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arXiv:2502.12901 · Physical Review Letters

Emergent Nonthermal Fluid from Jets in the Massive Schwinger Model Using Tensor Networks

Romuald A. Janik, Maciej A. Nowak, Marek M. Rams, Ismail Zahed A

Published Nov 20, 2025• 6 citations
arXivHTMLPDF

Abstract

We analyze the correlation between the energy, momentum, and spatial entanglement produced by two luminal jets in the massive Schwinger model. Using tensor network methods, we show that for m/g>1/π, in the vicinity of the strong- to weak-coupling transition, a nearly perfect and chargeless effective fluid behavior appears around the midrapidity region with a universal energy-pressure relationship. The evolution of energy and pressure is strongly correlated with the rise of the spatial entanglement entropy, indicating a key role of quantum dynamics. Some of these observations may be used to analyze high multiplicity jet fragmentation events, energy-energy and energy-charge correlators at current collider energies.

DOI
10.1103/gvr2-gqys

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We analyze the correlation between the energy, momentum, and spatial entanglement produced by two luminal jets in the massive Schwinger model. Using tensor network methods, we show that for m/g>1/π, in the vicinity of the strong- to weak-coupling transition, a nearly perfect and chargeless effective fluid behavior appears around the midrapidity region with a universal energy-pressure relationship. The evolution of energy and pressure is strongly correlated with the rise of the spatial entanglement entropy, indicating a key role of quantum dynamics. Some of these observations may be used to analyze high multiplicity jet fragmentation events, energy-energy and energy-charge correlators at current collider energies.
We investigate the emergence of thermalization in a quantum-field-theoretic model mimicking the production of jets in QCD: the massive lattice Schwinger model coupled to external sources. Specifically, we compute the expectation values of local operators as functions of time and compare them to their thermal counterparts, quantify the overlap between the evolving density matrix and the thermal one, and compare the dynamics of the energy-momentum tensor to predictions from relativistic hydrodynamics. Through these studies, we find that the system approaches thermalization at late times and elucidate the mechanisms by which quantum entanglement drives thermalization in closed field-theoretic systems. Our results show how thermodynamic behavior emerges in real time from unitary quantum dynamics.
The possible link between entanglement and thermalization, and the dynamics of hadronization are addressed by studying the real-time response of the massive Schwinger model coupled to external sources. This setup mimics the production and fragmentation of quark jets, as the Schwinger model and quantum chromodynamics (QCD) share the properties of confinement and chiral symmetry breaking. By using simulations of quantum dynamics on classical hardware, we study the entanglement between the produced jets, and observe the growth of the corresponding entanglement entropy in time. This growth arises from the increased number of contributing eigenstates of the reduced density matrix with sufficiently large and close eigenvalues. We also investigate the physical nature of these eigenstates, and find that at early times they correspond to fermionic Fock states. We then observe the transition from these fermionic Fock states to mesonlike bound states as a function of time. In other words, we observe how hadronization develops in real time. At late times, the local observables at midrapidity (such as the fermion density and the electric field) approach approximately constant values, suggesting the onset of equilibrium and approach to thermalization. Published by the American Physical Society 2024
We simulate the real-time dynamics of a massive Schwinger model using the Time-Evolving Block Decimation tensor network algorithm. Starting from a non-equilibrium initial state with localized energy excitation on top of vacuum, we track the subsequent evolution to investigate two distinct physical phenomena. First, by analyzing the system's energy-momentum tensor, we show that this system exhibits hydrodynamic behavior analogous to Bjorken flow at large coupling-to-mass ratio, a signature that diminishes as the coupling weakens, or mass increases. Second, by examining the evolution of the electric field and charge density, we observe the signal of spontaneous parity symmetry breaking phase transition in a dynamical system. The parity-restored regime is marked by ''string breaking'' and efficient charge screening, while the parity-broken regime displays stable propagation of nearly free charges and persistent electric fields connecting them.
In this work, we investigate the potential connection between entanglement and thermalization, as well as the dynamics of hadronization in QCD physics. Through simulations of quantum dynamics on classical hardware, we analyze the real-time response of the massive Schwinger model coupled to external sources, which serves as an analog for the production and fragmentation of quark jets. This analogy is justified by the shared properties of confinement and chiral symmetry breaking between the Schwinger model and QCD. We observe the growth of time-dependent entanglement between the produced jets, which is driven by the increasing number of contributing eigenstates of the reduced density matrix with sufficiently large and closely spaced eigenvalues. This growth is intrinsically linked to the process of thermalization. The emergent eigenstates are identified as meson-like bound states, and by tracking their temporal evolution, we are able to observe real-time hadronization. At mid-rapidity, the long-time values of local observables approach approximately constant values, indicating the onset of equilibrium and the approach to thermalization.
The onset of hydrodynamics in the hot medium created in relativistic heavy-ion collisions is a crucial theoretical question. A first-principle simulation requires a real-time, non-perturbative calculation of the quantum system. In this Letter, we perform such simulations using the tensor network method, which enables large-scale quantum many-body simulations by retaining only the most essential quantum states for collective behaviors. We focus on the massive Schwinger model, a low-dimensional analog of quantum chromodynamics (QCD), as they share important properties such as confinement and chiral symmetry breaking. Starting from an initial state that puts a localized excitation atop the vacuum and mimics the energy deposition from colliding nuclei, we observe hydrodynamic behavior consistent with Bjorken flow in all relevant degrees of freedom: energy density, fluid velocity, and bulk pressure. The time scale for hydrodynamic onset aligns with the thermalization time of the quantum distribution function.