002946648 001__ 2946648
002946648 005__ 20251024061554.0
002946648 0248_ $$aoai:cds.cern.ch:2946648$$pcerncds:FULLTEXT$$pcerncds:CERN:FULLTEXT$$pcerncds:CERN
002946648 037__ $$9arXiv:reportnumber$$aCERN-TH-2025-191
002946648 037__ $$9arXiv$$aarXiv:2510.16101$$cquant-ph
002946648 035__ $$9arXiv$$aoai:arXiv.org:2510.16101
002946648 035__ $$9Inspire$$aoai:inspirehep.net:3071118$$d2025-10-23T00:01:41Z$$h2025-10-24T02:01:19Z$$mmarcxml$$ttrue$$uhttps://inspirehep.net/api/oai2d
002946648 035__ $$9Inspire$$a3071118
002946648 041__ $$aeng
002946648 100__ $$aArtiaco, Claudia$$martiaco@kth.se$$uRoyal Inst. Tech., Stockholm$$vDepartment of Physics, KTH Royal Institute of Technology, Stockholm, 106 91 Sweden
002946648 245__ $$9arXiv$$aOut-of-Equilibrium Dynamics in a U(1) Lattice Gauge Theory via Local Information Flows: Scattering and String Breaking
002946648 269__ $$c2025-10-17
002946648 300__ $$a32 p
002946648 500__ $$9arXiv$$a32 pages, 17 figures
002946648 520__ $$9arXiv$$aWe introduce local information flows as a diagnostic tool for characterizing out-of-equilibrium quantum dynamics in lattice gauge theories. We employ the information lattice framework, a local decomposition of total information into spatial- and scale-resolved contributions, to characterize the propagation and buildup of quantum correlations in real-time processes. Focusing on the Schwinger model, a canonical $(1+1)$-dimensional U(1) lattice gauge theory, we apply this framework to two scenarios. First, in the near-threshold scattering of two vector mesons, we demonstrate that the emergence of correlations at a longer length scale in the information lattice marks the production of heavier scalar mesons. Second, in the dynamics of electric field strings, we clearly distinguish between the confining regime, which evolves towards a steady state with a static correlation profile, and the string-breaking sector. The latter is characterized by dynamic correlation patterns that reflect the sequential formation and annihilation of strings. This information-centric approach provides a direct, quantitative, and interpretable visualization of complex many-body phenomena, offering a promising tool for analyzing dynamics in higher-dimensional gauge theories and experiments on quantum hardware.
002946648 541__ $$aarXiv$$chepcrawl$$d2025-10-21T04:01:26.298788$$e10828862
002946648 540__ $$3preprint$$aarXiv nonexclusive-distrib 1.0$$uhttp://arxiv.org/licenses/nonexclusive-distrib/1.0/
002946648 595__ $$aCERN-TH
002946648 65017 $$2arXiv$$ahep-ph
002946648 65017 $$2SzGeCERN$$aParticle Physics - Phenomenology
002946648 65017 $$2arXiv$$ahep-lat
002946648 65017 $$2SzGeCERN$$aParticle Physics - Lattice
002946648 65017 $$2arXiv$$aquant-ph
002946648 65017 $$2SzGeCERN$$aGeneral Theoretical Physics
002946648 690C_ $$aCERN
002946648 690C_ $$aPREPRINT
002946648 700__ $$aBarata, João$$uCERN$$vEuropean Organization for Nuclear Research (CERN), Theoretical Physics Department, CH-1211 Geneva, Switzerland
002946648 700__ $$aRico, Enrique$$menrique.rico.ortega@cern.ch$$uCERN$$uBasque U., Bilbao$$uDonostia Intl. Phys. Ctr., San Sebastian$$vEuropean Organization for Nuclear Research (CERN), Theoretical Physics Department, CH-1211 Geneva, Switzerland$$vEHU Quantum Center and Department of Physical Chemistry, University of the Basque Country UPV/EHU, P.O. Box 644, 48080 Bilbao, Spain$$vDIPC -Donostia International Physics Center, Paseo Manuel de Lardizabal 4, 20018 San Sebastián, Spain$$vIKERBASQUE, Basque Foundation for Science, Plaza Euskadi 5, 48009 Bilbao, Spain
002946648 8564_ $$82790007$$s53370$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_25_k_1.3_plot.png$$y00014 Plots analogous to Fig.~\ref{fig:scattering_k07}, now for $ka=1.3$. We set $ga=1$, $ma=10^{-5}$, and $N=40$.
