Fisher zeros and conformality in lattice models

Abstract

Fisher zeros are the zeros of the partition function in the complex β = 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 β 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.

Renormalization Group and Phase Transitions in Spin, Gauge, and QCD Like Theories

“…It has inspired new concepts and computational methods in many branches of physics. In the lattice gauge theory, however, we recently found out that the conventional Migdal-Kadanoff (MK) method does not serve the purpose of high accuracy calculations due to its crude "bond sliding" approximation [57]. This inspired us to to develop new method to "block spin" the system accurately and provide a tool to study lattice models near conformality.…”

“…The corresponding polynomial expansion becomes R n (k) = a n,0 + a n,1 k 2 + a n,2 k 4 + ... + a n,lmax k 2lmax , 57) or equivalently (only even power of k is involved due to Z(2) symmetry of the φ field), R n (u) = a n,0 + a n,1 u + a n,2 u 2 + ... + a n,lmax u lmax . (2.58)…”

Fisher's zeros for SU(3) with Nf flavors and RG flows

“…We usually call the partition function zeros in the complex z = e 2β h plane Lee-Yang zeros [6,7,8] and those in the complex temperature plane Fisher zeros. Interestingly, one can perform finite-size scaling analysis on the zeros and extract critical exponents from them [9,10,11]. It was also argued that Fisher zeros act as separatrices for the RG flows in the complex coupling plane [12,13,14].…”