Normal transport and excitonic condensation in an incoherent semimetal

We study a two-band dispersive SYK model in $1+1$ dimension at half filling. We suggest a model that describes a semimetal with quadratic dispersion at half-filling. We compute the Green's function at the saddle point using a combination of analytical and numerical methods. Employing a scaling symmetry of the Schwinger Dyson equations that becomes transparent in the strongly dispersive limit, we show that the exact solution of the problem yields a distinct type of non-Fermi liquid with sublinear $\rho\propto T^{2/5}$ temperature dependence of the resistivity. A scaling analysis indicates that this state corresponds to the fixed point of the dispersive SYK model for a quadratic band touching semimetal. We examine the formation of indirect exciton condensation in a bilayer system constructed from the above model. We find that the condensation temperature scales as a fast power-law $T_{c}\propto g^{5}$, with $g$ the strength of the repulsive coupling between the layers.