# Mathematics Colloquia and Seminars

The ground states of two Hamiltonians $H(0)$ and $H(1)$ are said to be in the same phase if there exists a smooth family of uniformly gapped Hamiltonians $H(s)$ interpolating between the two systems. In this joint work with S.~Michalakis, B.~Nachtergaele and R.~Sims, we prove a general, non perturbative result that supports this definition, namely that it implies the unitary equivalence of the ground states. The latter can be implemented as an $s$-dependent flow of unitaries that we construct explicitly. In the case where the Hamiltonians $H(s)$ have a local structure, the analysis can be extended to the thermodynamic limit.