Repulsively Bound Hadrons in a $\mathbb{Z}_2$ Lattice Gauge Theory

Abstract

A paradigmatic model, the $\mathbb{Z}_2$ lattice gauge theory exhibits confinement mediated by the gauge field that binds pairs of particles into mesons, drawing connections to quantum chromodynamics. In the absence of any additional attractive interactions between particles, mesons are not known to bind in this model. Here, we show that resonant pair-production terms give rise to an additional repulsive binding mechanism that forms a stable ``hadron'' bound state of two mesons. A high-energy state, the hadron is stabilized by being off-resonantly coupled to a continuum. We study the dynamical formation of this bound state starting from local excitations. We use matrix product state techniques based on the time-evolving block decimation algorithm to perform our numerical simulations and analyze the effect of model parameters on hadron formation. Furthermore, we derive an effective model that explains its formation. Our findings are amenable to experimental observation on modern quantum hardware from superconducting qubits to trapped ions.

Figures (3)

• Figure 1: a. We show here examples of spin and matter configurations for the $\mathbb{Z}_2$ lattice gauge theory defined by the Hamiltonian Eq. \ref{['eq:H_fsp']}. The circles correspond to matter particles which are only present when there is a domain wall in gauge spins. We call two matter particles separated by $n$ spins an $n$-meson and four particles next to each other a tetraquark$\ket{q_4}$ state. b. The effective model showing all the relevant $3$-meson and tetraquark states occupying different center-of-mass positions $c$, and separated $1$-meson states with different relative separations $r$ along with the corresponding diagonal and off-diagonal matrix elements connecting them.
• Figure 2: Density plots of the local particle number and the local tetraquark number in the limit $m=h \gg J,K$ as a function of time, $Jt$ for the $3$-meson initial state in a chain of length $L=100$. We only show the central 50 sites. Top panel: Data for $K=0.1<J^2/h$ where we qualitatively see long-lived repulsively bound tetraquark and $3$-meson state. Middle panel: Data for $K=0.2 \sim J^2/h$. In this regime, we see enhanced tetraquark states at short times which decay to the continuum at longer time. Bottom panel: Data for $K=0.8>J^2/h$. In this regime, we see oscillations of bound $3$-meson and tetraquark.
• Figure 3: a,b. Long-time average of the total tetraquark number $\langle \overline{\hat{q}_{4}}\rangle$ and the total $3$-meson number $\langle \overline{\hat{m}_{3i}}\rangle$ as a function of $Kh/J^2$. The data points are computed using TEBD simulation of the full Hamiltonian while the black curve denotes simulation of the effective model. We see that as we increase $m=h$, the TEBD data approaches the effective model. c. The plot of $\langle \overline{\hat{m}_{3i}}\rangle+\langle \overline{\hat{q}_{4i}}\rangle$ shows that as we increase $Kh/J^2$, we nonmonotonically go from a bound hadronic state (repulsively bound) to enhanced dissociation to bound hadronic state again. Inset: We show the time series plots of $\langle \hat{q}_4\rangle$ and $\langle \hat{m}_3\rangle$ for different values of $K$ and setting $h=m=10J$ showing long-time oscillatory asymptote. For $m=h=8J,10J$ and $K\leq 0.25J$, we simulate (TEBD) up to $T=200J^{-1}$ because of slow oscillations.