Ground state energy of an interacting electron system in the background of two opposite magnetic strings

Motivated by our earlier work, we show in this paper rigorously that the ground state energy and degeneracy of an infinitely extended system of interacting electrons in the background of a homogeneous magnetic field and two separated magnetic strings of opposite strength is the same as for the system without strings. By using symmetry considerations we obtain further that the energy spectrum does not depend on the string separation distance for strictly positive distances. As a side effect of our considerations, we obtain a virial theorem for the two string system in the case of a homogeneous interaction potential which has the same form as the virial theorem without the strings.