Geometric structure of unnecessary quantum critical points and their connection to SPT phase boundaries

Distinct phases of quantum systems, such as different symmetry-protected topological (SPT) phases protected by a symmetry group, are always separated by critical points. However, critical points, dubbed unnecessary quantum critical points (UQCPs), can also appear within a single phase and do not separate two distinct phases. In this work, we investigate the geometric structure of such UQCPs in the parameter space of quantum systems and their relation to phase boundaries of SPT phases. We show, using 0D and 1D fermionic systems, that the locus of UQCPs originates from phase boundaries of SPT phases that emerge when the system acquires an enhanced symmetry on certain submanifolds of the parameter space. Specifically, if P is the parameter space of a system with symmetry G, and contains UQCPs, there exists a submanifold P'⊆ P with higher symmetry G' ⊇ G, whose distinct G'-protected SPT phases are separated by phase boundaries. The manifold of UQCPs in P then forms a geometric ``pencil'' whose axis coincides with this SPT phase boundary. We conjecture that this finding holds for all unnecessary critical manifolds found in other systems and higher dimensions.