Zero-energy Andreev surface bound states in the lattice model

The conditions for zero-energy Andreev surface bound states to exist are found for the lattice model of $d$-wave superconductor with arbitrary surface orientation. Both nearest neighbors and next nearest neighbors models are considered. It is shown that the results are very sensitive to the surface orientation. In particular, for half-filled $(hl0)$-surface zero-energy Andreev surface states only appear under the condition that $h$ and $l$ are odd simultaneously.