  Depending on the coupling to the environment, symmetries of open quantum
systems manifest in two distinct forms, the strong and the weak. We study the
spontaneous symmetry breaking among phases with different symmetries. Concrete
Liouvillian models with strong and weak symmetry are constructed, and different
scenarios of symmetry-breaking transitions are investigated from complementary
approaches. It is demonstrated that strong symmetry always spontaneously breaks
into the corresponding weak symmetry. For strong $U(1)$ symmetry, we show that
strong-to-weak symmetry breaking leads to gapless Goldstone modes dictating
diffusion of the symmetry charge in translational invariant systems. We
conjecture that this relation among strong-to-weak symmetry breaking, gapless
modes, and symmetry-charge diffusion is general for continuous symmetries. It
can be interpreted as an "enhanced Lieb-Schultz-Mattis (LSM) theorem" for open
quantum systems, according to which the gapless spectrum does not require
non-integer filling. We also investigate the scenario where the strong symmetry
breaks completely. In the symmetry-broken phase, we identify an effective
Keldysh action with two Goldstone modes, describing fluctuations of the order
parameter and diffusive hydrodynamics of the symmetry charge, respectively. For
a particular model studied here, we uncover a transition from a symmetric phase
with a "Bose surface" to a symmetry-broken phase with long-range order induced
by tuning the filling. It is also shown that the long-range order of $U(1)$
symmetry breaking is possible in spatial dimension $d\geq 3$, in both weak and
strong symmetry cases. Our work outline the typical scenarios of spontaneous
symmetry breaking in open quantum systems, and highlights their physical
consequences.
