Fisher's zeros as boundary of RG flows in complex coupling space

TL;DR

This paper explores the extension of renormalization group flows into complex coupling spaces, identifying Fisher's zeros as boundaries of attraction basins and providing numerical evidence across various models.

Contribution

It introduces the concept of Fisher's zeros as boundaries of RG flows in complex coupling space and supports this with numerical calculations in multiple models.

Findings

Fisher's zeros mark the boundary of IR fixed point basins in complex coupling space.

Zeros stabilize away from the real axis in large volume SU(2) lattice gauge theory.

Adding an adjoint term causes zeros to approach the real axis.

Abstract

We discuss the possibility of extending the RG flows to complex coupling spaces. We argue that the Fisher's zeros are located at the boundary of the complex basin of attraction of IR fixed points. We support this picture with numerical calculations at finite volume for2D O(N) models in the large-N limit and the hierarchical Ising model using the two-lattice matching method. We present numerical evidence supporting the idea that, as the volume increases, the Fisher's zeros of 4-dimensional pure gauge SU(2) lattice gauge theory with a Wilson action, stabilize at a distance larger than 0.1 from the real axis in the complex beta=4/g^2 plane. We show that when a positive adjoint term is added, the zeros get closer to the real axis. We compare the situation with the U(1) case. We discuss the implications of this new framework for proofs of confinement and searches for nontrivial IR fixed points in models beyond the standard model.