Anomalous flux state in higher-order topological superconductors

We investigate the anomalous flux state of interacting higher-order topological superconductors (HOTSC) protected by rotation symmetries. By introducing a $\pi$ superconducting flux in 2D HOTSC, we demonstrate the existence of a robust zero mode trapped at the flux center. Remarkably, the rotation symmetry and fermion parity display projective representation inside the $\pi$ flux with $N=2$ supersymmetry algebra. A similar gapless flux pattern also exists in 3D HOTSCs, the flux lines of which carry anomalous helical modes that cannot be realized on purely one-dimensional lattice models. Notably, these exotic phenomena can be manifested in a 2D frustrated quantum magnet whose low energy excitation characterizes emergent Majoranas with HOTSC band structure and the $Z_2$ flux exhibiting supersymmetries.