Oscillating-charged Andreev Bound States and Their Appearance in UTe$_2$

Surface Andreev bound states, including Majorana bound states in topological superconductors, are typically charge neutral. In this work, we demonstrate the emergence of unconventional charged Andreev bound states in a superconductor with a sublattice degree of freedom, where the sign of charge density of the Andreev bound states oscillates between the two sublattices. The oscillating-charged Andreev bound states lead to a complete breakdown of the proportionality among the electron part of the spectral function, the local density of states, and the tunneling conductance spectrum for energies below the superconducting gap. We also discuss the possible occurrence of these Andreev bound states in UTe$_2$ and locally noncentrosymmetric superconductors.