Nonlocal Fermions with Local Interactions and the SYK Model

One of the most promising routes to non-fermi liquids and strange metals has been through SYK models \cite{Sachdev:2010um}, which necessarily involve large flavor degrees of freedom and interactions with imposed disorder. We introduce an interacting model of nonlocal spinless fermions in which large flavor and effective disorder emerge spontaneously in the extreme nonlocal limit. This model may be thought of as an expansion in a dimensionless "locality scale," $\alpha$, with the limit $\alpha \rightarrow 0$ recovering a conventional local action. For finite $\alpha$, the resulting interacting nonlocal action exhibits a large number of low-energy degrees of freedom -- proportional to $\alpha$ -- and the structure factor mediating the local interactions between these fermions is effectively random for large $\alpha$. In one dimension, with finite $\alpha$, we show that interactions are marginal to the one loop level (as they are in the conventional local case), preserving a gapless phase. At large $\alpha$ the interaction strength becomes comparable to the bandwidth and we analyze this large flavor limit in a manner similar to the diagrammatic approach to SYK. As $\alpha$ is increased we argue that the gapless phase established by conventional RG possibly crosses over to a gapless SYK phase, although the "melon" diagrams are not exclusively dominant. We speculate on how this model might arise physically from an S-matrix connecting pure excited states -- rather than the vacuum -- to access finite temperature interacting fermions.