Counting the exponents of single transfer matrices

TL;DR

This paper develops a formula to count the exponents of transfer matrices in lattice models, linking decay lengths to matrix properties, with applications demonstrated on the Anderson model.

Contribution

It introduces a novel formula for counting transfer matrix exponents using duality and the Argument Principle, involving corner blocks of the inverse Hamiltonian.

Findings

Derived a counting function formula for transfer matrix exponents

Applied the formula to the quasi 1D Anderson model

Numerical evaluations confirm the theoretical results

Abstract

The eigenvalue equation of a band or a block tridiagonal matrix, the tight binding model for a crystal, a molecule, or a particle in a lattice with random potential or hopping amplitudes: these and other problems lead to three-term recursive relations for (multicomponent) amplitudes. Amplitudes n steps apart are linearly related by a transfer matrix, which is the product of n matrices. Its exponents describe the decay lengths of the amplitudes. A formula is obtained for the counting function of the exponents, based on a duality relation and the Argument Principle for the zeros of analytic functions. It involves the corner blocks of the inverse of the associated Hamiltonian matrix. As an illustration, numerical evaluations of the counting function of quasi 1D Anderson model are shown.