Abstract
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- degeneracy at the chain end. The two superfluids have the same symmetry and only differ by a spin- 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) . Such a CFT has a spin fractionalization: spin-1 excitation can decay into a spin- right mover and a spin- left mover. We determine the critical theory by solving the partition function based on emergent symmetries and modular invariance condition of CFTs.
- Received 1 November 2018
- Revised 24 April 2019
DOI:https://doi.org/10.1103/PhysRevLett.123.035301
© 2019 American Physical Society