Arbitrary order transfer matrix exceptional points and van Hove singularities

TL;DR

This paper reveals a fundamental link between van Hove singularities and exceptional points of arbitrary order in the transfer matrix of lattice models, providing new insights into band dispersion and spectral properties.

Contribution

It establishes a direct connection between VHSs and EPs of the transfer matrix, offering a method to generate and analyze high-order EPs and VHSs in lattice models.

Findings

VHSs are EPs of the same order in the transfer matrix.

Provided a general method to generate any order EP of TM.

Analyzed restrictions on EP orders for a given hopping range.

Abstract

In lattice models with quadratic finite-range Hermitian Hamiltonians, the inherently non-Hermitian transfer matrix (TM) governs the band dispersion. The van Hove singularities (VHSs) are special points in the band dispersion where the density of states (DOS) diverge. Considering a lattice chain with hopping of a finite range $n$, we find a direct fundamental connection between VHSs and exceptional points (EPs) of TM, both of arbitrary order. In particular, we show that VHSs are EPs of TM of the same order, thereby connecting two different types of critical points usually studied in widely different branches of physics. Consequently, several properties of band dispersion and VHSs can be analyzed in terms of spectral properties of TM. We further provide a general prescription to generate any order EP of the TM and therefore corresponding VHS. For a given range of hopping $n$, our analysis provides restrictions on allowed orders of EPs of TM. Finally, we exemplify all our results for the case $n=3$.