Analytic structure of Bloch functions for linear molecular chains

TL;DR

This paper analyzes the complex analytic structure of Bloch functions and energies in linear molecular chains, revealing their multi-valued nature and branch points, with applications to Green's functions and density matrix behavior.

Contribution

It provides a detailed description of the Riemann surface structure of Bloch functions and energies, including branch points and singularities, which was not previously characterized.

Findings

Identified the multi-valued analytic structure of Bloch functions and energies.

Located and characterized branch points and essential singularities.

Applied the analysis to Green's functions and density matrix asymptotics.

Abstract

This paper deals with Hamiltonians of the form $H=-{\bf \nabla}^2+v(\rr)$, with $v(\rr)$ periodic along the $z$ direction, $v(x,y,z+b)=v(x,y,z)$. The wavefunctions of $H$ are the well known Bloch functions $\psi_{n,\lambda}(\rr)$, with the fundamental property $\psi_{n,\lambda}(x,y,z+b)=\lambda \psi_{n,\lambda}(x,y,z)$ and $\partial_z\psi_{n,\lambda}(x,y,z+b)=\lambda \partial_z\psi_{n,\lambda}(x,y,z)$. We give the generic analytic structure (i.e. the Riemann surface) of $\psi_{n,\lambda}(\rr)$ and their corresponding energy, $E_n(\lambda)$, as functions of $\lambda$. We show that $E_n(\lambda)$ and $\psi_{n,\lambda}(x,y,z)$ are different branches of two multi-valued analytic functions, $E(\lambda)$ and $\psi_\lambda(x,y,z)$, with an essential singularity at $\lambda=0$ and additional branch points, which are generically of order 1 and 3, respectively. We show where these branch points come from, how they move when we change the potential and how to estimate their location. Based on these results, we give two applications: a compact expression of the Green's function and a discussion of the asymptotic behavior of the density matrix for insulating molecular chains.