Real-time scattering in ϕ {4} theory using matrix product states

We investigate the critical behavior and real-time scattering dynamics of the interacting ϕ^{4} quantum field theory in (1+1) dimensions using uniform matrix product states (uMPS) and the time-dependent variational principle (TDVP). A finite-entanglement scaling analysis at λ=0.8 bounds the critical mass-squared to μ_{c}^{2}∈]−0.2595,−0.2594[ and provides a quantitative map of the symmetric, near-critical, and spontaneously 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) using a sandwich geometry protocol. We find strongly inelastic scattering in the symmetric phase (P_{11→11}≃0.712, Δt≃−158 for μ^{2}=+0.2) and almost perfectly elastic collisions in the spontaneously broken phase (P_{11→11}≃1, Δt≃−108 for μ^{2}=−0.1 and P_{11→11}≃1, Δt≃−177.781 for μ^{2}=−0.5). Crucially, the scattering protocol exhibits a distinctive divergence near the critical coupling; we show that this behavior serves as a dynamical signature of the quantum critical point, arising directly from the closing of the mass gap. These results demonstrate that TDVP-based uMPS can effectively probe nonperturbative scattering and critical dynamics in lattice field theories with controlled entanglement truncation.