Hierarchical vector model of a ferromagnet in the method of collective variables. The Lee–Yang theorem

A hierarchical vector model of a ferromagnet that approximates the corresponding translationally symmetric model with two-body interaction of arbitrary range is proposed. For such a model of arbitrary vector dimension it is shown that the zeros of the partition function are localized on the imaginary axis of magnetic field values. A sufficient condition is also proved for block-spin distributions whose partition functions have such a property.