Spectral duality and distribution of exponents for transfer matrices of block tridiagonal Hamiltonians

TL;DR

This paper explores the spectral duality between block tridiagonal matrices and transfer matrices, revealing analytic properties of exponents and their relation to eigenvalues, with implications for transport models.

Contribution

It introduces a spectral duality for block tridiagonal matrices with complex boundary conditions, linking exponents to eigenvalues and analyzing their distribution and dynamics.

Findings

Spectral duality relates transfer matrix exponents to eigenvalues.

Counting function of exponents connects to winding numbers.

Distribution of exponents includes real bands and complex arcs.

Abstract

I consider a general block tridiagonal matrix and the corresponding transfer matrix. By allowing for a complex Bloch parameter in the boundary conditions, the two matrices are related by a spectral duality. As a consequence, I derive some analytic properties of the exponents of the transfer matrix in terms of the eigenvalues of the (non-Hermitian) block matrix. Some of them are the single-matrix analogue of results holding for Lyapunov exponents of an ensemble of block matrices, which occur in models of transport. The counting function of exponents is related to winding numbers of eigenvalues. I discuss some implications of duality on the distribution (real bands and complex arcs) and the dynamics of eigenvalues.