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The Sachdev-Ye-Kitaev model is a $(0+1)$-dimensional model describingMajorana fermions or complex fermions with random interactions. This model hasvarious interesting properties such as approximate local criticality (power lawcorrelation in time), zero temperature entropy, and quantum chaos. In thisarticle, we propose a higher dimensional generalization of theSachdev-Ye-Kitaev model, which is a lattice model with $N$ Majorana fermions ateach site and random interactions between them. Our model can be defined onarbitrary lattices in arbitrary spatial dimensions. In the large $N$ limit, thehigher dimensional model preserves many properties of the Sachdev-Ye-Kitaevmodel such as local criticality in two-point functions, zero temperatureentropy and chaos measured by the out-of-time-ordered correlation functions. Inaddition, we obtain new properties unique to higher dimensions such asdiffusive energy transport and a "butterfly velocity" describing thepropagation of chaos in space. We mainly present results for a$(1+1)$-dimensional example, and discuss the general case near the end.

Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models