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-$\frac{1}{2}$ chain. Here, we quantitatively analyse

this behaviour by establishing the relation between system size and correlation

length required for the fermionic Gaussianity to emerge. Importantly, we

demonstrate that this behaviour 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 realisations 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.