The Quantum Complexity of String Breaking in the Schwinger Model

TL;DR

This work addresses how confinement-related string breaking in the 1+1D Schwinger model can be illuminated by quantum information concepts. The authors simulate the system with Matrix Product States under static background charges, compute gauge-invariant measures such as entanglement entropy, mutual information, and magic (RoM, SRE), including nonlocal variants, to track the evolution of the flux tube as the external separation $d$ grows. They find a pronounced peak in both entanglement and nonlocal magic near the string-breaking distance ($d \approx 46$), with correlations along the string diminishing once two separate mesons form, demonstrating that quantum complexity provides a complementary lens on confinement and hadronization. The results also reveal subtle boundary and mass dependencies and point to extensions to dynamics, non-Abelian theories, higher dimensions, and potential experimental relevance to fragmentation in high-energy collisions. Overall, the paper establishes quantum complexity as a useful diagnostic for confinement phenomena in gauge theories and offers a route to connect fundamental dynamics to observables in future experiments.

Abstract

String breaking, the process by which flux tubes fragment into hadronic states, is a hallmark of confinement in strongly-interacting quantum field theories. We examine a suite of quantum complexity measures using Matrix Product States to dissect the string breaking process in the 1+1D Schwinger model. We demonstrate the presence of nonlocal quantum correlations along the string that may affect fragmentation dynamics, and show that entanglement and magic offer complementary perspectives on string formation and breaking beyond conventional observables.

Figures (16)

• Figure 1: The formation of hadrons in the string breaking process. The energy density, indicated by the heatmap color, shows the linearly increasing potential with the separation $d$ of external charges (red and blue circles). A flux tube connecting the external charges develops (red tubes). At a certain separation, a pair of hadronic bound states is energetically favorable. Dynamical charges (circles with dashed border) are extracted from the vacuum to screen the external charges.
• Figure 2: The charge density $q_n$ through the string breaking process. Left: the vacuum-subtracted charge density obtained with simulation parameters described in the text using $N=880$ and $a=1/4$, as a function of position $n$ and external charge separation $d$. Right: cross sections of the charge density for a selection of $d$'s. The inset shows the slope of the charge density at the center of the lattice, which has a local extremum at $d=48.5$.
• Figure 3: Bipartite measures of entanglement and quantum complexity in string breaking. The symmetric bipartition entanglement entropy, antiflatness and the upper bound on the nonlocal ${\cal M}_2$ using simulation parameters $N=880, a=0.25, g=0.09, m=0.04601$, as a function of separation between static sources in terms of physical spatial sites. These quantities all peak at $d=46.5$ (see Table \ref{['tab:peaks']}).
• Figure 4: Local and multipartite entanglement in string breaking. Top: the local vacuum-subtracted MI as a function of distance from the center $r$ and external charge separation $d$ for $N=880, a=0.25, g=0.09, m=0.04601$. The inset shows cross sections of the MI for a selection of $d$'s. Bottom: the vacuum-subtracted 2-tangle for the same parameters. The inset on the bottom panel shows the vacuum-subtracted 4-tangle. The black dashed lines indicate the peak of the charge distribution shown in Fig. \ref{['fig:chargespatial']}. The red dashed lines correspond to the $d$ where the respective quantity changes sign for the smallest $r$.
• Figure 5: Internal structure of the outgoing meson revealed by the RoM. Top: the vacuum-subtracted RoM of adjacent sites, $\Delta R$ in units of $a$, shown for for $n \geq 440$ as a function of $d$, for $N=880,a=1/4$ as in Table \ref{['tab:parameters']}. The inset shows the behavior of $\Delta R$ inside the meson region for $d=75$. Bottom: the charge density $q_n$ for $n\leq440$ for the same system.