Bound-state confinement after trap-expansion dynamics in integrable systems

Integrable systems possess stable families of quasiparticles, which are composite objects (bound states) of elementary excitations. Motivated by recent quantum computer experiments, we investigate bound-state transport in the spin-$1/2$ anisotropic Heisenberg chain ($XXZ$ chain). Specifically, we consider the sudden vacuum expansion of a finite region $A$ prepared in a non-equilibrium state. In the hydrodynamic regime, if interactions are strong enough, bound states remain confined in the initial region. Bound-state confinement persists until the density of unbound excitations remains finite in the bulk of $A$. Since region $A$ is finite, at asymptotically long times bound states are "liberated" after the "evaporation" of all the unbound excitations. Fingerprints of confinement are visible in the space-time profiles of local spin-projection operators. To be specific, here we focus on the expansion of the $p$-N\'eel states, which are obtained by repetition of a unit cell with $p$ up spins followed by $p$ down spins. Upon increasing $p$, the bound-state content is enhanced. In the limit $p\to\infty$ one obtains the domain-wall initial state. We show that for $p<4$, only bound states with $n>p$ are confined at large chain anisotropy. For $p\gtrsim 4$, also bound states with $n=p$ are confined, consistent with the absence of transport in the limit $p\to\infty$. The scenario of bound-state confinement leads to a hierarchy of timescales at which bound states of different sizes are liberated, which is also reflected in the dynamics of the von Neumann entropy.