  We study the excitation spectrum of the one-dimensional spin-1/2 XXZ chain
with antiferromagnetic Ising anisotropy across a magnetic quantum phase
transition induced by the application of a site-dependent transverse magnetic
field. Motivated by the chain antiferromagnet BaCo$_2$V$_2$O$_8$, we consider a
situation where the transverse magnetic field has a strong uniform component
and a weaker staggered part. To determine the nature of the excitations giving
rise to the spin dynamical structure factor, we use a combination of analytical
approaches and the numerically exact time-dependent matrix product state
method. We identify below the quantum phase transition high-energy many-body
two-magnon and three-magnon repulsively bound states which are clearly visible
due to the staggered component of the magnetic field. At high magnetic fields
and low temperature, single magnons dominate the dynamics. Our theory results
are in very good agreement with terahertz spectroscopy experimental results
presented in [Wang et al., Nature 631, 760 (2024)].
