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 $\gamma\mapsto H_\gamma$ 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?