However, we can neglect precise information about the bases of the matrices by approximating them with randompermutations or random rotations. �is seems very drastic, but it is sometimes exact!

Eigenvalues of sums from sums of eigenvalues: how accurate is free probability in calculating the density of states in disordered systems?Ramis Movassagh and Alan Edelman

Department of MathematicsMassachusetts Institute of Technology

Jiahao Chen, Eric Hontz, Jeremy Moix, Matthew Welborn, and Troy Van VoorhisDepartment of Chemistry

Massachusetts Institute of Technology

Alberto SuárezDepartamento de Ingeniería Informática

Universidad Autónoma de Madrid

FundingNSF SOLAR 1035400 ( J.C., E.H., M.W., T.V., R.M., A.E.), CHE1112825 ( J.M.), DMS 1016125 (A.E.)DARPA Grant No. N99001-10-1-4063 ( J.M.)Dirección General de Investigación, Project TIN2010-21575-C02-02 (A.S.)

Eigenvalues of sums of matrices

Explaining the di�erent behaviors of di�erent partitioningsHow well can we approximate the density of states in one-dimensional electronic systems? [4]Consider two possible partitionings of the Hamiltonian:

Why are disordered systems interesting?1. Unique physics, e.g.

state localization anomalous di�usion ergodicity breaking

2. Many applications

bulk heterojunction materials disordered metals defects in nanostructures

With thanks to

M.W.

T.V.E.H. J.C.

A.E. R.M. A.S. J.M.

useful discussions with

Sebastiaan Vlaming (MIT, Chemistry) Jonathan Novak (MIT, Mathematics) N Raj Rao (Michigan, Mathematics)

crystal atomic coordinateselectronic structure

dynamics observable

disordered system

ensemble-averagedobservable

sampling in configuration space

...

random permutation random rotation

In general, eigenvalues of matrix sums are not sums of eigenvalues!

In the limit of in�nitely large matrices, the density of states of A + B can be found by:

Exact if A and B commute, i.e. if relative orientationsof the eigenvectors are perfectly parallel.

Exact if A and B are free, i.e .their eigenvectors are ingeneric position, i.e. relative orientations are so randomthat they are e�ectively uniformly distributed over allpossible rotations (Q is uniform with Haar measure) [1,2].

Convolution of the eigenvalue densities of A and B Free convolution of the eigenvalue densities of A and B [2,3]

Random matrix theory can help us characterize the ensemble of random Hamiltonians and develop accurateapproximations to their eigenvalue spectra.

�e basic idea: take a Hamiltonian matrix with some (or all) random entries, break it up into pieces whoseeigenvalues can be easily calculated, then “add” then back together again.

Gives us ways to calculate eigenvalue spectra without ever diagonalizing a matrix!

Scheme 1 Scheme II

Application to disordered one-dimensional tight binding systems3. A challenge to model!

sampling in con�guration space diagonalize lots of Hamiltonians

For each piece, the eigenvalues can be calculated easily.

How well does the free convolution approximate the density of states?

Numerical convolution, Gaussian noise

random

Scheme I

exactScheme II

constant

low noise

moderate noise

high noise

Analytic convolution, semicircular noise

Scheme I shows universally good agreement with the exact density of states, whereas Scheme II worsensin the high noise regime.

How does Scheme I compare to perturbation theory?

Scheme I

B perturbs AA perturbs B

exact

References[1] D. Voiculescu, Invent. Math. 104, 201 (1991).[2] A. Nica and R. Speicher, Lectures on the Combinatorics of Free Probability, London Math. Soc. Lecture Note Ser. (2006).[3] D. Voiculescu, in Operator algebras and their connections with topology and ergodic theory, Lecture Notes in Mathematics, Vol. 1132, (Springer, 1985) pp. 556–588.[4] D. J. �ouless, Phys. Rep. 13, 93 (1974).[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).[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).

Our new result is to provide a quantative error analysis of the approximations from free probability.�is involves combining two known facts:

1. �e di�erence between two probability distributions can be quanti�ed by asymptotic moment expansions which generalize Edgeworth or Gram-Charlier series. [5, 6]

�e 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.�is involves combining two results:

2. Free probability implies a particular rule for calculating joint moments of the probability distribution:

�is gives us a way to calculate moments of the distribution produced from the free convolution bycalculating all the joint moments arising from the expansion of the moments of the sum:

�e noncommutative expansion of the trace is equivalent to the combinatorics of necklaces. [7]We can then �nd an n such that the leading order discrepancy between the exact and free distributions is

It turns out that Scheme I in the in�nite limit reduces to the coherent potential approximation, a self-consistentmean-�eld theory. [8] Our result provides an explanation for why the CPA works so well.

Unlike perturbation theory, where there is asymmetric treatment of A and B, Scheme I provides an excellentapproximation universally regardless of the strength of noise. But why?