I am trying to understand the notion of edge modes in the context of symmetry protected topological order (SPTO) and its relation (if there is any) with the virtual quantum register that sits at the virtual bond of the MPS representation of the SPTO quantum state. For concreteness, let us take the example of the 1D cluster Hamiltonian with open boundary conditions Hc=−∑i=2N−1Zi−1XiZi+1\displaystyle H_c= -\sum_{i=2}^{N-1}Z_{i-1}X_{i}Z_{i+1} which has a fourfold degeneracy. I have seen the following statements made in this context but am not sure what they precisely mean. Clarifying them further would significantly improve my comprehension. The model has a ground state with a symmetric bulk, while the action of the symmetry on the edges is said to fractionalize. In the non-trivial SPT phase, it gives rise to spin- 12\frac{1}{2} degrees of freedom localized on the edges of the chain and independent of the dynamics. On a related matter, the idea of "edge modes" (I am not sure what that term exactly means) seem to be inextricably linked to the existence of degeneracy. Does that mean if one added the terms like X1Z2X_1Z_2 and ZN−1XNZ_{N-1}X_N and removed the degeneracy of the ground state, the "edge modes" would be lost too ? If so, does the 1D cluster state itself have no edge modes associated with it? Finally, is there any relation between edge modes, projective representations and MPS virtual registers since they all seem to be used to encode and process quantum information in SPT phases? P.S. I apologize for not being more succinct in my question. However, this is the best I could manage, as I frequently encounter these related concepts but lack a comprehensive understanding of their interrelationship.