A topological phase is a phase of matter which cannot be characterized by a local order parameter. It has been shown that gapped symmetric phases in one-dimensional (1D) systems can be completely characterized using tools related to projective representations of the symmetry groups. We explain two ways to detect these symmetry protected topological phases in 1D. First, we give a numerical approach for directly extracting the projective representations from a matrix-product state representation. Second, we derive nonlocal order parameters for time-reversal and inversion symmetry, and discuss a generalized string order for local symmetries for which the regular string-order parameter cannot be applied. We furthermore point out that the nonlocal order parameter for these topological phases is actually related to topological surfaces.     «

A topological phase is a phase of matter which cannot be characterized by a local order parameter. It has been shown that gapped symmetric phases in one-dimensional (1D) systems can be completely characterized using tools related to projective representations of the symmetry groups. We explain two ways to detect these symmetry protected topological phases in 1D. First, we give a numerical approach for directly extracting the projective representations from a matrix-product state representation....     »