Density of states of interacting quantum wires with impurities: a Dyson equation approach

We calculate the density of states for an interacting quantum wire in the presence two impurities of arbitrary potential strength. To perform this calculation, we describe the Coulomb interactions in the wire within the Tomonaga-Luttinger liquid theory. After establishing and solving the Dyson equation for the fermionic retarded Green's functions, we study how the profile of the local density of states is affected by the interactions in the entire range of impurity potentials. Same as in the non-interacting case, when increasing the impurity strength, the central part of the wire becomes more and more disconnected from the semi-infinite leads, and discrete localized states begin to form; the width of the corresponding peaks in the spectrum depends on the interaction strength. As expected from the Luttinger liquid theory, impurities also induce a reduction of the local density of states at small energies. Two other important aspects are highlighted: the appearance of an extra-modulation in the density of states at non-zero Fermi momentum when interactions are present, and the fact that forward scattering must be taken into account in order to recover the Coulomb blockade regime for strong impurities.