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

TL;DR

We address real-time scattering and critical dynamics in the $(1+1)$-D φ^4 theory using uniform matrix product states (uMPS) and the time-dependent variational principle (TDVP). Finite-entanglement scaling at fixed $λ=0.8$ yields a bracket for the critical mass-squared $μ_c^2 ∈ [-0.3190,-0.3185] and maps the symmetric, near-critical, weakly broken, and deeply broken regimes. Two-particle collisions in a sandwich geometry reveal strong inelastic scattering in the symmetric phase (e.g., $P_{11\to 11}(E) \approx 0.63$) and near-elastic scattering in the broken phase (e.g., $P_{11\to 11}(E) \approx 0.998$ or $1$), with a dynamical signature of criticality as the protocol breaks down at $μ_c^2$ due to diverging correlation length. This work demonstrates that TDVP-based uMPS can access nonperturbative scattering and critical dynamics in lattice φ^4 theory with controlled entanglement truncation, providing a template for studying more complex quantum field theories and guiding extensions to larger bond dimensions and refined observables.

Abstract

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 and the time-dependent variational principle. A finite-entanglement scaling analysis at $λ= 0.8$ bounds the critical mass-squared to $μ_c^2 \in [-0.3190,-0.3185]$ 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 $P_{11\to 11}(E)$ 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 ($P_{11\to 11} \simeq 0.63$, $Δt \simeq -180$ for $μ^2 = 0.2$), almost perfectly elastic collisions in the spontaneously broken phase ($P_{11\to 11} \simeq 0.998$, $Δt \simeq -270$ for $μ^2=-0.2$ and $P_{11\to 11} \simeq 1$, $Δt \simeq -177.781$ for $μ^2=-0.5$), 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 $φ^4$ theory with controlled entanglement truncation.

Figures (3)

• Figure 1: Finite-entanglement scaling near the critical point of the $\phi^4$ theory at $\lambda = 0.8$. Ground-state uMPS data with finite bond dimension $D$ are used to extract the effective central charge $c_{\mathrm{eff}}(r)$, a local order parameter, and the entropy--correlation-length relation near $r_c = \mu_c^2$. The peak in $c_{\mathrm{eff}}(r)$ and the linear dependence $\mathcal{S} \simeq \frac{c}{6} \log \xi_D + \mathrm{const.}$ close to $r_c$ yield the bracket $\mu_c^2 \in [-0.3190,-0.3185]$ and confirm Ising universality.
• Figure 2: Excitation spectra of the $\phi^4$ theory at $\lambda = 0.8$ across the phase diagram. Each panel shows the single- and multi-particle excitation energies $d E(p)$ obtained from the uMPS transfer matrix as a function of momentum $p$ for representative values of $\mu^2$. The gap closes near (b) and reopens with a massive single-particle band in (c),(d), providing a spectral signature of the phase transition and mass generation.
• Figure 3: Real-time two-particle scattering in the $\phi^4$ theory at $\lambda = 0.8$, shown as space--time plots of $\langle \phi_n(t) \rangle$ in the sandwich geometry. Panels (a)–(d) correspond to the same values of $\mu^2$ as in Fig. \ref{['fig::spectra']}. The gapped phases, (a),(c),(d), display the characteristic "X" pattern of wave packets approaching, colliding, and separating as outgoing quasiparticles, with strong inelasticity in the symmetric case (a) and almost perfectly elastic scattering in the broken phases (c),(d). At the critical point (b) the "X" pattern is absent and a slow drift of the profile indicates a breakdown of the sandwich protocol due to the diverging correlation length.