  We study the effect of bond dissipation on the one-dimensional
antiferromagnetic spin-$1/2$ Heisenberg model. In analogy to the spin-Peierls
problem, the dissipative bath is described by local harmonic oscillators that
modulate the spin exchange coupling, but instead of a single boson frequency we
consider a continuous bath spectrum $\propto \omega^s$. Using an exact quantum
Monte Carlo method for retarded interactions, we show that for $s<1$ any finite
coupling to the bath induces valence-bond-solid order, whereas for $s>1$ the
critical phase of the isolated chain remains stable up to a finite critical
coupling. We find that, even in the presence of the gapless bosonic spectrum,
the spin-triplet gap remains well defined for any system size, from which we
extract a dynamical critical exponent of $z=1$. We provide evidence for a
Berezinskii-Kosterlitz-Thouless quantum phase transition that is governed by
the SU(2)$_1$ Wess-Zumino-Witten model. Our results suggest that the critical
properties of the dissipative system are the same as for the spin-Peierls
model, irrespective of the different interaction range, i.e., power-law vs.
exponential decay, of the retarded dimer-dimer interaction, indicating that the
spin-Peierls criticality is robust with respect to the bosonic density of
states.
