Interacting One-Dimensional Fermionic Symmetry-Protected Topological Phases

In free fermion systems with given symmetry and dimension, the possible topological phases are labeled by elements of only three types of Abelian groups, 0, ${\mathbb{Z}}_2$, or $\mathbb{Z}$. For example, noninteracting one-dimensional fermionic superconducting phases with $S_z$ spin rotation and time-reversal symmetries are classified by $\mathbb{Z}$. We show that with weak interactions, this classification reduces to ${\mathbb{Z}}_4$. Using group cohomology, one can additionally show that there are only four distinct phases for such one-dimensional superconductors even with strong interactions. Comparing their projective representations, we find that all these four symmetry-protected topological phases can be realized with free fermions. Further, we show that one-dimensional fermionic superconducting phases with $Z_n$ discrete $S_z$ spin rotation and time-reversal symmetries are classified by ${\mathbb{Z}}_4$ when $n$ is even and ${\mathbb{Z}}_2$ when $n$ is odd; again, all these strongly interacting topological phases can be realized by noninteracting fermions. Our approach can be applied to systems with other symmetries to see which one-dimensional topological phases can be realized with free fermions.

Figure 1

Phase diagram when varying the parameters ${\Delta }_s$ and ${\Delta }_p$: The phase boundaries are ${\Delta }_s=\pm {\Delta }_p$, which separate three phases denoted by $N=+1$, 0, and $-1$. We can make ${\Delta }_s$ arbitrarily large without closing the gap: this limit describes on-site pairing where any cut cleanly separates the system into two without leaving edge states—allowing identification of this trivial $N=0$ phase.

Figure 2

We stack two chains in the same nontrivial phase with positive (or negative) index to see if their edge states are stable. With the first chain $a=1$ and the second $b=2$, $M_{ab}$ is any coupling between them. We find that all possible couplings are forbidden by our system symmetries, and so the two similar modes are stable and form the $N=2$ phase.

Figure 3

Without interactions $V_{1234}=0$, the ground state degeneracy for four zero modes is $2^4=16$. With interactions $V_{1234}\ne 0$, the two states are split by $\delta E=\pm |V_{1234}|$, making the ground state nondegenerate and this $N=4$ phase trivial. As this $N=4$ phase is now smoothly connected to the trivial $N=0$ phase, our classification reduces from $\mathbb{Z}$ to ${\mathbb{Z}}_4$ with interactions.