I will discuss a pair of coupled partial differential equations identified as the Bogoliubov-de Gennes equations known in superconductivity but with Dirac operators replacing the usual Laplace operators and show that the Dirac version appears naturally for inhomogeneous quantum liquids. The equations feature an effective local gap due to inhomogeneities, coupling right- and left-moving degrees of freedom, leading to scattering, and so far had not been solved in general. I will show that one can compute exact analytical solutions using Magnus expansions and study their properties to obtain information about the dynamics. The main physical motivation comes from so-called inhomogeneous Tomonaga-Luttinger liquids, but the equations also arise in descriptions of, for instance, superconductor-normal-metal interfaces, polymer chains, and coupled fractional quantum Hall edges.

Contact: Jérôme Dubail