Two-dimensional Asymptotic Generalized Brillouin Zone Theory

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.