Emergence of gaussianity in the thermodynamic limit of interacting fermions

Systems of interacting fermions can give rise to ground states whose correlations become effectively free-fermion-like in the thermodynamic limit, as shown by Baxter for a class of integrable models that include the one-dimensional XYZ spin-½ chain. Here, we quantitatively analyze this behavior by establishing the relation between system size and correlation length required for the fermionic gaussianity to emerge. Importantly, we demonstrate that this behavior can be observed through the applicability of Wick's theorem, and thus, it is experimentally accessible. To establish the relevance of our results to possible experimental realizations of XYZ-related models, we demonstrate that the emergent gaussianity is insensitive to weak variations in the range of interactions, coupling inhomogeneities, and local random potentials.