Exact solution of corner-modified banded block-Toeplitz eigensystems

TL;DR

This paper presents an exact algorithm for solving the eigensystems of corner-modified banded block-Toeplitz matrices, modeling boundary effects in fermionic systems, with applications to topological superconductors.

Contribution

It introduces a novel, analytically guaranteed method to determine spectra and eigenvectors of corner-modified matrices, capturing boundary effects in quantum models.

Findings

Eigenvectors can exhibit power-law corrections to exponential decay.

The algorithm guarantees full eigensystem solutions independent of system size.

Application to the Kitaev Majorana chain demonstrates boundary effects in fermionic models.

Abstract

Motivated by the challenge of seeking a rigorous foundation for the bulk-boundary 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 finite-dimensional 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.