Complex renormalization group flows for 2D nonlinear O(N) sigma models

Motivated by recent attempts to find nontrivial infrared fixed points in 4-dimensional lattice gauge theories, we discuss the extension of the renormalization group transformations to complex coupling spaces for O(N) models on L x L lattices, in the large-N limit. We explain the Riemann sheet structure and singular points of the finite L mappings between the mass gap and the 't Hooft coupling. We argue that the Fisher's zeros appear on "strings" ending approximately near these singular points. We show that for the spherical model at finite N and L, the density of states is stripwise polynomial in the complex energy plane. We compare finite volume complex flows obtained from the rescaling of the ultraviolet cutoff in the gap equation and from the two-lattice matching. In both cases, the flows are channelled through the singular points and end at the strong coupling fixed points, however strong scheme dependence appear when the Compton wavelength of the mass gap is larger than the linear size of the system. We argue that the Fisher's zeros control the global properties of the complex flows. We briefly discuss the implications for perturbation theory, proofs of confinement, and searches for nontrivial infrared fixed points in models beyond the standard model.