Generalization of Bloch’s Theorem to Systems with Boundary

We formulate and prove a generalization of Bloch’s theorem to finite-range lattice systems of independent fermions, in which translation symmetry is broken solely due to arbitrary boundary conditions. This generalization, which is made possible mainly by allowing the crystal momentum to take complex values, provides exact analytic expressions for all energy eigenvalues and eigenvectors of the system Hamiltonian. A remarkable consequence of this theorem is the predicted emergence of localized excitations, whose amplitude decays in space exponentially with a power-law prefactor. We leverage this generalization of Bloch’s theorem to design an algorithm for computing energy eigenvalues and eigenstates of systems under consideration, and also to construct an indicator of bulk-boundary correspondence. We spell out the connections of the generalized Bloch theorem with the well-known transfer matrix method. We discuss how higher-dimensional systems and interfaces can be analyzed by extending generalized Bloch theorem.