002946648 8564_ $$82790008$$s93834$$uhttps://cds.cern.ch/record/2946648/files/I_l_distribution_plot_g_1.0_charge_2.0.png$$y00058 Partially integrated information per scale $\bar{I}(\ell)$ (see main text) for the time evolution illustrated in Fig.~\ref{fig:string_full_iln_2}.
002946648 8564_ $$82790009$$s93345$$uhttps://cds.cern.ch/record/2946648/files/Q2p8_Efield.png$$y00020 Time evolution of the expectation value of the onsite electric field $L(n,t)-L(n,0)$ and the bipartite entanglement entropy $\mathcal{S}(n)$ for quenches with injected external charges using $Q=\{2.8,4\}$, $ga=0.5$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790010$$s43145$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_36_g_1.0_Q_3.6_plot.png$$y00056 Snapshots of the information lattice for the time evolution with injected external charges as in Fig.~\ref{fig:string_full_iln}, now for $Q=\{1.6,2.4,3,6\}$, $ga=1$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790011$$s35062$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_6_g_1.0_Q_3.6_plot.png$$y00054 Snapshots of the information lattice for the time evolution with injected external charges as in Fig.~\ref{fig:string_full_iln}, now for $Q=\{1.6,2.4,3,6\}$, $ga=1$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790012$$s62881$$uhttps://cds.cern.ch/record/2946648/files/I_l_distribution_plot_g_1.0_charge_3.6.png$$y00060 Partially integrated information per scale $\bar{I}(\ell)$ (see main text) for the time evolution illustrated in Fig.~\ref{fig:string_full_iln_2}.
002946648 8564_ $$82790013$$s38484$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_6_g_0.5_Q_3.4_plot.png$$y00038 Snapshots of the information lattice for the time evolution with injected external charges for $Q=\{2, 3.4, 4\}$. We set $ga=0.5$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790014$$s98942$$uhttps://cds.cern.ch/record/2946648/files/Q1_Entropy.png$$y00029 Time evolution of the expectation value of the onsite electric field and bipartite entanglement entropy for $Q=\{1,2.6,3.2\}$ for the quench in which the initially injected external charges are removed at time $t=12 \, a$, indicated by the vertical white dashed line. We set $ga=1$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790015$$s39315$$uhttps://cds.cern.ch/record/2946648/files/scalar_only.png$$y00005 Information lattice for $\ell \leq 6$ for the ground, vector meson, and scalar meson states in the Schwinger model for $ga=1$, $ma=10^{-5}$, and $N=40$. \textbf{Left}: The ground state's $i(n,\ell)$ distribution is dominated by correlations at $\ell \approx 1$ with a decaying tail. Note that the exact strong coupling vacuum is a product state, and thus, there one would have correlations only at $\ell=0$. \textbf{Right}: Difference between the $i(n,\ell)$ distributions for the first excited state and the ground state. The vector meson state is dominated by higher-level correlations for $\ell \gtrsim 3$. \textbf{Center}: The difference between the ground state and the scalar meson $i(n,\ell)$ distributions, which shows that this excited state has a strong enhancement of correlations at $\ell \approx \{3,4\}$.
002946648 8564_ $$82790016$$s59183$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_36_g_0.5_Q_2.0_plot.png$$y00037 Snapshots of the information lattice for the time evolution with injected external charges for $Q=\{2, 3.4, 4\}$. We set $ga=0.5$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790017$$s44539$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_20_g_1.0_Q_2.0_plot.png$$y00066 Snapshots of the information lattice for the quench in which the charges are removed at $t=12 \, a$ as in Fig.~\ref{fig:E_field_Entropy_stopped} for $Q=\{1,2,2.6,3.2\}$, $ga=1$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790018$$s72822$$uhttps://cds.cern.ch/record/2946648/files/I_l_distribution_plot_g_0.5_charge_2.0.png$$y00044 Partially integrated information per scale $\bar{I}(\ell)$ (see main text) for the time evolution illustrated in Fig.~\ref{fig:string_full_iln}.
