Master's Defense: Niels Gustav Nortvig Willesen
Field theory modelling of non-interacting junctions with induced potentials
Nanowires are an instrumental component of modern nano-technology and so a network of leads forming a star graph is considered. The leads are being modelled as being one dimensional, and we consider each channel within the lead to be induced by a potential, which physically is manifested as the energies of transverse modes in the lead. We consider arbitrary values of these induced potentials. At the vertex of the graph is a point-like and non-interacting general scatterer – and by finding a self-adjoint extension to the Hamiltonian, we are able to explicitly construct the unitary scattering matrix. The scattering matrix depend on the boundary conditions of the wavefunctions in each lead, as well as the energy of the incoming electron. By connecting each lead to a thermal reservoir, the system is away from equilibrium and it admits a non-equilibrium steady state.
Equipped with the scattering matrix formalism, we construct a quantum field theory on the star graph and calculate the electric current, particle density, differential conductance and the heat current. The $x$-dependence of the particle density is manifested as Friedel oscillations. Furthermore, the electric current is constant along edges, and thus the state constitute a steady state. Finally, we study the quantum quench protocol for switching from an equilibrium state without potentials, to suddenly enter a non-equilibrium regime and turning on the potentials.