Generalization of Bloch's theorem for arbitrary boundary conditions: Theory

We present 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, by providing exact, analytic expressions for all energy eigenvalues and eigenstates. Starting with a reordering of the fermionic basis that transforms the single-particle Hamiltonian into a corner-modified banded block-Toeplitz matrix, a key step is a Hamiltonian-dependent bipartition of the lattice, which splits the eigenvalue problem into a system of bulk and boundary equations. The eigensystem inherits most of its solutions from an auxiliary, infinite translation-invariant Hamiltonian that allows for nonunitary representations of translation-hence complex values of crystal momenta with specific localization properties. A reformulation of the boundary equation in terms of a boundary matrix ensures compatibility with the boundary conditions, and determines the allowed energy eigenstates in the form of generalized Bloch states. We show how the boundary matrix quantitatively captures the interplay between bulk and boundary properties, leading to the construction of efficient indicators of bulk-boundary correspondence. Remarkable consequences of our generalized Bloch theorem are the engineering of Hamiltonians that host perfectly localized, robust zero-energy edge modes, and the predicted emergence, for instance, in Kitaev's Majorana chain, of localized excitations whose amplitudes decay in space exponentially with a power-law prefactor. We further show how the theorem may be used to construct numerical and algebraic diagonalization algorithms for the class of Hamiltonians under consideration, and use the proposed bulk-boundary indicator to characterize the topological response of a multiband time-reversal invariant s-wave topological superconductor under twisted boundary conditions, showing how a fractional Josephson effect can occur without entailing a fermionic parity switch. Finally, we establish connections to the transfer matrix method and demonstrate, using the paradigmatic Kitaev's chain example, that a defective (nondiagonalizable) transfermatrix signals the presence of solutions with a power-law prefactor.