002946648 8564_ $$82790019$$s143662$$uhttps://cds.cern.ch/record/2946648/files/Q2p4_Efield.png$$y00024 Plots analogous to Fig.~\ref{fig:11_full_charges}, now for $ga=1$. We use $Q=\{ 2.4,3.6 \}$, and set $ma=0.25$ and $N=100$.
002946648 8564_ $$82790020$$s131583$$uhttps://cds.cern.ch/record/2946648/files/I_l_distribution_plot_g_1.0_k_1.0.png$$y00017 Integrated information distribution for the central region of the scattering process shown in Fig.~\ref{fig:entropy} (right) for $ka=\{0.7,1,1.2,1.3\}$. Thicker lines highlight times multiples of $5a$.
002946648 8564_ $$82790021$$s40892$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_12_g_1.0_Q_1.0_plot.png$$y00062 Snapshots of the information lattice for the quench in which the charges are removed at $t=12 \, a$ as in Fig.~\ref{fig:E_field_Entropy_stopped} for $Q=\{1,2,2.6,3.2\}$, $ga=1$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790022$$s37088$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_6_g_1.0_Q_1.6_plot.png$$y00048 Snapshots of the information lattice for the time evolution with injected external charges as in Fig.~\ref{fig:string_full_iln}, now for $Q=\{1.6,2.4,3,6\}$, $ga=1$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790023$$s50813$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_25_k_0.7_plot.png$$y00010 Snapshots of the information lattice for the scattering of two wave packets as in Eq.~\eqref{eq:jet_init} for $ka=0.7$. The selected times correspond to the dashed vertical lines shown in Fig.~\ref{fig:entropy}. We set $ga=1$, $ma=10^{-5}$, and $N=40$.
002946648 8564_ $$82790024$$s46522$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_30_g_1.0_Q_1.0_plot.png$$y00064 Snapshots of the information lattice for the quench in which the charges are removed at $t=12 \, a$ as in Fig.~\ref{fig:E_field_Entropy_stopped} for $Q=\{1,2,2.6,3.2\}$, $ga=1$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790025$$s78614$$uhttps://cds.cern.ch/record/2946648/files/Q2p6_Entropy.png$$y00031 Time evolution of the expectation value of the onsite electric field and bipartite entanglement entropy for $Q=\{1,2.6,3.2\}$ for the quench in which the initially injected external charges are removed at time $t=12 \, a$, indicated by the vertical white dashed line. We set $ga=1$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790026$$s38425$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_12_g_1.0_Q_3.2_plot.png$$y00071 Snapshots of the information lattice for the quench in which the charges are removed at $t=12 \, a$ as in Fig.~\ref{fig:E_field_Entropy_stopped} for $Q=\{1,2,2.6,3.2\}$, $ga=1$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790027$$s42333$$uhttps://cds.cern.ch/record/2946648/files/S_l_vs_l_all_groups.png$$y00002 \textbf{Left:} Mass spectrum as a function of the expectation value of the squared pseudo momentum operator $P =-i \sum_n \left( \sigma_{n}^- \sigma^z_{n+1} \sigma^+_{n+2}- {\rm h.c.} \right)$ for $ga=1$. Here, $\mathcal{M}_i= E_i-E_{\rm vac}$ is the energy gap to the vacuum of the $i$-th state. The vector ($i=1$) and scalar ($i=20$) states (gold stars) are identified by having the minimal momentum and exhibiting a mass gap. Their identification was further confirmed by checking their parity. Blue circular markers denote finite momentum excitations of the vector meson, which appear only in the lattice theory. The results for mass gaps of the identified states agree quantitatively with those reported in Ref.~\cite{Papaefstathiou:2024zsu}. \textbf{Right:} $I(\ell)$ distribution for the vacuum, vector, and scalar meson states identified in the left panel. $I(\ell)$ is also shown for the same states at $ga=2$, which are identified through the same DMRG procedure. Note that for $ga=2$ the curves of the vector and scalar meson states overlap. We set $ma=10^{-5}$ and $N=40$ in both panels.
