Real-Time Scattering in Ising Field Theory using Matrix Product States (2411.13645v1)

Abstract: 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 non-integrable 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 post-collision wavefunction, particle tracks from all final-state channels occur in superposition; processes of interest can be isolated by projecting the wavefunction 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.

• The paper demonstrates that matrix product states enable real-time simulations of scattering in the Ising Field Theory, capturing key elastic time delays and inelastic processes.
• It employs the time-dependent variational principle to analyze scattering within strongly coupled, non-integrable quantum field theories.
• Numerical results validate analytical predictions from perturbation theory and Zamolodchikov’s integrable models, revealing resonance behaviors and high-energy transitions.

Overview of "Real-Time Scattering in Ising Field Theory using Matrix Product States"

The paper "Real-Time Scattering in Ising Field Theory using Matrix Product States" presents an analytical and numerical investigation of scattering processes in the Ising Field Theory (IFT). This work uses matrix product states (MPS) to explore the scattering phenomena within strongly coupled non-integrable quantum field theories in 1+1 dimensions.

Key Aspects and Methodology

The Ising Field Theory is a one-parameter family that transitions between massive free fermion theory and Zamolodchikov's massive $E_8$ theory. The paper focuses on elastic and inelastic scattering processes, which are analyzed through the framework of matrix product states and the time-dependent variational principle.

A crucial methodological advancement in this research is the application of MPS to simulate the real-time evolution of quantum states during scattering. By numerically modeling the scattering events, the paper offers insights into the Ising Field Theory that are not easily accessible through other computational techniques such as Euclidean lattice computations or Feynman diagrams.

The real-time scattering in IFT is of particular physical interest due to its relevance in understanding fundamental particle physics and its applicability to other gapped quantum field theories. The paper thoroughly investigates the time delay of elastic scattering and the probability of inelastic particle production as functions of collision energy. The researchers employ simulated numerical results to validate theoretical expectations derived from form-factor perturbation theory, especially near the two integrable endpoints—the free fermion theory and $E_8$ theory.

Results and Analysis

The primary results discussed in the paper include determining the time delay for elastic scattering events and assessing the high-energy behavior of two-to-two particle scattering probabilities. Near the integrable limits, critical insights are gained by comparing numerical simulations with analytical conjectures, notably Zamolodchikov’s predictions.

The paper identifies and characterizes the presence of resonances—unstable particle excitations above the mass threshold that manifest as poles in the S-matrix. This allows for an estimation of resonance masses and decay widths, particularly focusing on the third and fourth resonances in the $E_8$ domain, thereby confirming the theoretical predictions from perturbation theory.

Furthermore, the paper explores high-energy scattering behavior across varying values of the $\eta$ parameter in IFT. It reports a transition in scattering regimes, characterized by alterations in the dominance of elastic or inelastic behavior at high energies, consistent with Zamolodchikov's conjecture of a critical $\eta_c$ where the high-energy scattering outcome undergoes a qualitative change.

Implications for Quantum Field Theory and Future Research

The numerical methods employed have broader implications, confirming that tensor-network approaches can effectively simulate real-time dynamics in strongly coupled regimes where traditional perturbative methods fail. This paper enriches our understanding of the scattering matrix structure in non-integrable models, paving the way for future explorations of similarly complex field theories.

Future advancements based on this research could leverage quantum computational methods to explore high entanglement domains unreachable by classical MPS simulations, potentially enhancing the analysis of non-integrable quantum field theories. By applying these findings to other models like the $O(3)$ sigma model or the three-state Potts model, we can further clarify the complex landscape of scattering processes in low-dimensional quantum systems.

This research, therefore, not only provides detailed insights into the Ising Field Theory but also opens avenues for more exhaustive explorations into real-time quantum field evolutions, highlighting the strength of MPS as a dynamic computational tool in modern theoretical physics.