  Symmetry provides important insight in understanding the nature of phase
transitions. In the presence of crystalline symmetries, new phenomena in phase
transition can emerge, such as intertwined orders and emergent symmetries. In
this work, we present a group-theoretical diagnosis of these phenomena in 32
crystallographic point groups. For each group, we classify the possible
combinations of intertwined order and analyze different symmetry-broken phases.
We find that the symmetries before and after the phase transition uniquely
determine the irreducible representation (irrep) of the order parameter and the
allowed form of Ginzburg-Landau free energy, from which the nature of phase
transition can be inferred, including the order of phase transition and the
existence of intermediate phases and emergent symmetries. Our finding will be
helpful for the experimental diagnosis of order parameters from symmetry
breaking.

Mesoscale and Nanoscale Physics;Strongly Correlated Electrons