Effects of interactions on the topological classification of free fermion systems

We describe in detail a counterexample to the topological classification of free fermion systems. We deal with a one-dimensional chain of Majorana fermions with an unusual $T$ symmetry. The topological invariant for the free fermion classification lies in $\mathbb{Z}$, but with the introduction of interactions the $\mathbb{Z}$ is broken to ${\mathbb{Z}}_8$. We illustrate this in the microscopic model of the Majorana chain by constructing an explicit path between two distinct phases whose topological invariants are equal modulo 8, along which the system remains gapped. The path goes through a strongly interacting region. We also find the field-theory interpretation of this phenomenon. There is a second-order phase transition between the two phases in the free theory, which can be avoided by going through the strongly interacting region. We show that this transition is in the two-dimensional Ising universality class, where a first-order phase transition line, terminating at a second-order transition, can be avoided by going through the analog of a high-temperature paramagnetic phase. In fact, we construct the full phase diagram of the system as a function of the thermal operator (i.e., the mass term that tunes between the two phases in the free theory) and two quartic operators, obtaining a first-order Peierls transition region, a second-order transition region, and a region with no transition.

Figure 1

(Color online) Eigenvalues of $H=tT+\left(1-t\right)W_{\text{tot}}$ as a function of $t$. The system remains gapped throughout the path.

Figure 2

(Color online) Eigenvalues of $\tilde{H}=V+tT+w{W}_{\text{tot}}$ along the path $\left(t,w\right)=\left(0.1\text{cos}\theta ,0.1\text{sin}\theta \right),\theta ∊\left[0,\pi \right]$. The degeneracy is broken and the system remains gapped.

Figure 3

Smoothly connecting the $m>0$ and $m<0$ phases through an Ising transition region.

Figure 4

RG flow for $B$ ($x$ axis) and $A$ ($y$ axis).

Figure 5

Phase diagram for the transition from negative to positive $m$, indicating dependence on $B$ ($x$ axis) and $A$ ($y$ axis).