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 non-unitary 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 multi-band 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 (non-diagonalizable) transfer matrix signals the presence of solutions with a power-law prefactor.

We present a procedure for exactly diagonalizing finite-range quadratic fermionic Hamiltonians with arbitrary boundary conditions in one of D dimensions, and periodic in the remaining D − 1. The key is a Hamiltoniandependent separation of the bulk from the boundary. By combining information from the two, we identify a matrix function that fully characterizes the solutions, and may be used to construct an efficiently computable indicator of bulk-boundary correspondence. As an illustration, we show how our approach correctly describes the zero-energy Majorana modes of a time-reversal-invariant s-wave two-band superconductor in a Josephson ring configuration, and predicts that a fractional 4π-periodic Josephson effect can only be observed in phases hosting an odd number of Majorana pairs per boundary.Developing a quantitative understanding of the physical properties of fermionic systems in the presence of non-trivial boundaries has widespread significance from both a fundamental and applied perspective. Not only has the behavior of fermions at a boundary informed leading materialcharacterization techniques like angle-resolved photoemission spectroscopy [1] and the revolution in metrology brought about by the integer quantum Hall effect [2]; nowadays, surface states of topological insulators and Majorana boundary modes of topological superconductors [3,4] play a central role in state-of-the-art proposals ranging from coherent spintronics [5,6] to topological quantum computation [7,8].All of the above phenomena are linked by a common theme: topologically non-trivial band structures [4]. Band structure theory, including the topological classification of mean-field fermionic systems [9], rests on a manifestation of crystal translational symmetry, the Bloch theorem. Since translational symmetry is broken by the presence of a boundary, it is remarkable that there exists a connection between the topological nature of the bulk and the boundary physics -the bulkboundary (BB) correspondence [4,10]. This principle states that a topologically non-trivial bulk mandates the emergence of fermionic states localized on the boundary, when boundary conditions (BCs) are changed from periodic to open, and that such states are distinguished by their robustness against symmetry-preserving local perturbations. While this heuristics has been numerically validated in a variety of cases, and rigorous results exist for discrete-time systems described by one-dimensional quantum walks [11], no general analytic insight is available as yet. Allowing for arbitrary BCs is necessary for any theory of BB correspondence to capture the robustness of the emerging localized modes to different perturbations [12]. Further motivation stems from studies of quantum quenches [13,14], where robustness against changes of the BCs has been argued to control the (quasi)local symmetries that characterize the stationary properties in the bulk. Tackling these issues calls for a procedure to determine energy eigenstates of lattice Hamiltonians with arbitrary BCs, compa...

Abstract. Motivated by the challenge of seeking a rigorous foundation for the bulkboundary correspondence for free fermions, we introduce an algorithm for determining exactly the spectrum and a generalized-eigenvector basis of a class of banded block quasi-Toeplitz matrices that we call corner-modified. Corner modifications of otherwise arbitrary banded block-Toeplitz matrices capture the effect of boundary conditions and the associated breakdown of translational invariance. Our algorithm leverages the interplay between a non-standard, projector-based method of kernel determination (physically, a bulk-boundary separation) and families of linear representations of the algebra of matrix Laurent polynomials. Thanks to the fact that these representations act on infinite-dimensional carrier spaces in which translation symmetry is restored, it becomes possible to determine the eigensystem of an auxiliary projected block-Laurent matrix. This results in an analytic eigenvector Ansatz, independent of the system size, which we prove is guaranteed to contain the full solution of the original finitedimensional problem. The actual solution is then obtained by imposing compatibility with a boundary matrix, also independent of system size. As an application, we show analytically that eigenvectors of short-ranged fermionic tight-binding models may display power-law corrections to exponential decay, and demonstrate the phenomenon for the paradigmatic Majorana chain of Kitaev.

We describe a method for exactly diagonalizing clean D-dimensional lattice systems of independent fermions subject to arbitrary boundary conditions in one direction, as well as systems composed of two bulks meeting at a planar interface. The specification of boundary conditions and interfaces can be easily adjusted to describe relaxation, reconstruction, or disorder away from the clean bulk regions of the system. Our diagonalization method builds on the generalized Bloch theorem [A. Alase et al., Phys. Rev. B 96, 195133 (2017)] and the fact that the bulk-boundary separation of the Schrödinger equation is compatible with a partial Fourier transform operation. Bulk equations may display unusual features because they are relative eigenvalue problems for non-Hermitian, bulk-projected Hamiltonians. Nonetheless, they admit a rich symmetry analysis that can simplify considerably the structure of energy eigenstates, often allowing a solution in fully analytical form. We illustrate our extension of the generalized Bloch theorem to multicomponent systems by determining the exact Andreev bound states for a simple SNS junction. We then analyze the Creutz ladder model, by way of a conceptual bridge from one to higher dimensions. Upon introducing a new Gaussian duality transformation that maps the Creutz ladder to a system of two Majorana chains, we show how the model provides a first example of a short-range chiral topological insulators hosting topological zero modes with a power-law profile. Additional applications include the complete analytical diagonalization of graphene ribbons with both zigzag-bearded and armchair boundary conditions, and the analytical determination of the edge modes in a chiral p + ip twodimensional topological superconductor. Lastly, we revisit the phenomenon of Majorana flat bands and anomalous bulk-boundary correspondence in a two-band gapless s-wave topological superconductor. Beside obtaining sharp indicators for the presence of Majorana modes through the use of the boundary matrix, we analyze the equilibrium Josephson response of the system, showing how the presence of Majorana flat bands implies a substantial enhancement in the 4π-periodic supercurrent. arXiv:1808.07555v1 [cond-mat.stat-mech]

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