Volledigheid van de classificatie van symmetriebeschermde topologische toestanden

This thesis investigates the structure of topological phases of quantum lattice systems. We focus on the classification of unique gapped ground states and symmetric short-range entangled states in one and two spatial dimensions. Our main results establish the completeness of well-known classification indices, proving that they form complete invariants with respect to stable equivalence of states. The principal technical challenge lies in constructing the corresponding adiabatic paths that connect equivalent states, both in the presence and absence of symmetries. In the first part of this thesis, we study unique gapped ground states of one-dimensional quantum spin systems. We first show that such systems are topologically trivial in the bulk, in the sense that their ground states are short-range entangled. We then incorporate symmetries modelled by compact topological groups and classify the resulting symmetry-protected topological (SPT) phases of spin chains. In particular, we prove that the well-known cohomological classification is complete under stable equivalence: if two SPT states share the same index, then they can be connected by an adiabatic evolution generated by a local, symmetric Hamiltonian. In the second part, we investigate two-dimensional SPT phases and their relation to one-dimensional locality-preserving symmetries (LPSs). We show that stable equivalence classes of LPSs form a group that is isomorphic to the third cohomology group for a finite symmetry group G. Moreover, LPSs emerge at the boundaries of two-dimensional SPT states that are entangled by globally symmetric finite-depth quantum circuits, and we prove that such a subclass of SPT states is completely classified by stable equivalence classes of boundary LPSs.