  We investigate the fate of dissipative phase transitions in quantum many-body
systems when the individual constituents are qudits ($d$-level systems) instead
of qubits. As an example system, we employ a permutation-invariant $XY$ model
of $N$ infinite-range interacting $d$-level spins undergoing individual and
collective dissipation. In the mean-field limit, we identify a dissipative
phase transition, whose critical point is independent of $d$ after a suitable
rescaling of parameters. When the decay rates between all adjacent levels are
identical and $d\geq 4$, the critical point expands, in terms of the ratio
between dissipation and interaction strengths, to a critical region in which
two phases coexist and which increases as $d$ grows. In addition, a larger $d$
leads to a more pronounced change in spin expectation values at the critical
point. Numerical investigations for finite $N$ reveal symmetry breaking
signatures in the Liouvillian spectrum at the phase transition. The phase
transition is furthermore marked by maximum entanglement negativity and a
significant purity change of the steady state, which become more pronounced as
$d$ increases. Considering qudits instead of qubits thus opens new perspectives
on accessing rich phase diagrams in open many-body systems.
