Physical meaning of gapped path between Hamiltonians in the same phase

I’m reading this famous paper about the classification of quantum phases, and I’m wondering about the physical meaning of the definition of phases the authors use.

They say that two Hamiltonians $H_0$ and $H_1$ are in the same phase if there exists a continuous path $gammamapsto H_gamma$ of gapped Hamiltonians interpolating them. In the discussion, they say that this is also equivalent to the existance of a set of quasilocal unitaries mapping the ground state of $H_0$ to the ground state of $H_1$.

What does this definition mean, physically? More precisely, suppose we have to Hamiltonians for which either of these two statements are true

• All continuous paths from $H_0$ to $H_1$ must have a gapless point

• It is not possible to go from one ground state to the other using only quasilocal unitaries

why should I think of them as being in two different phases?