Quadratic band touching with long range interactions in and out of equilibrium

Motivated by recent advances in cold atomic systems, we study the equilibrium and quench properties of two dimensional fermions with quadratic band touching at the Fermi level, in the presence of infinitely long range interactions. Unlike when only short range interactions are present, both nematic and quantum anomalous Hall (QAH) states state appear at weak interactions, separated by a narrow coexistence region, whose boundaries mark second and third order quantum phase transitions. After an interaction quench, the QAH order exhibits three distinct regions: persistent or damped oscillations and exponential decay to zero. In contrast, the nematic order always reaches a non-zero stationary value through power law damped oscillations, due to the interplay of the symmetry of the interaction and the specific topology of the quadratic band touching.