Out-of-Equilibrium Dynamics in a U(1) Lattice Gauge Theory via Local Information Flows: Scattering and String Breaking

TL;DR

The paper introduces the information lattice as a local, scale-resolved diagnostic for real-time, out-of-equilibrium dynamics in lattice gauge theories. It defines local information flows via a nonnegative decomposition of the total information into $(n,\ell)$ components and demonstrates, in the (1+1)D Schwinger model, how near-threshold vector-meson scattering and electric-string dynamics imprint distinct, interpretable patterns of information propagation. Through tensor-network simulations, the authors show that scalar-meson production corresponds to emergent long-scale information, while string confinement yields static, finite-range correlations whereas string breaking produces cyclical, multi-scale correlations. The results offer a robust, information-centric view of complex many-body phenomena with potential extensions to higher dimensions and quantum hardware experiments. The framework thus provides a practical bridge between microscopic gauge theories and observable, information-based diagnostics in real-time dynamics.

Abstract

We introduce local information flows as a diagnostic tool for characterizing out-of-equilibrium quantum dynamics in lattice gauge theories. We employ the information lattice framework, a local decomposition of total information into spatial- and scale-resolved contributions, to characterize the propagation and buildup of quantum correlations in real-time processes. Focusing on the Schwinger model, a canonical $(1+1)$-dimensional U(1) lattice gauge theory, we apply this framework to two scenarios. First, in the near-threshold scattering of two vector mesons, we demonstrate that the emergence of correlations at a longer length scale in the information lattice marks the production of heavier scalar mesons. Second, in the dynamics of electric field strings, we clearly distinguish between the confining regime, which evolves towards a steady state with a static correlation profile, and the string-breaking sector. The latter is characterized by dynamic correlation patterns that reflect the sequential formation and annihilation of strings. This information-centric approach provides a direct, quantitative, and interpretable visualization of complex many-body phenomena, offering a promising tool for analyzing dynamics in higher-dimensional gauge theories and experiments on quantum hardware.

Figures (17)

• Figure 1: (a) Information lattice for a product state of qubits, e.g., $\ket{\Omega_\mathrm{s.c.}}$ in Eq. \ref{['eq:vac_SC']}. (b) Illustration of the formula for local information in Eq. \ref{['eq:local_info_formula']} for $i(3.5,3)$; red denotes positive contributions and blue negative ones. The underlying green area shows the subsytem $\mathcal{C}^{3}_{3.5}$. (c) Information per scale for the ground state $\ket{\Omega_{\mathrm{s.c.}}}$ and first excited states $\ket{1_{\mathrm{V,S}}}$ of the Schwinger model in the strong coupling limit in Eqs. \ref{['eq:vac_SC']} and \ref{['eq:vector_meson_strong_coupling']}, respectively, with $N=20$. Only scales $\ell < 7$ are shown.
• Figure 2: Left: Mass spectrum as a function of the expectation value of the squared pseudo momentum operator $P =-i \sum_n \left( \sigma_{n}^- \sigma^z_{n+1} \sigma^+_{n+2}- {\rm h.c.} \right)$ for $ga=1$. Here, $\mathcal{M}_i= E_i-E_{\rm vac}$ is the energy gap to the vacuum of the $i$-th state. The vector ($i=1$) and scalar ($i=20$) states (gold stars) are identified by having the minimal momentum and exhibiting a mass gap. Their identification was further confirmed by checking their parity. Blue circular markers denote finite momentum excitations of the vector meson, which appear only in the lattice theory. The results for mass gaps of the identified states agree quantitatively with those reported in Ref. Papaefstathiou:2024zsu. Right:$I(\ell)$ distribution for the vacuum, vector, and scalar meson states identified in the left panel. $I(\ell)$ is also shown for the same states at $ga=2$, which are identified through the same DMRG procedure. Note that for $ga=2$ the curves of the vector and scalar meson states overlap. We set $ma=10^{-5}$ and $N=40$ in both panels.
• Figure 3: Information lattice for $\ell \leq 6$ for the ground, vector meson, and scalar meson states in the Schwinger model for $ga=1$, $ma=10^{-5}$, and $N=40$. Left: The ground state's $i(n,\ell)$ distribution is dominated by correlations at $\ell \approx 1$ with a decaying tail. Note that the exact strong coupling vacuum is a product state, and thus, there one would have correlations only at $\ell=0$. Right: Difference between the $i(n,\ell)$ distributions for the first excited state and the ground state. The vector meson state is dominated by higher-level correlations for $\ell \gtrsim 3$. Center: The difference between the ground state and the scalar meson $i(n,\ell)$ distributions, which shows that this excited state has a strong enhancement of correlations at $\ell \approx \{3,4\}$.
• Figure 4: Bipartite entanglement entropy across a cut between sites $n$ and $n+1$, i.e., for the region from site 1 to $n$. We set $ga=1$, $ma=10^{-5}$, and $N=40$. Left: Scattering below particle threshold ($ka=0.7$). Vertical white lines indicate the time slices used in Figs. \ref{['fig:scattering_k07']} and \ref{['fig:scattering_k13']}. Right: Scattering above threshold ($ka=1.3$). The white box highlights the region used to compute the distributions in Fig. \ref{['fig:cut_Iln_scattering']}.
• Figure 5: Snapshots of the information lattice for the scattering of two wave packets as in Eq. \ref{['eq:jet_init']} for $ka=0.7$. The selected times correspond to the dashed vertical lines shown in Fig. \ref{['fig:entropy']}. We set $ga=1$, $ma=10^{-5}$, and $N=40$.