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Densities of States of Disordered Systems from Free Probability
Matt Welborn
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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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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!
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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:
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Previous Work: 1D tight-binding with diagonal disorder
J J J J Chen et al. arXiv:
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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
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1D with 4 Neighbors
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1D with 4 Neighbors Solid: Exact Boxes: Free
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2D Grid
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2D Grid Solid: Exact Boxes: Free
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2D Honeycomb Lattice on a Torus
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2D Honeycomb Lattice on a Torus
Solid: Exact Boxes: Free
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3D Grid
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3D Grid Solid: Exact Boxes: Free
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1D with off-diagonal disorder
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1D with off-diagonal disorder
Solid: Exact Boxes: Free
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Chen and Edelman. arXiv:1204.2257
Error Analysis Expand the error in moments of the approximant: Chen and Edelman. arXiv:
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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:
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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
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<ABABABAB> gi-1 gi gi+1 < > Jgi Jgi+1 Jgi Jgi+1
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<ABABABAB> gi-1 gi gi+1 < > Jgi Jgi+1 Jgi Jgi-1
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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
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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
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Random Off-Diagonal gi-1 gi gi+1
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