A continuous topological phase transition between two 1D anti-ferromagnetic spin-1 boson superfluids with the same symmetry

Spin-1 bosons on a 1-dimensional chain, at incommensurate filling with anti-ferromagnetic spin interaction between neighboring bosons, may form a spin-1 boson condensed state that contains both gapless charge and spin excitations. We argue that the spin-1 boson condensed state is unstable, and will become one of two superfluids by opening a spin gap. One superfluid must have a spin-1 ground state on a ring if it contains an odd number of bosons and has no degenerate states at the chain end. The other superfluid has a spin-0 ground state on a ring for any numbers of bosons and has a spin-$\frac{1}{2}$ degeneracy at the chain end. The two superfluids have the same symmetry and only differ by a spin-$SO(3)$ symmetry protected topological order. Although Landau theory forbids a continuous phase transition between two phases with the same symmetry, the phase transition between the two superfluids can be generically continuous, which is described by a conformal field theory (CFT) $su(2)_2\oplus u(1)_4 \oplus \overline{su(2)}_2\oplus \overline{u(1)}_4$. Such a CFT has a spin fractionalization: spin-1 excitation can decay into a spin-$\frac{1}{2}$ right mover and a spin-$\frac{1}{2}$ left mover. We determine the critical theory by solving the partition function based on emergent symmetries and modular invariance condition of CFTs.