Instanton size dependence on fermion energy spectra in a ℂP2 fermionic sigma model

Abstract

Spectral flow is a fascinating behavior of fermion’s energy levels that cross zero as a parameter varies. In fermion-soliton system, it owes the topology of the soliton. We examine the spectral flow in a (2 + 1)-dimensional ℂP2 fermionic sigma model. We employ an axisymmetric instanton solution of the ℂP2 non-linear sigma model as a background bosonic field to the Dirac fermion. The explicit form of the solutions contains two parameters concerning the size. We show that some energy levels flow from positive (negative) to negative (positive) as the parameters vary. We propose that the behavior of the energy levels can easily be understood by carefully examining the topological property of the instanton in the entire range of parameters, including the limit of zero and infinity.