Real-time scattering in the lattice Schwinger model

TL;DR

The paper uses tensor-network methods to simulate real-time meson-meson scattering in the lattice Schwinger model, focusing on the strong-coupling regime. By preparing two vector-meson wavepackets and evolving them with a controlled time-step scheme, it identifies a momentum threshold for opening the inelastic channel and observes production of scalar bound states. It proposes entanglement measures, electric-flux correlators, and a four-body projector as practical signatures of collision outcomes, with results aligning with variational MPS predictions. The study provides a pathway toward quantum-simulation realizations and highlights challenges in approaching the continuum limit for real-time dynamics in lattice gauge theories.

Abstract

Tensor network methods have demonstrated their suitability for the study of equilibrium properties of lattice gauge theories, even close to the continuum limit. We use them in an out-of-equilibrium scenario, much less explored so far, by simulating the real-time collisions of composite mesons in the lattice Schwinger model. Constructing wave-packets of vector mesons at different incoming momenta, we observe the opening of the inelastic channel in which two heavier mesons are produced and identify the momentum threshold. To detect the products of the collision in the strong coupling regime we propose local quantitites that could be measured in current quantum simulation platforms.

Figures (6)

• Figure 1: The upper panel shows the initial state preparation following Eq. \ref{['eq: initial state with O operators']}. The lower panel shows the dispersion relation of the adimensional Hamiltonian of Eq. \ref{['eq: Schwinger spin']} as $E=f(\langle O_{P}^{2} \rangle)$, with $O_{P}=-ix\sum_{n}(\sigma_{n}^{-}\sigma_{n+1}^{z} \sigma_{n+2}^{+}-h.c.)$ the dimensionless momentum operator for the fermion field (see banuls2013mass). The lower branch corresponds to the vector meson and the upper one to the scalar, for which only the lowest momentum excitations are shown. The ground state energy, $E_{\mathrm{GS}}/g$, is subtracted from all the energies plotted. We consider a finite system with lattice size $N=100$ and physical volume $N /\sqrt{x} =100$. The mass is $m/g = 10^{-5}$. The energies were calculated with variational MPS as in banuls2013mass with small bond dimension, $D=50$.
• Figure 2: The plots show the two-site entanglement entropy $S(n,t)$, defined in the main text \ref{['eq:ent_entropy']}, with the entanglement entropy of the vacuum subtracted from all the plots. The fermion mass is fixed to $m/g=10^{-5}$. We consider a finite system with lattice size $N=100$ and physical volume $N/\sqrt{x} =100$. The standard deviation of the wavepackets is $\sigma=4$. In the left plot, the momentum magnitude of the incoming mesons is $k=1$ and only the elastic channel is present. The elastic channel corresponds to two mesons exiting the collision (yellow colour) with the same momentum as the momentum of the incoming mesons. In the middle plot, the momentum of the incoming mesons is $k=1.13$, slightly above the threshold predicted by the variational MPS calculation $k_{\mathrm{thr}}^{\mathrm{MPS}}$\ref{['eq: threshold DMRG']}. At this point, the inelastic channel opens, with two bound states produced with velocities close to 0. The signal of the two bound states that were created after the collision is visible a bit below $gt_{\mathrm{phys}} \sim 80$, with the two outgoing mesons coming from the elastic channel also being visible. The momentum of the incoming mesons for the plot on the right is $k=1.5$, also above the momentum threshold. Both the elastic and inelastic channels are open. The two bound states created from the collision (inelastic channel) have non-zero velocities with opposite signs and less magnitude than the momenta of the outgoing mesons of the elastic channel. The elastic channel is represented by the light blue cone-like shape, after the collision, whereas the velocities of the bound states are shown with the two arrows. The parameters used in the numerical simulation were bond dimension $D=50$, Trotter step $\delta=0.1$ and maximum electric flux $L_{trunc}=8$.
• Figure 3: The plot shows the two-site entanglement entropy $S(n,t)$\ref{['eq:ent_entropy']} minus the entanglement entropy of the vacuum, $S_{vac}$, on a lattice with size $N=100$ and physical volume $N /\sqrt{x}=100$, for the cases where the initial particles have momenta below, slightly above and above the momentum threshold, at $k=1$, $k= 1.13$ and $k=1.5$. The mass is $m/g=10^{-5}$, and the standard deviation of the wavepackets $\sigma=4$. As in Fig. \ref{['entanglement']}, the parameters used in the numerical simulation were bond dimension $D=50$, Trotter step $\delta=0.1$ and maximum electric flux $L_{trunc}=8$. To demonstrate the numerical convergence, for the cases $k=1$ and $k=1.13$ we show also the results for $D=80$ (black dash lines) while keeping all the other parameters the same.
• Figure 4: This plot shows the electric flux correlator, $C^{mn}_{l}$, of a system size $N=100$, corresponding to physical volume $N/\sqrt{x} =100$ at lattice spacing $x=1$, for times after the meson-meson collision, $g t_{\mathrm{phys}}=100$. The initial momentum of the mesons, $k=1$, is below the momentum threshold. The fermion mass is $m/g=10^{-5}$, and the standard deviation of the initial wavepackets $\sigma=4$. We set $C^{mn}=0$ for $|m-n|\leq 1$, so that $L^{m,n}$ do not overlap. The insets show the enclosed area rescaled by a factor, to highlight the signal of the correlations between the two outgoing meson wavepackets of the elastic channel. The bond dimension used for the calculations is $D=50$ and the maximum electric flux $L_{trunc}=8$.
• Figure 5: This plot shows the expectation value of the four-body projector, $P_{n}(t)$, of a system with lattice size $N=100$ and physical volume $N /\sqrt{x} =100$. For the first plot the initial momenta of the mesons are $k=1.13$ and for the second plot are $k=1.5$. The mass is $m/g=10^{-5}$, and the standard deviation of the wavepackets $\sigma=4$. The bond dimension is $D=50$ and the maximum electric flux $L_{trunc}=8$.