Title: The phase transitions between $Z_n\times Z_n$ bosonic topological phases in 1+1 D, and a constraint on the central charge for the critical points between bosonic symmetry protected topological phases,
Abstract: The study of continuous phase transitions triggered by spontaneous symmetry
breaking has brought revolutionary ideas to physics. Recently, through the
discovery of symmetry protected topological phases, it is realized that
continuous quantum phase transition can also occur between states with the same
symmetry but different topology. Here we study a specific class of such phase
transitions in 1+1 dimensions -- the phase transition between bosonic
topological phases protected by $Z_n\times Z_n$. We find in all cases the
critical point possesses two gap opening relevant operators: one leads to a
Landau-forbidden symmetry breaking phase transition and the other to the
topological phase transition. We also obtained a constraint on the central
charge for general phase transitions between symmetry protected bosonic
topological phases in 1+1D.