2017/10/25 by Samuel Savitz, Savitz, Samuel, Marcus Bintz +1
Mathematics · Physics and Astronomy · #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.other #math-ph #math.CO #math.MP #quant-ph
paper · pdf · doi:10.48550/arxiv.1710.10260
11 pages, 4 figures, 3 tables, Submitting to Communications in Mathematical Physics, Comments welcome
arxiv created 2017/10/25 · arxiv updated 2017/10/31
The three exceptional lattices, E6, E7, and E8, 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 E6 and E7. 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.