002946648 8564_ $$82790028$$s57915$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_36_g_1.0_Q_1.6_plot.png$$y00050 Snapshots of the information lattice for the time evolution with injected external charges as in Fig.~\ref{fig:string_full_iln}, now for $Q=\{1.6,2.4,3,6\}$, $ga=1$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790029$$s40860$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_12_g_1.0_Q_2.4_plot.png$$y00052 Snapshots of the information lattice for the time evolution with injected external charges as in Fig.~\ref{fig:string_full_iln}, now for $Q=\{1.6,2.4,3,6\}$, $ga=1$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790030$$s93561$$uhttps://cds.cern.ch/record/2946648/files/I_l_distribution_plot_g_1.0_charge_2.4.png$$y00059 Partially integrated information per scale $\bar{I}(\ell)$ (see main text) for the time evolution illustrated in Fig.~\ref{fig:string_full_iln_2}.
002946648 8564_ $$82790031$$s99925$$uhttps://cds.cern.ch/record/2946648/files/Q3p2_Efield.png$$y00032 Time evolution of the expectation value of the onsite electric field and bipartite entanglement entropy for $Q=\{1,2.6,3.2\}$ for the quench in which the initially injected external charges are removed at time $t=12 \, a$, indicated by the vertical white dashed line. We set $ga=1$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790032$$s38316$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_6_g_0.5_Q_2.0_plot.png$$y00035 Snapshots of the information lattice for the time evolution with injected external charges for $Q=\{2, 3.4, 4\}$. We set $ga=0.5$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790033$$s53842$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_36_g_1.0_Q_2.4_plot.png$$y00053 Snapshots of the information lattice for the time evolution with injected external charges as in Fig.~\ref{fig:string_full_iln}, now for $Q=\{1.6,2.4,3,6\}$, $ga=1$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790034$$s72413$$uhttps://cds.cern.ch/record/2946648/files/schematic.png$$y00000 (a) Information lattice for a product state of qubits, e.g., $\ket{\Omega_\mathrm{s.c.}}$ in Eq.~\eqref{eq:vac_SC}. (b) Illustration of the formula for local information in Eq.~\eqref{eq:local_info_formula} for $i(3.5,3)$; red denotes positive contributions and blue negative ones. The underlying green area shows the subsytem $\mathcal{C}^{3}_{3.5}$. (c) Information per scale for the ground state $\ket{\Omega_{\mathrm{s.c.}}}$ and first excited states $\ket{1_{\mathrm{V,S}}}$ of the Schwinger model in the strong coupling limit in Eqs.~\eqref{eq:vac_SC} and \eqref{eq:vector_meson_strong_coupling}, respectively, with $N=20$. Only scales $\ell < 7$ are shown.
002946648 8564_ $$82790035$$s47679$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_0_k_1.3_plot.png$$y00012 Plots analogous to Fig.~\ref{fig:scattering_k07}, now for $ka=1.3$. We set $ga=1$, $ma=10^{-5}$, and $N=40$.
002946648 8564_ $$82790036$$s43901$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_12_g_0.5_Q_2.0_plot.png$$y00036 Snapshots of the information lattice for the time evolution with injected external charges for $Q=\{2, 3.4, 4\}$. We set $ga=0.5$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790037$$s41900$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_12_g_0.5_Q_4.0_plot.png$$y00042 Snapshots of the information lattice for the time evolution with injected external charges for $Q=\{2, 3.4, 4\}$. We set $ga=0.5$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790038$$s82582$$uhttps://cds.cern.ch/record/2946648/files/Lavg2sites_minus_L0_central_vs_time_with_colorbands.png$$y00034 Time evolution of the average electric field $\overline{L}(t)$ at the center of the lattice (see main text) minus its initial value at $t=0$ for the quench in which the external charges are removed at $t=12 \, a$ as in Fig.~\ref{fig:E_field_Entropy_stopped}. Here we use $ma=0.25$ and $ga=\{0.5,1\}$, with varying values of the external charge $Q$. The plot shows the results from the instant at which the external charges are removed from the system.
