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

Abstract

We investigate the critical behavior and real-time scattering dynamics of the interacting $\phi^4$ 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 $\lambda = 0.8$ bounds the critical mass-squared to $\mu_c^2 \in ]-0.2595,-0.2594[$ 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 $P_{11\to 11}(E)$ and Wigner time delay $\Delta t(E)$ using a sandwich geometry protocol. We find strongly inelastic scattering in the symmetric phase ($P_{11\to 11} \simeq 0.712$, $\Delta t \simeq -158$ for $\mu^2 = +0.2$) and almost perfectly elastic collisions in the spontaneously broken phase ($P_{11\to 11} \simeq 1$, $\Delta t \simeq -108$ for $\mu^2=-0.1$ and $P_{11\to 11} \simeq 1$, $\Delta t \simeq -177.781$ for $\mu^2=-0.5$). 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.