Exceptional Lattice Green's Functions

TL;DR

This paper investigates the electronic properties of the exceptional lattices E6, E7, and E8 using numerical methods to analyze their Green's functions, densities of states, and walk probabilities, revealing Van Hove singularities and moments.

Contribution

It provides the first detailed numerical analysis of Green's functions and densities of states for the exceptional lattices E6, E7, and E8, including Van Hove singularities and walk enumeration.

Findings

Identified Van Hove singularities in densities of states.

Tabulated densities of states for E6, E7, and E8.

Counted moments of densities of states via walk enumeration.

Abstract

The three exceptional lattices, $E_6$, $E_7$, and $E_8$, have attracted much attention due to their anomalously dense and symmetric structures which are of critical importance in modern theoretical physics. Here, we study the electronic band structure of a single spinless quantum particle hopping between their nearest-neighbor lattice points in the tight-binding limit. Using Markov chain Monte Carlo methods, we numerically sample their lattice Green's functions, densities of states, and random walk return probabilities. We find and tabulate a plethora of Van Hove singularities in the densities of states, including degenerate ones in $E_6$ and $E_7$. Finally, we use brute force enumeration to count the number of distinct closed walks of length up to eight, which gives the first eight moments of the densities of states.