Symmetry protected topological (SPT) phases are characterized by robust boundary features, which do not disappear unless passing through a phase transition. These boundary features can be quantified by a topological invariant which, in some cases, is related to a physical quantity, such as the spin conductivity for the quantum spin Hall insulators. In other cases, the boundary features give rise to new physics, such as the Majorana fermion. In all cases the boundary features can be analyzed with the help of an entanglement spectrum and their robustness make them promising candidates for storing quantum information. The topological invariant characterizing SPT phases is strictly only invariant under deformations which respect a certain
symmetry. For example, the boundary currents of the quantum spin Hall insulator are only robust against non-magnetic, i.e. time-reversal invariant, impurities. In this thesis we study the SPT phases of spin chains.
As a result of our work we find a topological invariant for SPT phases of spin chains which are protected by continuous symmetries. By means of a non-local order parameter we find a way to extract this invariant from the ground state wave function of the system. Using density-matrix-renormalization-group techniques we verify that this invariant is a tool to detect transitions between different topological phases. We find a non-local transformation that maps SPT phases to conventional phases characterized by a local order parameter. This transformation suggests an analogy between topological phases and conventional phases and thus give a deeper understanding of the role of topology in spin systems