002946648 8564_ $$82790039$$s44287$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_20_g_1.0_Q_2.6_plot.png$$y00069 Snapshots of the information lattice for the quench in which the charges are removed at $t=12 \, a$ as in Fig.~\ref{fig:E_field_Entropy_stopped} for $Q=\{1,2,2.6,3.2\}$, $ga=1$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790040$$s88674$$uhttps://cds.cern.ch/record/2946648/files/I_l_distribution_plot_g_0.5_charge_3.4.png$$y00046 Partially integrated information per scale $\bar{I}(\ell)$ (see main text) for the time evolution illustrated in Fig.~\ref{fig:string_full_iln}.
002946648 8564_ $$82790041$$s37019$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_6_g_1.0_Q_2.4_plot.png$$y00051 Snapshots of the information lattice for the time evolution with injected external charges as in Fig.~\ref{fig:string_full_iln}, now for $Q=\{1.6,2.4,3,6\}$, $ga=1$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790042$$s116877$$uhttps://cds.cern.ch/record/2946648/files/Entropy_k_0.7.png$$y00006 Bipartite entanglement entropy across a cut between sites $n$ and $n+1$, i.e., for the region from site 1 to $n$. We set $ga=1$, $ma=10^{-5}$, and $N=40$. \textbf{Left}: Scattering below particle threshold ($ka=0.7$). Vertical white lines indicate the time slices used in Figs.~\ref{fig:scattering_k07} and \ref{fig:scattering_k13}. \textbf{Right}: Scattering above threshold ($ka=1.3$). The white box highlights the region used to compute the distributions in Fig.~\ref{fig:cut_Iln_scattering}.
002946648 8564_ $$82790043$$s109663$$uhttps://cds.cern.ch/record/2946648/files/Entropy_k_1.3.png$$y00007 Bipartite entanglement entropy across a cut between sites $n$ and $n+1$, i.e., for the region from site 1 to $n$. We set $ga=1$, $ma=10^{-5}$, and $N=40$. \textbf{Left}: Scattering below particle threshold ($ka=0.7$). Vertical white lines indicate the time slices used in Figs.~\ref{fig:scattering_k07} and \ref{fig:scattering_k13}. \textbf{Right}: Scattering above threshold ($ka=1.3$). The white box highlights the region used to compute the distributions in Fig.~\ref{fig:cut_Iln_scattering}.
002946648 8564_ $$82790044$$s111133$$uhttps://cds.cern.ch/record/2946648/files/I_l_distribution_plot_g_1.0_k_1.2.png$$y00018 Integrated information distribution for the central region of the scattering process shown in Fig.~\ref{fig:entropy} (right) for $ka=\{0.7,1,1.2,1.3\}$. Thicker lines highlight times multiples of $5a$.
002946648 8564_ $$82790045$$s148313$$uhttps://cds.cern.ch/record/2946648/files/I_l_distribution_plot_g_1.0_k_1.3.png$$y00019 Integrated information distribution for the central region of the scattering process shown in Fig.~\ref{fig:entropy} (right) for $ka=\{0.7,1,1.2,1.3\}$. Thicker lines highlight times multiples of $5a$.
002946648 8564_ $$82790046$$s43529$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_20_g_1.0_Q_3.2_plot.png$$y00072 Snapshots of the information lattice for the quench in which the charges are removed at $t=12 \, a$ as in Fig.~\ref{fig:E_field_Entropy_stopped} for $Q=\{1,2,2.6,3.2\}$, $ga=1$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790047$$s95183$$uhttps://cds.cern.ch/record/2946648/files/Q2p8_Entropy.png$$y00021 Time evolution of the expectation value of the onsite electric field $L(n,t)-L(n,0)$ and the bipartite entanglement entropy $\mathcal{S}(n)$ for quenches with injected external charges using $Q=\{2.8,4\}$, $ga=0.5$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790048$$s41054$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_12_g_1.0_Q_1.6_plot.png$$y00049 Snapshots of the information lattice for the time evolution with injected external charges as in Fig.~\ref{fig:string_full_iln}, now for $Q=\{1.6,2.4,3,6\}$, $ga=1$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790049$$s90588$$uhttps://cds.cern.ch/record/2946648/files/I_l_distribution_plot_g_0.5_charge_4.0.png$$y00047 Partially integrated information per scale $\bar{I}(\ell)$ (see main text) for the time evolution illustrated in Fig.~\ref{fig:string_full_iln}.
