Densities of States of Disordered Systems from Free Probability Matt Welborn

The Electronic Structure Problem For a fixed set of nuclear coordinates, solve the Schrödinger equation: which is a “simple” eigenvalue problem Two main costs: Finding the elements of H Diagonalizing

Disordered systems The previous equation describes the system at a fixed set of nuclear coordinates In a disordered system, we need to capture Static disorder Molecules don’t pack into a nice crystal Bigger matrices! Dynamic disorder Molecules move around at non-zero temperatures More matrices!

Approximate with Free Probability Assume distribution of Hamiltonians Partition Hamiltonian into two easily-diagonalizable parts: Use free probability to approximate the spectrum of H from that of A and B:

Previous Work: 1D tight-binding with diagonal disorder J J J J Chen et al. arXiv:1202.5831

Moving towards reality We’d like to look at real systems Extend the 1D tight-binding model: 2nd,3rd, etc. Nearest Neighbors 2D/3D Tight Binding Off-Diagonal Disorder

1D with 4 Neighbors

1D with 4 Neighbors Solid: Exact Boxes: Free

2D Grid

2D Grid Solid: Exact Boxes: Free

2D Honeycomb Lattice on a Torus

2D Honeycomb Lattice on a Torus Solid: Exact Boxes: Free

3D Grid

3D Grid Solid: Exact Boxes: Free

1D with off-diagonal disorder

1D with off-diagonal disorder Solid: Exact Boxes: Free

Chen and Edelman. arXiv:1204.2257 Error Analysis Expand the error in moments of the approximant: Chen and Edelman. arXiv:1204.2257

Finding the difference in moments For the ith moment, check that all joint centered moments of order i are 0: Example - for the fourth moment, check: ? Chen and Edelman. arXiv:1204.2257

Error Coefficients Lattice Moment Word Error Coefficient 1D/1NN 8 ABABABAB 1D/2NN 1D/3NN 1D/4NN 2D Grid 2D Hex 3D Grid 1D ODD 6 ABBABB

<ABABABAB> gi-1 gi gi+1 < > Jgi Jgi+1 Jgi Jgi+1

<ABABABAB> gi-1 gi gi+1 < > Jgi Jgi+1 Jgi Jgi-1

Why ABABABAB? allows hopping to more neighbors, but centering removes self-loops is diagonal with i.i.d. elements of mean zero Need four hops to collect squares of two elements of is the shortest such word

Error Coefficients Lattice Moment Word Error Coefficient 1D/1NN 8 ABABABAB 1D/2NN 1D/3NN 1D/4NN 2D Grid 2D Hex 3D Grid 1D ODD 6 ABBABB

Random Off-Diagonal gi-1 gi gi+1