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Lines of Fisher's zeros as separatrices for complex renormalization group flows

We extend the renormalization group transformation based on the two-lattice matching to the complex inverse temperature plane for Dyson's hierarchical Ising model. We consider values of the dimensional parameter above, below and exactly at the critical value where the ordered low temperature phase becomes impossible for a real positive temperature. We show numerically that, as the volume increases, the Fisher's zeros appear to accumulate along lines that separate the flows ending on different fixed points. We justify these findings in terms of finite size scaling. We argue that the location of the Fisher's zeros at large volume determine the phase diagram in the complex plane. We discuss the implications for nontrivial infrared fixed points in lattice gauge theory.