Fisher zeros and conformality in lattice models

TL;DR

This paper investigates Fisher zeros in various lattice models to understand phase transitions, confinement, and conformality, combining numerical results with analytical insights into RG flows and boundary effects.

Contribution

It provides new numerical results on Fisher zeros in multiple lattice models and explores analytical interpretations using RG flow approximations and boundary condition effects.

Findings

Fisher zeros pinching the real axis indicate phase transitions.

Gaps in Fisher zeros signal confinement and boundaries of the conformal window.

RG flow analysis reveals complex behavior near bulk transitions.

Abstract

Fisher zeros are the zeros of the partition function in the complex beta=2N_c/g^2 plane. When they pinch the real axis, finite size scaling allows one to distinguish between first and second order transition and to estimate exponents. On the other hand, a gap signals confinement and the method can be used to explore the boundary of the conformal window. We present recent numerical results for 2D O(N) sigma models, 4D U(1) and SU(2) pure gauge and SU(3) gauge theory with N_f=4 and 12 flavors. We discuss attempts to understand some of these results using analytical methods. We discuss the 2-lattice matching and qualitative aspects of the renormalization group (RG) flows in the Migdal-Kadanoff approximation, in particular how RG flows starting at large beta seem to move around regions where bulk transitions occur. We consider the effects of the boundary conditions on the nonperturbative part of the average energy and on the Fisher zeros for the 1D O(2) model.