Nematic Order in Topological SYK models

Motivated by ongoing experiments on diverse quantum materials, we study a class of multi-orbital Sachdev-Ye-Kitaev (SYK) models on the lattice which exhibit non-Fermi liquid physics at high temperature and develop spontaneous nematic order at low temperature. Starting with a non-interacting two-orbital model for higher Chern number topological insulators (proposed by Vanderbos, Hu and Kane), we study the impact of SYK interactions on these lattice models which feature quadratic band touching points at the topological-to-trivial transition. The resulting SYK lattice model displays average symmetries upon disorder averaging, including a charge conjugation symmetry, so they are topological class D interacting models enriched by crystalline rotation symmetry. The phase diagram of this model features a non-Fermi liquid at high temperature and an exciton condensate with nematic order at low temperature. We also explore multi-orbital variants of such models on 2D and 3D lattices which can be described by Potts nematic order. These models also exhibit intriguing orbital-selective behavior. This order has consequences for the optical conductivity, which can be computed exactly in these models.