq 0.998$, $Δt simeq -270$ for $μ²=-0.2$ and $P_{11→ 11} simeq 1$, $Δt simeq -177.781$ for $μ²=-0.5$, and a breakdown of the sandwich evolution precisely at the critical coupling, which provides a dynamical signature of the quantum critical point. These results demonstrate that TDVP-based uniform matrix product states can probe nonperturbative scattering and critical dynamics in lattice φ⁴ theory with controlled entanglement truncation.