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Real-time scattering in Ο† 4 theory using matrix product states

Bahaa Al Sayegh, Wissam A Chemissany

Published Jan 1, 2026β€’ 0 citations
Publisher

Abstract

We investigate the critical behavior and real-time scattering dynamics of the interacting πœ™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 πœ†=0.8 bounds the critical mass-squared to πœ‡2π‘βˆˆ]βˆ’0.2595,βˆ’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 𝑃11β†’11⁑(𝐸) and Wigner time delay Δ⁒𝑑⁑(𝐸) using a sandwich geometry protocol. We find strongly inelastic scattering in the symmetric phase (𝑃11β†’11≃0.712, Ξ”β’π‘‘β‰ƒβˆ’158 for πœ‡2=+0.2) and almost perfectly elastic collisions in the spontaneously broken phase (𝑃11β†’11≃1, Ξ”β’π‘‘β‰ƒβˆ’108 for πœ‡2=βˆ’0.1 and 𝑃11β†’11≃1, Ξ”β’π‘‘β‰ƒβˆ’177.781 for πœ‡2=βˆ’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.

DOI
10.17169/refubium-52745

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We investigate the critical behavior and real-time scattering dynamics of the interacting πœ™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 πœ†=0.8 bounds the critical mass-squared to πœ‡2π‘βˆˆ]βˆ’0.2595,βˆ’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 𝑃11β†’11⁑(𝐸) and Wigner time delay Δ⁒𝑑⁑(𝐸) using a sandwich geometry protocol. We find strongly inelastic scattering in the symmetric phase (𝑃11β†’11≃0.712, Ξ”β’π‘‘β‰ƒβˆ’158 for πœ‡2=+0.2) and almost perfectly elastic collisions in the spontaneously broken phase (𝑃11β†’11≃1, Ξ”β’π‘‘β‰ƒβˆ’108 for πœ‡2=βˆ’0.1 and 𝑃11β†’11≃1, Ξ”β’π‘‘β‰ƒβˆ’177.781 for πœ‡2=βˆ’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.
We investigate the critical behavior and real-time scattering dynamics of the interacting Ο†4Ο†^4Ο†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 Ξ»=0.8Ξ»= 0.8Ξ»=0.8 bounds the critical mass-squared to ΞΌc2∈]βˆ’0.2595,βˆ’0.2594[ΞΌ_c^2 \in ]-0.2595,-0.2594[ΞΌc2β€‹βˆˆ]βˆ’0.2595,βˆ’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 P11β†’11(E)P_{11\to 11}(E)P11β†’11​(E) and Wigner time delay Ξ”t(E)Ξ”t(E)Ξ”t(E) using a sandwich geometry protocol. We find strongly inelastic scattering in the symmetric phase (P11β†’11≃0.712P_{11\to 11} \simeq 0.712P11β†’11​≃0.712, Ξ”tβ‰ƒβˆ’158Ξ”t \simeq -158Ξ”tβ‰ƒβˆ’158 for ΞΌ2=+0.2ΞΌ^2 = +0.2ΞΌ2=+0.2) and almost perfectly elastic collisions in the spontaneously broken phase (P11β†’11≃1P_{11\to 11} \simeq 1P11β†’11​≃1, Ξ”tβ‰ƒβˆ’108Ξ”t \simeq -108Ξ”tβ‰ƒβˆ’108 for ΞΌ2=βˆ’0.1ΞΌ^2=-0.1ΞΌ2=βˆ’0.1 and P11β†’11≃1P_{11\to 11} \simeq 1P11β†’11​≃1, Ξ”tβ‰ƒβˆ’177.781Ξ”t \simeq -177.781Ξ”tβ‰ƒβˆ’177.781 for ΞΌ2=βˆ’0.5ΞΌ^2=-0.5ΞΌ2=βˆ’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.
We study scattering in Ising field theory (IFT) using matrix product states and the time-dependent variational principle. IFT is a one-parameter family of strongly coupled nonintegrable quantum field theories in 1 + 1 dimensions, interpolating between massive free fermion theory and Zamolodchikov's integrable massive E 8 theory. Particles in IFT may scatter either elastically or inelastically. In the postcollision wave function, particle tracks from all final-state channels occur in superposition; processes of interest can be isolated by projecting the wave function onto definite particle sectors, or by evaluating energy density correlation functions. Using numerical simulations we determine the time delay of elastic scattering and the probability of inelastic particle production as a function of collision energy. We also study the mass and width of the lightest resonance near the E 8 point in detail. Close to both the free fermion and E 8 theories, our results for both elastic and inelastic scattering are in good agreement with expectations from form-factor perturbation theory. Using numerical computations to go beyond the regime accessible by perturbation theory, we find that the high-energy behavior of the two-to-two particle scattering probability in IFT is consistent with a conjecture of Zamolodchikov. Our results demonstrate the efficacy of tensor-network methods for simulating the real-time dynamics of strongly coupled quantum field theories in 1 + 1 dimensions.
