densities of states of disordered systems from free probability
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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