  In this work, we propose a theory on the two-dimensional non-Hermitian skin
effect by resolving two representative minimal models. Specifically, we show
that for any given non-Hermitian Hamiltonian, (i) the corresponding region
covered by its open boundary spectrum on the complex energy plane should be
independent of the open boundary geometry; and (ii) for any given open boundary
eigenvalue $E_0$ , its corresponding two-dimensional asymptotic generalized
Brillouin zone is determined by a series of geometry-independent
Bloch/non-Bloch Fermi points and geometry-dependent non-Bloch equal frequency
contours that connect them. A corollary of our theory is that most
symmetry-protected exceptional semimetals should be robust to variations in OBC
geometry. Our theory paves the way to the discussion on the higher dimensional
non-Bloch band theory and the corresponding non-Hermitian bulk-boundary
correspondence.
