Continuous Topological Phase Transition between Two 1D Antiferromagnetic Spin-1 Boson Superfluids with the Same Symmetry

Spin-1 bosons on a one-dimensional chain, at incommensurate filling with an antiferromagnetic spin interaction between neighboring bosons, may form a spin-1 boson condensed state that contains both a 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 number 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 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.

Figure 1

For right-moving fermions with charge $q$ and $U_t(1)$-charge $q_t$ on a ring, adding a $2\pi /q$ flux of $U(1)$ through the ring will create a fermion and add a $U_t(1)$-charge $q_t$. So the $U(1)\times U_t(1)$ mixed anomaly for the right-moving fermions is given by $q{q}_t$, i.e., $2\pi$ flux of $U(1)$ will create a $U_t(1)$-charge $q{q}_t$. Similarly, for left-moving fermions with charge $q$ and $U_t(1)$-charge $-q_t$, the $U(1)\times U_t(1)$ mixed anomaly is $q{q}_t$.