Effects of interactions on the topological classification of free fermion systems Can someone who read this paper (ArXiv, APS) by Fidkowski and Kitaev explain to me the two highlighted statements in the following    extract?

This is because for four Majorana chains the only possible quartic
  interaction $W$ is proportional to $\hat{c}_1 \hat{c}_2 \hat{c}_3 \hat{c}_4$, and regardless of its sign leaves a doubly degenerate
  ground state for each group of four corresponding sites, leading to a
  gapless system at the midpoint of the path when the kinetic terms are
  turned off.

 A: Four Majorana modes $c_1,c_2,c_3,c_4$ together make two fermionic modes and hence a $2^2$-dimensional Hilbert space. Since we are describing the edge of our topological material, we are thus describing a state with a fourfold degeneracy on one of the edges. Let us now impose an interaction term $H = c_1c_2c_3c_4$ on this space. Note that this term squares to one, so the only energy eigenvalues are $\pm 1$. In fact due to the symmetric nature of this term it is not hard to see that there must be as many positive as negative eigenvalues. So in conclusion this term has the eigenvalues $+1,+1,-1,-1$ on our four dimensional space. So now we have lifted our fourfold degeneracy on the edge to a twofold degeneracy for the ground state of our interaction term.
As long as the bulk has a gap, we are justified in just looking at the edge degrees of freedom. But since the above is the only term that can be added, there will always be a twofold degeneracy on the edge. So if we ever want to interpolate our quantum state to a trivial product state, then we will have to close the bulk gap. I.e. there will have to be a phase transition! (The statement in bold is then a special case: they are discussing a path they would try to construct to avoid any phase transition, an intermediate state of which involves taking the kinetic terms going to zero, but as we have just argued any path will have to close the bulk gap.)
