Effect of size on quantized conductance in a dirty quantum wire

Making use of the Zubarev correlation functions, we calculate and discuss the quantized conductance for a Tomonaga-Luttinger (TL) liquid coupled to phonons in a dirty quantum wire. In this system, the electron-phonon (e-p) coupling increases the conductance, while the impurities in the quantum wire reduce the conductance. The quantum wire length satisfies a certain condition, which determines whether e-p scattering will dominate over impurity scattering. However, the conductance is destroyed under certain conditions: namely, there is a size effect.