Local Density of States in a Helical Tomonaga-Luttinger Liquid of Loop and Josephson Junction Geometries

The local density of states (LDOS) in a one-dimensional helical channel of finite length is studied within a Tomonaga-Luttinger model at zero temperature. Two particular cases of loop and Josephson junction geometries are considered. The LDOS, as a function of energy $\omega$ measured from the Fermi level, consists of equally spaced spikes of the $\delta$-function type, and electron-electron interactions modify their relative height. It is shown that, in the loop geometry, the height of spikes decreases as $\omega \to 0$ everywhere in the system. It is also shown that, in the Josephson junction, the behavior of the LDOS significantly depends on the spatial position. At the end points of the junction, the height increases as $\omega \to 0$ and its variation is more pronounced than that in the loop case. Away from the end points, the height of spikes shows a non-monotonic $\omega$-dependence, which disappears in the long-junction limit.