  We consider multiple non-interacting quantum mechanical two-level systems
coupled to a common bosonic bath and study its quantum phase transition with
Monte Carlo simulations using a continuous imaginary time cluster algorithm.
The common bath induces an effective ferromagnetic interaction between the
otherwise independent two-level systems, which can be quantified by an
effective interaction strength. For degenerate energy levels above a critical
value of the bath coupling strength $\alpha$ all two-level systems freeze into
the same state and the critical value $\alpha_c$ decreases asymptotically as
$1/N$ with increasing $N$. For a finite number, $N$, of two-level systems the
quantum phase transition (at zero temperature) is in the same universality
class as the single spin-boson model, in the limit $N\to\infty$ the system
shows mean-field critical behavior independent of the power of the spectral
function of the bosonic bath. We also study the influence of a spatial
separation of the spins in a bath of bosonic modes with linear dispersion
relation on the location and characteristics of the phase transition as well as
on correlations between the two-level systems.