002946648 8564_ $$82790050$$s126051$$uhttps://cds.cern.ch/record/2946648/files/I_l_distribution_plot_g_1.0_k_0.7.png$$y00016 Integrated information distribution for the central region of the scattering process shown in Fig.~\ref{fig:entropy} (right) for $ka=\{0.7,1,1.2,1.3\}$. Thicker lines highlight times multiples of $5a$.
002946648 8564_ $$82790051$$s41361$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_12_g_1.0_Q_2.0_plot.png$$y00065 Snapshots of the information lattice for the quench in which the charges are removed at $t=12 \, a$ as in Fig.~\ref{fig:E_field_Entropy_stopped} for $Q=\{1,2,2.6,3.2\}$, $ga=1$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790052$$s186971$$uhttps://cds.cern.ch/record/2946648/files/Q3p6_Entropy.png$$y00027 Plots analogous to Fig.~\ref{fig:11_full_charges}, now for $ga=1$. We use $Q=\{ 2.4,3.6 \}$, and set $ma=0.25$ and $N=100$.
002946648 8564_ $$82790053$$s30715$$uhttps://cds.cern.ch/record/2946648/files/vacuum_only.png$$y00003 Information lattice for $\ell \leq 6$ for the ground, vector meson, and scalar meson states in the Schwinger model for $ga=1$, $ma=10^{-5}$, and $N=40$. \textbf{Left}: The ground state's $i(n,\ell)$ distribution is dominated by correlations at $\ell \approx 1$ with a decaying tail. Note that the exact strong coupling vacuum is a product state, and thus, there one would have correlations only at $\ell=0$. \textbf{Right}: Difference between the $i(n,\ell)$ distributions for the first excited state and the ground state. The vector meson state is dominated by higher-level correlations for $\ell \gtrsim 3$. \textbf{Center}: The difference between the ground state and the scalar meson $i(n,\ell)$ distributions, which shows that this excited state has a strong enhancement of correlations at $\ell \approx \{3,4\}$.
002946648 8564_ $$82790054$$s118435$$uhttps://cds.cern.ch/record/2946648/files/Q1_Efield.png$$y00028 Time evolution of the expectation value of the onsite electric field and bipartite entanglement entropy for $Q=\{1,2.6,3.2\}$ for the quench in which the initially injected external charges are removed at time $t=12 \, a$, indicated by the vertical white dashed line. We set $ga=1$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790055$$s42822$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_12_g_0.5_Q_3.4_plot.png$$y00039 Snapshots of the information lattice for the time evolution with injected external charges for $Q=\{2, 3.4, 4\}$. We set $ga=0.5$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790056$$s47361$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_30_g_1.0_Q_2.0_plot.png$$y00067 Snapshots of the information lattice for the quench in which the charges are removed at $t=12 \, a$ as in Fig.~\ref{fig:E_field_Entropy_stopped} for $Q=\{1,2,2.6,3.2\}$, $ga=1$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790057$$s57140$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_36_g_0.5_Q_3.4_plot.png$$y00040 Snapshots of the information lattice for the time evolution with injected external charges for $Q=\{2, 3.4, 4\}$. We set $ga=0.5$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790058$$s93459$$uhttps://cds.cern.ch/record/2946648/files/Q4_Entropy.png$$y00023 Time evolution of the expectation value of the onsite electric field $L(n,t)-L(n,0)$ and the bipartite entanglement entropy $\mathcal{S}(n)$ for quenches with injected external charges using $Q=\{2.8,4\}$, $ga=0.5$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790059$$s93765$$uhttps://cds.cern.ch/record/2946648/files/Q2p6_Efield.png$$y00030 Time evolution of the expectation value of the onsite electric field and bipartite entanglement entropy for $Q=\{1,2.6,3.2\}$ for the quench in which the initially injected external charges are removed at time $t=12 \, a$, indicated by the vertical white dashed line. We set $ga=1$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790060$$s50160$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_15_k_0.7_plot.png$$y00009 Snapshots of the information lattice for the scattering of two wave packets as in Eq.~\eqref{eq:jet_init} for $ka=0.7$. The selected times correspond to the dashed vertical lines shown in Fig.~\ref{fig:entropy}. We set $ga=1$, $ma=10^{-5}$, and $N=40$.
