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Densities of States of Disordered Systems from Free Probability

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.

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Densities of States of Disordered Systems from Free Probability

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  1. Densities of States of Disordered Systems from Free Probability Matt Welborn

  2. 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

  3. 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 nicecrystal • Bigger matrices! • Dynamic disorder • Molecules move around at non-zero temperatures • More matrices!

  4. 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:

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

  6. 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

  7. 1D with 4 Neighbors

  8. 1D with 4 Neighbors Solid: Exact Boxes: Free

  9. 2D Grid

  10. 2D Grid Solid: Exact Boxes: Free

  11. 2D Honeycomb Lattice on a Torus

  12. 2D Honeycomb Lattice on a Torus Solid: Exact Boxes: Free

  13. 3D Grid

  14. 3D Grid Solid: Exact Boxes: Free

  15. 1D with off-diagonal disorder

  16. 1D with off-diagonal disorder Solid: Exact Boxes: Free

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

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

  19. Error Coefficients

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

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

  22. 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

  23. Error Coefficients

  24. Random Off-Diagonal gi-1 gi+1 gi

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