The fermionic DGFF and its scaling limit logCFT

TL;DR

This paper demonstrates that the two-dimensional fermionic discrete Gaussian free field (fDGFF) converges in the scaling limit to a logarithmic conformal field theory, specifically symplectic fermions, linking discrete models to continuous CFTs.

Contribution

It establishes a precise correspondence between the fDGFF and symplectic fermions CFT, showing convergence of correlation functions and linking discrete models to logarithmic CFTs.

Findings

fDGFF correlation functions converge to CFT correlation functions

established a one-to-one correspondence between local observables and local fields

interpreted observables in spanning tree and sandpile models as CFT fields

Abstract

In this paper, we identify the scaling limit of the fermionic discrete Gaussian free field (fDGFF) as a logarithmic conformal field theory (CFT) in two dimensions. We first establish a one-to-one correspondence between the space of local observables of the fDGFF and the space of local fields of the symplectic fermions CFT, a logarithmic CFT with central charge $c = - 2$. This correspondence is meaningful in the sense that, when appropriately renormalised, the fDGFF correlation functions converge to corresponding CFT correlation functions in the scaling limit. As an application to these results, we interpret (the scaling limit of) certain local observables in the uniform spanning tree and the Abelian sandpile model as local fields of the symplectic fermions.