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

TL;DR

This work demonstrates real-time, non-perturbative scattering in Ising Field Theory using matrix product states and TDVP, bridging the gap between integrable limits (free fermion and E8) and strongly coupled non-integrable regimes. By constructing carefully localized two-particle wave packets and evolving them within a finite spatial window, the authors extract elastic and inelastic scattering observables, time delays, and resonance properties, validating results against form-factor perturbation theory near integrable points. The study reveals resonance structures near the E8 point, quantifies the high-energy behavior of P_{11→11}(E) consistent with Zamolodchikov’s conjecture, and showcases the efficacy of tensor-network methods for real-time dynamics in 1+1D QFTs. These results offer a non-perturbative toolkit for probing S-matrix features and may inform S-matrix bootstrap efforts, with potential extensions to other field theories and quantum computing implementations.

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.

Figures (22)

• Figure 2: IFT interpolates between the integrable $E_8$ theory ($\eta=0$) and massive free field theory ($\eta=\infty$). As $\eta$ increases, the number of stable particles changes from 3 to 2 at $\eta_3 \approx 0.022$ and changes from 2 to 1 at $\eta_2 \approx 0.333$Delfino:2005bh.
• Figure 3: Pictorial representation of MPS states. MPS tensors are represented by blue squares (vacuum $A$ tensor) or red circles (excitation $B$ tensor). The horizontal lines represent the contraction of the bond dimension indices. The vertical black lines correspond to spin degrees of freedom. Upper figure: A single excitation tensor affects expectation values about a correlation length away from its insertion locus. Lower figure: Our scattering states are wave packets built from superpositions of localized basis states with $\sigma \gg \ell$.
• Figure 4: Inelastic particle scattering is visible as the presence of many tracks in the post-collision region (log-scale plot of excess energy on the left). Not every combination of tracks is a possible scattering outcome. Instead, the energy density is strongly correlated between subregions of the post-collision region (right). A comparison with the rescaled energy expectation values, shown as solid blue backgrounds, indicates that only subsets of the outgoing tracks are correlated. The simulation are run with $D = 64$ width a dynamical window of $2000$ sites. Above: $\eta_\text{latt} = .700$$(g_x = 1.06, g_z = 0.01)$, $E = 3.9$. Below: $\eta_\text{latt} = 1.97$$(g_x = 1.4, g_z =0.05)$, $E = 3.15$.
• Figure 5: If we detect the position of one of the three particles (one of the red circles), then for a specified total energy we automatically know where the other two particles are. The relative probability of encountering particles in these various configurations -- which can be parameterized by the velocity of the middle particle -- is plotted on the top. For this figure the total incoming energy is $3.33 m_1$.
• Figure 6: (a) The simulated production probability at three different values of the lattice coupling parameter $\xi_\text{latt}$. The solid lines are single parameter fits of the theoretical production probability at first order in FF perturbation theory, Eqs. \ref{['NearFFS']} and \ref{['eq:SnFF_branchcut_correction']}. (b) The simulated time delay at the same values of $\xi_\text{latt}$. The theoretical predictions from FF perturbation theory are completely fixed by the probability fits. The shaded regions show the expected magnitude of the second-order correction.