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We show that an ensemble of $^2\Sigma$ molecules in the rotationally groundstate trapped on an optical lattice exhibits collective spin excitations thatcan be controlled by applying superimposed electric and magnetic fields. Inparticular, we show that the lowest energy excitation of the molecular ensembleat certain combinations of electric and magnetic fields leads to the formationof a magnetic Frenkel exciton. The exciton bandwidth can be tuned by varyingthe electric or magnetic fields. We show that the exciton states can belocalized by creating vacancies in the optical lattice. The localizationpatterns of the magnetic exciton states are sensitive to the number anddistribution of vacancies, which can be exploited for engineering many-bodyentangled spin states. We consider the dynamics of magnetic exciton wavepacketsand show that the spin excitation transfer between molecules in an opticallattice can be accelerated or slowed down by tuning an external magnetic orelectric field.

External field control of collective spin excitations in an optical lattice of $^2\Sigma$ molecules