Quantum Wires and Field Theory

摘要

Quantum graphs are networks of one-dimensional wires connected at nodes. The interest for such structures increased these recent times with the development of nano-scale technology. We focus our attention on star graphs made of n edges with one junction. The related bosonic fields propagate in the bulk, either freely or submitted to a four-fermion interaction, and interact at the vertex, which can be considered as a defect. Hereafter, a quantum field theoretical framework is developed and applied to the computation of physical quantities, such that the electric and spin conductance. More precisely, our approach combines results from the spectral theory of the Schrodinger operator on quantum graphs with an algebraic technique for dealing with quantum fields with defects (impurities). At the vertex, all possible interactions preserving unitarity are taken into account, but special attention is given to scale-invariant ones, which lead to the critical properties of the system. Then bosonisation and vertex operators on quantum graphs are investigated to solve exactly, for scale invariant boundary conditions, the four-fermion bulk interaction (Tomonaga-Luttinger model). At this point, we are in position to derive the charge and spin transport, and establish a simple relationship among them.