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

Abstract

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.