We make the first testable predictions for the local two-particle spectral function of a disordered s-wave superconductor, probed by scanning Josephson spectroscopy (sjs), providing complementary information to scanning tunneling spectroscopy (sts). We show that sjs provides a direct map of the local superconducting order parameter that is found to be anticorrelated with the gap map obtained by sts. Furthermore, this anticorrelation increases with disorder. For the momentum resolved spectral function, we find the Higgs mode shows a nondispersive subgap feature at low momenta, spectrally separated from phase modes, for all disorder strengths. The amplitude-phase mixing remains small at low momenta even when disorder is large. Remarkably, even for large disorder and high momenta, the amplitude-phase mixing oscillates rapidly in frequency and hence do not affect significantly the purity of the Higgs and phase dominated response functions.

The sequence of the zeroth Landau levels (LLs) between filling factors ν=-6 to 6 in ABA-stacked trilayer graphene (TLG) is unknown because it depends sensitively on the nonuniform charge distribution on the three layers of ABA-stacked TLG. Using the sensitivity of quantum Hall data on the electric field and magnetic field, in an ultraclean ABA-stacked TLG sample, we quantitatively estimate the nonuniformity of the electric field and determine the sequence of the zeroth LLs. We also observe anticrossings between some LLs differing by 3 in LL index, which result from the breaking of the continuous rotational to C_{3} symmetry by the trigonal warping.

A recent study have 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 non-local 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 maybe realized on the edge of an integer or a fractional quantum Hall state. We show that the TDOS enhancement in a LL quantum wire (QW) due to superconducting correlation at the junction can be farther amplified in presence of appropriate choice of non-local interaction parameters, which is demonstrated through a neat algebraic expression in the weak interaction limit. We also show that, in the full parameter regime comprising both the local and the non-local interaction, the region of enhanced TDOS for LL junction with "superconducting" boundary condition and that of "nonsuperconducting charge conserving" boundary condition (discussed in 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 non-superconducting 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 'TDOS(x)/TDOS(x → 0)' and the 'pair-potential(x)/pair-potential(x → 0)' are distinct function of the various local and non-local interaction parameters, which implies that the TDOS enhancement can not be directly attributed to the proximity induced pair potential in the LL.

No abstract