String Breaking and Glueball Dynamics in $2+1$D Quantum Link Electrodynamics

TL;DR

This paper analyzes flux-string dynamics in a $2+1$D $U(1)$ quantum link model with a spin-$1$ gauge representation using tensor networks, revealing rich confinement and string-breaking phenomena not present in lower-spin formulations. It shows that static charges of magnitude $q=1$ yield single-stage breaking, while $q=2$ enables a two-stage process, with a finite plaquette term $g_B$ essential for genuine $2+1$D dynamics. The authors extend the study to out-of-equilibrium quenches, uncovering resonant and off-resonant dynamics, including glueball formation when non-minimal strings are excited, and they propose efficient qudit circuits for digital quantum simulation on trapped-ion platforms. These findings illuminate how gauge-field truncation and higher-spin representations shape string behavior and provide practical pathways toward quantum-field-theory-limit simulations. The work thus bridges tensor-network studies, quantum simulation, and gauge theory phenomenology, offering concrete routes to observe these effects in state-of-the-art hardware.

Abstract

At the heart of quark confinement and hadronization, the physics of flux strings has recently become a focal point in the field of quantum simulation of high-energy physics (HEP). Despite considerable progress, a detailed understanding of the behavior of flux strings in quantum simulation-relevant lattice formulations of gauge theories has remained limited to the lowest truncations of the gauge field, which are severely limited in their ability to draw conclusions about the quantum field theory limit. Here, we employ tensor network simulations to investigate the behavior of flux strings in a quantum link formulation of $2+1$D quantum electrodynamics (QED) with a spin-$1$ representation of the gauge field. We first map out the ground-state phase diagram of this model in the presence of two spatially separated static charges, revealing distinct microscopic processes responsible for string breaking, including a two-stage breaking mechanism not possible in the spin-$\frac{1}{2}$ formulation. Starting in different initial product state string configurations, we then explore far-from-equilibrium quench dynamics across various parameter regimes, demonstrating genuine $2+1$D real-time string breaking and glueball-like bound state formation, with the latter not possible in the spin-$\frac{1}{2}$ formulation. In and out of equilibrium, we consider different values and placements of the static charges. Finally, we provide efficient qudit circuits for a quantum simulation experiment in which our results can be observed in state-of-the-art ion-trap setups. Our findings lay the groundwork for quantum simulations of flux strings towards the quantum field theory limit.

Figures (11)

