Symmetry-protected topological states of interacting fermions and bosons

We study the classification for a large class of interacting fermionic and bosonic symmetry-protected topological (SPT) states, focusing on the cases where interaction reduces the classification of free-fermion SPT states. We define a SPT state as whether it is separated from the trivial state through a bulk phase transition, which is a general definition applicable to SPT states with or without spatial symmetries. We show that in all dimensions short-range interactions can reduce the classification of free-fermion SPT states, and we demonstrate these results by making a connection between the fermionic and bosonic SPT states. We first demonstrate that our formalism gives the correct classification for several known SPT states, with or without interaction, and then we generalize our method to SPT states that involve the spatial inversion symmetry.

Figure 1

Lattice model of the Kitaev Majorana chain.

Figure 2

Phase diagram of the interacting Majorana chain at $\nu =8$. The bulk criticality at the origin can be avoided if interaction is allowed. Each horizontal chain is four copies of the Majorana chain, equivalent to a Haldane chain. The vertical bound is the on-site spin-spin interaction. $A$ and $B$ labels the sites in a unit cell. Gray ovals mark out the spin-singlet dimers.

Figure 3

Many-body energy levels of the two-site segment by exact diagonalization. From the weak interaction limit (left) to the strong interaction (right) limit, the many-body gap never closes.