002946648 8564_ $$82790061$$s37409$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_12_g_1.0_Q_3.6_plot.png$$y00055 Snapshots of the information lattice for the time evolution with injected external charges as in Fig.~\ref{fig:string_full_iln}, now for $Q=\{1.6,2.4,3,6\}$, $ga=1$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790062$$s97267$$uhttps://cds.cern.ch/record/2946648/files/Q4_Efield.png$$y00022 Time evolution of the expectation value of the onsite electric field $L(n,t)-L(n,0)$ and the bipartite entanglement entropy $\mathcal{S}(n)$ for quenches with injected external charges using $Q=\{2.8,4\}$, $ga=0.5$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790063$$s38288$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_6_g_0.5_Q_4.0_plot.png$$y00041 Snapshots of the information lattice for the time evolution with injected external charges for $Q=\{2, 3.4, 4\}$. We set $ga=0.5$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790064$$s55610$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_36_g_0.5_Q_4.0_plot.png$$y00043 Snapshots of the information lattice for the time evolution with injected external charges for $Q=\{2, 3.4, 4\}$. We set $ga=0.5$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790065$$s47949$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_30_g_1.0_Q_2.6_plot.png$$y00070 Snapshots of the information lattice for the quench in which the charges are removed at $t=12 \, a$ as in Fig.~\ref{fig:E_field_Entropy_stopped} for $Q=\{1,2,2.6,3.2\}$, $ga=1$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790066$$s3727247$$uhttps://cds.cern.ch/record/2946648/files/2510.16101.pdf$$yFulltext
002946648 8564_ $$82790067$$s47649$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_30_g_1.0_Q_3.2_plot.png$$y00073 Snapshots of the information lattice for the quench in which the charges are removed at $t=12 \, a$ as in Fig.~\ref{fig:E_field_Entropy_stopped} for $Q=\{1,2,2.6,3.2\}$, $ga=1$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790068$$s47629$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_0_k_0.7_plot.png$$y00008 Snapshots of the information lattice for the scattering of two wave packets as in Eq.~\eqref{eq:jet_init} for $ka=0.7$. The selected times correspond to the dashed vertical lines shown in Fig.~\ref{fig:entropy}. We set $ga=1$, $ma=10^{-5}$, and $N=40$.
002946648 8564_ $$82790069$$s108232$$uhttps://cds.cern.ch/record/2946648/files/Q2p4_Entropy.png$$y00025 Plots analogous to Fig.~\ref{fig:11_full_charges}, now for $ga=1$. We use $Q=\{ 2.4,3.6 \}$, and set $ma=0.25$ and $N=100$.
002946648 8564_ $$82790070$$s82535$$uhttps://cds.cern.ch/record/2946648/files/Q3p2_Entropy.png$$y00033 Time evolution of the expectation value of the onsite electric field and bipartite entanglement entropy for $Q=\{1,2.6,3.2\}$ for the quench in which the initially injected external charges are removed at time $t=12 \, a$, indicated by the vertical white dashed line. We set $ga=1$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790071$$s194225$$uhttps://cds.cern.ch/record/2946648/files/Q3p6_Efield.png$$y00026 Plots analogous to Fig.~\ref{fig:11_full_charges}, now for $ga=1$. We use $Q=\{ 2.4,3.6 \}$, and set $ma=0.25$ and $N=100$.
002946648 8564_ $$82790072$$s81146$$uhttps://cds.cern.ch/record/2946648/files/I_l_distribution_plot_g_0.5_charge_2.8.png$$y00045 Partially integrated information per scale $\bar{I}(\ell)$ (see main text) for the time evolution illustrated in Fig.~\ref{fig:string_full_iln}.
002946648 8564_ $$82790073$$s40664$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_12_g_1.0_Q_2.6_plot.png$$y00068 Snapshots of the information lattice for the quench in which the charges are removed at $t=12 \, a$ as in Fig.~\ref{fig:E_field_Entropy_stopped} for $Q=\{1,2,2.6,3.2\}$, $ga=1$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790074$$s78773$$uhttps://cds.cern.ch/record/2946648/files/I_l_distribution_plot_g_1.0_charge_1.6.png$$y00057 Partially integrated information per scale $\bar{I}(\ell)$ (see main text) for the time evolution illustrated in Fig.~\ref{fig:string_full_iln_2}.
