Why number of left-moving and right-moving edge states on a finite lattice system is equal?

Question

I read an arguments about number of left-movers and right-mover in finite system in paper titled as Antichiral Edge States in a Modified Haldane Nanoribbon. In second paragraph of introduction, it says

In this Letter, we ask the following question: Is it possible to have a 2D fermionic system with copropagating edge modes, illustrated in Fig. 1(c)? A simple consideration shows that such “antichiral” edge modes cannot exist in a system with a full bulk gap. This is because the number of left- and right-moving modes in any finite system defined on the lattice must be the same.

Intuitively, the total velocity (or momentum) of the system should be 0, since it is a finite system standing still in the space. Thus, if two copropagating edge modes exist in bulk gap, it will lead to a nonzero total velocity. I understand this argument in such intuitive way above. First, I am not quite sure about my reasoning. Even if it is true, I am looking for a rigorous reasoning or explicit derivation to show the correctness of the argument in the paper.

Fig1.(c)

Answer 1

This paper https://arxiv.org/abs/1904.05491 proves that there can not be any net energy current in equilibrium state on a lattice system, which implies that left-moving and right-moving modes must balance.