Effects on a metal that is proximately coupled to hyperbolic photon modes

The hyperbolic mode (HM) refers to a polariton mode in a polar insulator where the dielectric function is negative in some direction of propagation. Within a frequency window the light occupies a greatly expanded region in momentum space. The HM in hexagonal Boron Nitride (hBN) has been under intense study and we consider placing a metal directly on top of hBN and ask whether its physical properties can be strongly affected. While the problem resembles superficially the electron phonon coupling problem, there are important differences. Due to the longitudinal nature of the HM mode the coupling is driven by time dependent charge fluctuations which results in a coupling that strongly increases with the energy difference of the initial and final states. We find a significant frequency and momentum dependence of the self energy which allows us to identify the dimensionless coupling constant $\lambda_0$ that controls this effect. There is a suppression of the quasi-particle weight but it turns out that the leading correction to the velocity renormalization is canceled. We compute the single particle spectral function which shows a side band that can be measured experimentally. The virtual exchange of HM leads to a repulsive interaction which is ineffective to leading order because of the energy dependence. We are motivated to seek pairing by going to second order. Unfortunately we find that the leading contributions exactly cancel. This cancellation is not an accident and we give an explanation of why this cancellation would take place.