002946648 8564_ $$82790075$$s28490$$uhttps://cds.cern.ch/record/2946648/files/lmax_vs_time_all_charges_g_0.5.png$$y00061 Peak position $\ell_{\mathrm{max}}$ for the $\bar{I}(\ell)$ distribution (see main text) for $ga=0.5$ and several $Q$ values as a function of time. The dotted dashed gray line indicates the slope for a ballistic propagation at half the light speed on the lattice. We set $ma=0.25$ and $N=100$.
002946648 8564_ $$82790076$$s54604$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_35_k_1.3_plot.png$$y00015 Plots analogous to Fig.~\ref{fig:scattering_k07}, now for $ka=1.3$. We set $ga=1$, $ma=10^{-5}$, and $N=40$.
002946648 8564_ $$82790077$$s44903$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_20_g_1.0_Q_1.0_plot.png$$y00063 Snapshots of the information lattice for the quench in which the charges are removed at $t=12 \, a$ as in Fig.~\ref{fig:E_field_Entropy_stopped} for $Q=\{1,2,2.6,3.2\}$, $ga=1$, $ma=0.25$, and $N=100$.
002946648 8564_ $$82790078$$s19308$$uhttps://cds.cern.ch/record/2946648/files/mass_spectrum.png$$y00001 \textbf{Left:} Mass spectrum as a function of the expectation value of the squared pseudo momentum operator $P =-i \sum_n \left( \sigma_{n}^- \sigma^z_{n+1} \sigma^+_{n+2}- {\rm h.c.} \right)$ for $ga=1$. Here, $\mathcal{M}_i= E_i-E_{\rm vac}$ is the energy gap to the vacuum of the $i$-th state. The vector ($i=1$) and scalar ($i=20$) states (gold stars) are identified by having the minimal momentum and exhibiting a mass gap. Their identification was further confirmed by checking their parity. Blue circular markers denote finite momentum excitations of the vector meson, which appear only in the lattice theory. The results for mass gaps of the identified states agree quantitatively with those reported in Ref.~\cite{Papaefstathiou:2024zsu}. \textbf{Right:} $I(\ell)$ distribution for the vacuum, vector, and scalar meson states identified in the left panel. $I(\ell)$ is also shown for the same states at $ga=2$, which are identified through the same DMRG procedure. Note that for $ga=2$ the curves of the vector and scalar meson states overlap. We set $ma=10^{-5}$ and $N=40$ in both panels.
002946648 8564_ $$82790079$$s38044$$uhttps://cds.cern.ch/record/2946648/files/vector_only.png$$y00004 Information lattice for $\ell \leq 6$ for the ground, vector meson, and scalar meson states in the Schwinger model for $ga=1$, $ma=10^{-5}$, and $N=40$. \textbf{Left}: The ground state's $i(n,\ell)$ distribution is dominated by correlations at $\ell \approx 1$ with a decaying tail. Note that the exact strong coupling vacuum is a product state, and thus, there one would have correlations only at $\ell=0$. \textbf{Right}: Difference between the $i(n,\ell)$ distributions for the first excited state and the ground state. The vector meson state is dominated by higher-level correlations for $\ell \gtrsim 3$. \textbf{Center}: The difference between the ground state and the scalar meson $i(n,\ell)$ distributions, which shows that this excited state has a strong enhancement of correlations at $\ell \approx \{3,4\}$.
002946648 8564_ $$82790080$$s51786$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_15_k_1.3_plot.png$$y00013 Plots analogous to Fig.~\ref{fig:scattering_k07}, now for $ka=1.3$. We set $ga=1$, $ma=10^{-5}$, and $N=40$.
002946648 8564_ $$82790081$$s53177$$uhttps://cds.cern.ch/record/2946648/files/i_l_time_35_k_0.7_plot.png$$y00011 Snapshots of the information lattice for the scattering of two wave packets as in Eq.~\eqref{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 $N=40$.
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