Abstract
We consider different methods of calculating the (fractional) fermion number of solitons based on the heat kernel expansion. We derive a formula for the localized eta function that provides a more systematic version of the derivative expansion for spectral asymmetry and compute the fermion number in a multi flavor extension of the Goldstone–Wilczek model. We also propose an improved expansion of the heat kernel that allows the tackling of the convergence issues and permits an automated computation of the coefficients.
Highlights
More than 40 years ago Jackiw and Rebbi [1] discovered that the Fermi number of solitonic ground states may be halfinteger
A crucial step in understanding the fermion fractionization in solitonic backgrounds was given by Niemi and Semenoff in Reference [4] where it was reinterpreted in the context of QFT anomalies
The link between the existence of the fermion fractionization phenomenon in condensed matter physics and relativistic field theory was established in Reference [7] giving rise to one more connection between these two apparently distant areas of Physics
Summary
More than 40 years ago Jackiw and Rebbi [1] discovered that the Fermi number of solitonic ground states may be halfinteger. One needs certain kinds of the derivative expansion [2,9] This method is not free of problems. The purpose of this work is to reconsider and improve the calculation methods of the fractional fermion number of solitons by systematically using heat kernel techniques We shall concentrate on combinatorial aspect, derive a new expression for the fractional fermion number in a model with matrix-valued fields, and analyze the convergence issues in an improved version of the heat kernel expansion. We derive localized expressions for the η function and for the spectral asymmetry This allows us to use the standard heat kernel expansion and derive an expression for the fermion number in a non-abelian (multi flavor) generalization of the Goldstone– Wilczek model.
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