Fractional Fermi numbers in Dirac systems with topological point defects

By revisiting the fractional Fermi number through the perspective of spectral asymmetry, a unified approach to fractional Fermi numbers in the presence of topological point defects is established based on complex analysis. When applied to the generic Dirac system, our approach shows that the fractional Fermi number is ultimately expressed as a surface integral evaluated over a sphere surrounding the defect at spatial infinity. This result clearly demonstrates that fractional Fermi number is determined by the defect's global topology, rather than its local details. We further apply our approach to concrete Dirac systems in various dimensions. For the one-dimensional case, it directly reproduces the seminal Goldstone-Wilczek formula. As for the two-dimensional case, it yields a fractional Fermi number proportional to the topological charge of the defect (${N}_{0}$), even when the chiral symmetry is broken by an additional mass term. If the chiral symmetry is restored as the mass approaches zero, the fractional Fermi number is quantized to $\ifmmode\pm\else\textpm\fi{}{N}_{0}/2$ with the sign inherited from the mass even in this limit. Additionally, we provide a supersymmetric interpretation about the bound states. For the three-dimensional massive Dirac system, our method yields an elegant formula of fractional Fermi number proportional to the topological charge of the defect (${N}_{1}$). If the chiral symmetry is restored, the fractional Fermi number is quantized to $\ifmmode\pm\else\textpm\fi{}{N}_{1}/2$.