Charge of the vacuum in (1+1)-dimensional fermion theories with external fields

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Abstract
In the framework (1+1)-dimensional fermion theories with scalar, pseudoscalar, and vector external fields we discuss the relation between the charge of the vaccum and the topological properties of the external fields. For theories with a gap above zero in the particle spectrum we prove the known relation between the charge of the vacuum and the behavior at infinity of the scalar and pseudoscalar external fields, and show that a vector potential alters the charge of the vacuum only if it is generated by a nonvanishing external charge. Some solvable models are also presented and discussed.