We study the second-order quantum phase transition of massive real scalar field theory with a quartic interaction in (1+11+11+1) dimensions on an infinite spatial lattice using matrix product states. We introduce and apply a naive variational conjugate gradient method, based on the time-dependent variational principle for imaginary time, to obtain approximate ground states, using a related ansatz for excitations to calculate the particle and soliton masses and to obtain the spectral density. We also estimate the central charge using finite-entanglement scaling. Our value for the critical parameter agrees well with recent Monte Carlo results, improving on an earlier study which used the related density matrix normalization group method, verifying that these techniques are well-suited to studying critical field systems. We also obtain critical exponents that agree, as expected, with those of the transverse Ising model. Additionally, we treat the special case of uniform product states (mean field theory) separately, showing that they may be used to investigate noncritical quantum field theories under certain conditions.
A bstract We present a first real-time study of hadronic scattering in a (1+1)-dimensional SU(2) lattice gauge theory with fundamental fermions using tensor-network techniques. Working in the gaugeless Hamiltonian formulation β€” where the gauge field is exactly integrated out and no truncation of the electric flux is required β€” we investigate scattering processes across sectors of fixed global baryon number B = 0, 1, 2. These correspond respectively to meson-meson, meson-baryon, and baryon-baryon collisions. At strong coupling, the B = 0 and B = 2 channels exhibit predominantly elastic dynamics closely resembling those of the U(1) Schwinger model. In contrast, the mixed B = 1 sector shows qualitatively new behavior: meson and baryon wavepackets become entangled during the collision, and depending on their initial kinematics, the slower state becomes spatially delocalized while the faster one propagates ballistically. We characterize these processes through local observables, entanglement entropy, and the information-lattice, which together reveal how correlations build up and relax during the interaction. Our results establish a first benchmark for non-Abelian real-time scattering from first principles and open the path toward quantum-simulation studies of baryon-number dynamics and inelastic processes in gauge theories.
A scattering event in a quantum field theory is a coherent superposition of all processes consistent with its symmetries and kinematics. While real-time simulations have progressed toward resolving individual channels, existing approaches rely on knowledge of the asymptotic particle wavefunctions. This work introduces an experimentally inspired method to isolate scattering channels in Matrix Product State simulations based on the entanglement structure of the late-time wavefunction. Schmidt decompositions at spatial bipartitions of the post-scattering state identify elastic and inelastic contributions, enabling deterministic detection of outgoing particles of specific species. This method may be used in settings beyond scattering and is applied to detect heavy particles produced in a collision in the one-dimensional Ising field theory. Natural extensions to quantum simulations of other systems and higher-order processes are discussed.
We simulate, using nonperturbative methods, the real-time dynamics of small bubbles of "false vacuum" in a quantum spin chain near criticality, where the low-energy physics is described by a relativistic (1+1)-dimensional quantum field theory. We consider bubbles whose walls are kink and antikink quasiparticle excitations, so that wall collisions are kink-antikink scattering events. To construct these bubbles in the presence of strong correlations, we extend a recently proposed matrix product state (MPS) ansatz for quasiparticle wavepackets. We simulate dynamics within a window of about 1000 spins embedded in an infinite chain at energies of up to about 5 times the mass gap. By choosing the wavepacket width and the bubble size appropriately, we avoid strong lattice effects and observe relativistic kink-antikink collisions. We use the MPS quasiparticle ansatz to detect scattering outcomes. (i) In the Ising model, with transverse and longitudinal fields, we do not observe particle production despite nonintegrability (supporting recent observations of nonthermalizing states in this model). (ii) Switching on an additional interaction, we see production of confined and unconfined particle pairs. We characterize the amount of entanglement generated as a function of energy and time and conclude that our classical simulation methods will ultimately fail as these increase. We anticipate that kink-antikink scattering in 1+1 dimensions will be an instructive benchmark problem for future quantum computers and analog quantum simulators.