• Figure 1: Schematic illustrations and real-time snapshots of equilibrium and out-of-equilibrium string breaking and dynamics in the $2+1$D spin-$1$$\mathrm{U}(1)$ QLM. Equilibrium string breaking: (a) Schematic showing the breaking of a string of electric flux (illustrated in green) connecting opposite static charges of magnitude $q=1$, where increasing the electric field strength $g_E$ beyond a critical value $g_{E_1}$ leads to the creation of a pair of dynamical matter particles comprising of a positron and an electron. (b) Schematic of two-stage string breaking between opposite static charges of magnitude $q=2$. The minimal possible string is a closed loop connecting static charges. At the first critical field value $g_{E_1}$, the closed string partially breaks into a single string, producing a positron-electron pair, followed by a complete string breaking at the second critical field value $g_{E_2}$ with an additional positron-electron pair. Out-of-equilibrium dynamics: (c) Real-time snapshots of the dynamical breaking of an initial L-string at resonance $2m=g_E$. In the absence of the plaquette term ($g_B/\kappa=0$), string breaking is restricted to the initial configuration, resulting in effectively $1+1$D dynamics. When $g_B/\kappa \neq 0$, genuine $2+1$D string breaking occurs beyond the initial minimal string. (d) Snapshots illustrating the resonant breaking of a rectangular string connecting opposite static charges of magnitude $q=2$. For $g_B = 0$, the string breaks only along its initial configuration, whereas for $g_B \neq 0$, breaking also occurs outside the initial string. (e) Snapshots illustrating dynamical glueball formation (B1,B2) from a snake initial string quenched off resonance ($2m\neq g_E$).
• Figure 2: Equilibrium string breaking between static charges of magnitude $q=1$ with $+q$ placed at the odd lattice site $(6,1)$ and $-q$ placed at the even lattice site $(10,4)$ on a rectangular lattice of dimension $L_x=16$ and $L_y=6$ for the $2+1$D spin-$1$ U$(1)$ QLM as the electric coupling $g_E$ is increased at fixed $g_B$. String breaking is signaled by (a) a sharp increase in the vacuum-subtracted matter density $\langle \hat{n} \rangle - \langle \hat{n}_{\mathrm{vac}} \rangle$, corresponding to the creation of a positron-electron pair, and (b) a rapid drop in the entanglement entropy $\mathcal{S}$. A finite magnetic coupling $g_B$ stabilizes the confining string, shifting the breaking point to larger values of $g_E$. (c) Representative string state (showing only the central region containing the $\pm q$ static charges, indicated by red and blue circles, respectively) b) prior to breaking, computed at $g_E/\kappa=3.0$ and $g_B/\kappa = 0.1$, where the two static charges are connected by a superposition of electric flux string configurations. (d) Broken-string state at $g_E/\kappa = 5.0$ and $g_B/\kappa = 0.1$, in which the string disappears and an electron is created at the site $(6,1)$ and positron at the site $(10,4)$ ensuring full screening of the static charges. For all cases, we use $m = 6$.
• Figure 3: Equilibrium string breaking between static charges of magnitude $q=2$ with $+q$ placed at the odd lattice site $(6,1)$ and $-q$ placed at the even lattice site $(10,4)$ on a rectangular lattice of dimension $L_x=16$ and $L_y=6$ for the $2+1$D spin-$1$ U$(1)$ QLM as the electric coupling $g_E$ is increased at fixed $g_B$. The two-stage string breaking process is reflected in (a) two successive jumps in the vacuum-subtracted matter density $\langle \hat{n} \rangle - \langle \hat{n}_{\mathrm{vac}} \rangle$, each corresponding to the creation of a positron-electron pair, and (b) two corresponding jumps in the entanglement entropy $\mathcal{S}$. A finite magnetic coupling $g_B$ stabilizes the confining strings, shifting both breaking points to larger values of $g_E$. (c) Representative string state (showing only the central patch containing static charges) prior to breaking, computed at $g_E/\kappa = 3.0$ and $g_B/\kappa = 0.1$, where the static charges are connected by a superposition of electric flux string configurations. (d) Intermediate state after the first string breaking event, in which an electron is created at the site (6,1) and a positron at the site (10,4), while these two sites remain connected by a residual string. (e) Final fully broken string state after the second breaking event, in which the remaining string vanishes completely. The additional electron is distributed over the four odd lattice sites surrounding the lattice site $(6,1)$, while the additional positron is distributed over the four even lattice sites surrounding the lattice site $(10,4)$. All results are obtained for mass $m = 6$.
• Figure 4: Resonant dynamics of an L-string (see inset) between static charges of magnitude $q=1$ with $+q$ placed at the odd lattice site $(1,2)$ and $-q$ placed at the even lattice site $(4,4)$ on a $6\times6$ square lattice. The system is studied at $m/\kappa=12$ and $g_E/\kappa=24$ for different values of the magnetic coupling $g_B/\kappa$ in the $2+1$D spin-$1$ U$(1)$ QLM. (a) The fidelity $\mathcal{F}$ with respect to the initial string state. (b) The total overlap $\mathcal{P}_{\gamma\neq\gamma_{\text{i}}}$ with all minimal string configurations excluding the initial string. (c) The total matter occupation $\langle\hat{n}\rangle$ computed within the minimal patch containing the two static charges.
• Figure 5: Resonant dynamics of a rectangular string (see inset) between static charges of magnitude $q=2$ with $+q$ placed at the odd lattice site $(1,2)$ and $-q$ placed at the even lattice site $(4,4)$ on a $6\times6$ square lattice. The system is studied at $m/\kappa=12$ and $g_E/\kappa=24$ for different values of the magnetic coupling $g_B/\kappa$ in the $2+1$D spin-$1$ U$(1)$ QLM. (a) The fidelity $\mathcal{F}$ with respect to the initial string state. (b) The total overlap $\mathcal{P}_{\gamma\neq\gamma_{\text{i}}}$ with all minimal string configurations excluding the initial string. (c) The total matter occupation $\langle\hat{n}\rangle$ computed within the minimal patch containing the two static charges.