In this paper, we study the geometry of reduced density matrices for states with symmetry-protected topological (SPT) order. We observe ruled surface structures on the boundary of the convex set of low-dimensional projections of the reduced density matrices. In order to signal the SPT order using ruled surfaces, it is important that we add a symmetry-breaking term to the boundary of the system—no ruled surface emerges in systems without a boundary or when we add a symmetry-breaking term representing a thermodynamic quantity. Although the ruled surfaces only appear in the thermodynamic limit where the ground-state degeneracy is exact, we analyze the precision of our numerical algorithm and show that a finite-system calculation suffices to reveal the ruled surface structures.

• Received 28 September 2015

DOI:https://doi.org/10.1103/PhysRevA.93.012309