Exceptional Lattice Green's Functions
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 nearestneighbor lattice points in the tightbinding 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2017
 DOI:
 10.48550/arXiv.1710.10260
 arXiv:
 arXiv:1710.10260
 Bibcode:
 2017arXiv171010260S
 Keywords:

 Mathematical Physics;
 Condensed Matter  Other Condensed Matter;
 Mathematics  Combinatorics;
 Quantum Physics
 EPrint:
 11 pages, 4 figures, 3 tables, Submitting to Communications in Mathematical Physics, Comments welcome