Title: Nematic superconductivity in Cu$_{x}$Bi$_{2}$Se$_{3}$: The surface Andreev bound states,
Abstract: We study theoretically the topological surface states (TSSs) and the possible
surface Andreev bound states (SABSs) of Cu$_{x}$Bi$_{2}$Se$_{3}$ which is known
to be a topological insulator at $x=0$. The superconductivity (SC) pairing of
this compound is assumed to have the broken spin-rotation symmetry, similar to
that of the A-phase of $^{3}$He as suggested by recent nuclear-magnetic
resonance experiments. For both spheroidal and corrugated cylindrical Fermi
surfaces with the hexagonal warping terms, we show that the bulk SC gap is
rather anisotropic; the minimum of the gap is negligibly small as comparing to
the maximum of the gap. This would make the fully-gapped pairing effectively
nodal. For a clean system, our results indicate the bulk of this compound to be
a topological superconductor with the SABSs appearing inside the bulk SC gap.
The zero-energy SABSs which are Majorana fermions, together with the TSSs not
gapped by the pairing, produce a zero-energy peak in the surface density of
states (SDOS). The SABSs are expected to be stable against short-range
nonmagnetic impurities, and the local SDOS is calculated around a nonmagnetic
impurity. The relevance of our results to experiments is discussed.