Zeroes in the Complex β Plane Of 2D Ising Block Spin Boltzmannians

Effective Boltzmannians in the sense of the block spin renormalization group are computed for the 2D Ising model. The blocking is done with majority and Kadanoff rules for blocks of size 2 by 2. Transfer matrix techniques allow the determination of the effective Boltzmannians as polynomials in u = exp(4β) for lattices of up to 4 by 4 blocks. The zeroes of these polynomials are computed for all non-equivalent block spin configurations. Their distribution in the complex β plane reflects the regularity structure of the block spin transformation. In the case of the Kadanoff rule spurious zeroes approach the positive real β axis at large values of β. They might be related to the renormalization group pathologies discussed in the literature.