Abstract: Theoretical studies of localization, anomalous diffusion and ergodicity breaking require solving the electronic structure of disordered systems. We use free probability to approximate the ensemble-averaged density of states without exact diagonalization. We present an error analysis that quantifies the accuracy using a generalized moment expansion, allowing us to distinguish between different approximations. We identify an approximation that is accurate to the eighth moment across all noise strengths, and contrast this with the perturbation theory and isotropic entanglement theory. Presented at the national meeting of the American Physical Society in Boston, MA, Feb 26, 2012, Poster No. C1.333.

1. Eigenvalues of sums from sums of eigenvalues: how accurate is free probability in calculating the density of states in disordered systems?
Jiahao Chen, Eric Hontz, Jeremy Moix, Matthew Welborn, and Troy Van Voorhis Alberto Suárez Ramis Movassagh and Alan Edelman
Department of Chemistry Departamento de Ingeniería Informática Department of Mathematics
Massachusetts Institute of Technology Universidad Autónoma de Madrid Massachusetts Institute of Technology
Why are disordered systems interesting? Application to disordered one-dimensional tight binding systems Explaining the different behaviors of different partitionings
1. Unique physics, e.g. 2. Many applications 3. A challenge to model! How well can we approximate the density of states in one-dimensional electronic systems? [4] Our new result is to provide a quantative error analysis of the approximations from free probability.
Consider two possible partitionings of the Hamiltonian: This involves combining two known facts:
state localization bulk heterojunction materials sampling in configuration space
anomalous diffusion disordered metals diagonalize lots of Hamiltonians 1. The difference between two probability distributions can be quantified by asymptotic moment expansions
ergodicity breaking defects in nanostructures random which generalize Edgeworth or Gram-Charlier series. [5, 6]
constant
The moment expansion is completely parameterized by the cumulants of the two distributions.
Our new result is to provide a quantative error analysis of the approximations from free probability.
This involves combining two results:
electronic structure
crystal atomic coordinates observable
dynamics 2. Free probability implies a particular rule for calculating joint moments of the probability distribution:
disordered system This gives us a way to calculate moments of the distribution produced from the free convolution by
calculating all the joint moments arising from the expansion of the moments of the sum:
The noncommutative expansion of the trace is equivalent to the combinatorics of necklaces. [7]
For each piece, the eigenvalues can be calculated easily. We can then find an n such that the leading order discrepancy between the exact and free distributions is
ensemble-averaged
observable
How well does the free convolution approximate the density of states?
...
Numerical convolution, Gaussian noise
sampling in configuration space
Scheme I low noise high noise
Random matrix theory can help us characterize the ensemble of random Hamiltonians and develop accurate Scheme II
approximations to their eigenvalue spectra. exact Scheme 1 Scheme II
The basic idea: take a Hamiltonian matrix with some (or all) random entries, break it up into pieces whose
eigenvalues can be easily calculated, then “add” then back together again.
It turns out that Scheme I in the infinite limit reduces to the coherent potential approximation, a self-consistent
Eigenvalues of sums of matrices moderate noise mean-field theory. [8] Our result provides an explanation for why the CPA works so well.
In general, eigenvalues of matrix sums are not sums of eigenvalues!
References
[1] D. Voiculescu, Invent. Math. 104, 201 (1991).
Scheme I shows universally good agreement with the exact density of states, whereas Scheme II worsens [2] A. Nica and R. Speicher, Lectures on the Combinatorics of Free Probability, London Math. Soc. Lecture Note Ser. (2006).
in the high noise regime. [3] D. Voiculescu, in Operator algebras and their connections with topology and ergodic theory, Lecture Notes in Mathematics, Vol. 1132,
However, we can neglect precise information about the bases of the matrices by approximating them with random (Springer, 1985) pp. 556–588.
permutations or random rotations. This seems very drastic, but it is sometimes exact! [4] D. J. Thouless, Phys. Rep. 13, 93 (1974).
random permutation random rotation How does Scheme I compare to perturbation theory? [5] A. Stuart and J. K. Ord, Kendall’s advanced theory of statistics. (Edward Arnold, London, 1994).
[6] D. Wallace, Ann. Math. Stat. 29, 635 (1958).
Analytic convolution, semicircular noise [7] J. Sawada, SIAM J. Comput. 31, 259 (2001).
[9] P. Neu and R. Speicher, Z. Phys. B 95, 101 (1994); J. Phys. A 79, L79 (1995); J. Stat. Phys. 80, 1279 (1995).
Exact if A and B commute, i.e. if relative orientations Exact if A and B are free, i.e .their eigenvectors are in
Scheme I
of the eigenvectors are perfectly parallel. generic position, i.e. relative orientations are so random
that they are effectively uniformly distributed over all
A perturbs B With thanks to
B perturbs A
possible rotations (Q is uniform with Haar measure) [1,2].
exact E.H. J.C. T.V.
In the limit of infinitely large matrices, the density of states of A + B can be found by:
M.W. A.E. R.M. A.S. J.M.
Convolution of the eigenvalue densities of A and B Free convolution of the eigenvalue densities of A and B [2,3]
useful discussions with
Sebastiaan Vlaming (MIT, Chemistry)
Jonathan Novak (MIT, Mathematics)
N Raj Rao (Michigan, Mathematics)
Unlike perturbation theory, where there is asymmetric treatment of A and B, Scheme I provides an excellent
approximation universally regardless of the strength of noise. But why?
Funding
NSF SOLAR 1035400 ( J.C., E.H., M.W., T.V., R.M., A.E.), CHE1112825 ( J.M.), DMS 1016125 (A.E.)
Gives us ways to calculate eigenvalue spectra without ever diagonalizing a matrix! DARPA Grant No. N99001-10-1-4063 ( J.M.)
Dirección General de Investigación, Project TIN2010-21575-C02-02 (A.S.)