Tunneling density of states in a Luttinger liquid in proximity to a superconductor: Effect of nonlocal interaction

A recent study has shown that it is possible to have enhancement, in contrast to an expected suppression, in tunneling density of states (TDOS) in a Luttinger liquid (LL) which is solely driven by the nonlocal density-density interactions. Also, it is well known that a LL in proximity to a superconductor (SC) shows enhancement in TDOS in the vicinity of the junction in the zero-energy limit. In this paper, we study the interplay of nonlocal density-density interaction and superconducting correlations in the TDOS in the vicinity of the SC-LL junction, where the LL may be realized on the edge of an integer or a fractional quantum Hall state. We show that the interplay of superconducting proximity effect and nonlocal interactions can give rise to enhancement in TDOS in the weak interaction limit, beyond what was previously observed. We also show that, in the full parameter regime comprising both the local and the nonlocal interactions, the region of enhanced TDOS for LL junction with ``superconducting'' boundary condition and that of ``nonsuperconducting charge-conserving'' boundary condition [discussed in A. Ratnakar and S. Das, Phys. Rev. B 104, 045402 (2021)] are mutually exclusive. We show that this fact can be understood in terms of symmetry relation established between the superconducting and nonsuperconducting sectors of the theory. We compare the dependence of the proximity-induced pair potential and TDOS as a function of distance $x$ from the junction. We demonstrate that the dependence of the spatial power-law exponent for the ``$\mathrm{TDOS}(x)/\mathrm{TDOS}(x\ensuremath{\rightarrow}0)$'' and the ``pair potential($x$)/pair potential($x\ensuremath{\rightarrow}0$)'' are distinct function of the various local and nonlocal interaction parameters, which implies that the TDOS enhancement can not be directly attributed to the proximity-induced pair potential in the LL.