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

TL;DR

This paper explores a continuous topological phase transition between two distinct 1D anti-ferromagnetic spin-1 boson superfluids with the same symmetry, characterized by a conformal field theory with spin fractionalization.

Contribution

It identifies a novel continuous phase transition between two symmetry-protected topological superfluids in 1D spin-1 boson systems, described by a specific conformal field theory.

Findings

Two superfluids differ by a spin-$SO(3)$ topological order.

The phase transition is described by a specific CFT with spin fractionalization.

The critical theory is determined via partition function and symmetry analysis.

Abstract

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.