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:1. Finding the elements of H2. 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

Chen et al. arXiv:1202.5831

G G GGGJ J J J

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: ExactBoxes: Free

2D Grid

2D Grid

Solid: ExactBoxes: Free

2D Honeycomb Lattice on a Torus

2D Honeycomb Lattice on a Torus

Solid: ExactBoxes: Free

3D Grid

3D Grid

Solid: ExactBoxes: Free

1D with off-diagonal disorder

1D with off-diagonal disorder

Solid: ExactBoxes: Free

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 CoefficientsLattice Moment Word Error

Coefficient1D/1NN 8 ABABABAB1D/2NN 8 ABABABAB1D/3NN 8 ABABABAB1D/4NN 8 ABABABAB2D Grid 8 ABABABAB2D Hex 8 ABABABAB3D Grid 8 ABABABAB1D ODD 6 ABBABB

<ABABABAB>

< >Jgi Jgi+1 Jgi Jgi+1

gi-1 gi gi+1

<ABABABAB>

< >Jgi Jgi+1 Jgi Jgi-1

gi+1gi-1 gi

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 CoefficientsLattice Moment Word Error

Coefficient1D/1NN 8 ABABABAB1D/2NN 8 ABABABAB1D/3NN 8 ABABABAB1D/4NN 8 ABABABAB2D Grid 8 ABABABAB2D Hex 8 ABABABAB3D Grid 8 ABABABAB1D ODD 6 ABBABB

Random Off-Diagonal

gi-1 gi gi+1