Eigenvalues of sums from sums of eigenvalues: how accurate is free probability in calculating the...
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However, we can neglect precise information about the bases of the matrices by approximating them with random permutations 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 Mathematics Massachusetts Institute of Technology Jiahao Chen, Eric Hontz, Jeremy Moix, Matthew Welborn, and Troy Van Voorhis Department of Chemistry Massachusetts Institute of Technology Alberto Suárez Departamento de Ingeniería Informática Universidad Autónoma de Madrid Funding NSF 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 different behaviors of different partitionings How 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 diffusion 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 coordinates electronic structure dynamics observable disordered system ensemble-averaged observable sampling in configuration space ... random permutation random rotation In general, eigenvalues of matrix sums are not sums of eigenvalues! In the limit of infinitely large matrices, the density of states of A + B can be found by: Exact if A and B commute, i.e. if relative orientations of the eigenvectors are perfectly parallel. Exact if A and B are free, i.e .their eigenvectors are in generic position, i.e. relative orientations are so random that they are effectively uniformly distributed over all possible 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 accurate approximations to their eigenvalue spectra. e 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. Gives us ways to calculate eigenvalue spectra without ever diagonalizing a matrix! Scheme 1 Scheme II Application to disordered one-dimensional tight binding systems 3. A challenge to model! sampling in configuration 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 exact Scheme 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 worsens in the high noise regime. How does Scheme I compare to perturbation theory? Scheme I B perturbs A A 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 difference between two probability distributions can be quantified 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 by calculating 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 find an n such that the leading order discrepancy between the exact and free distributions is It turns out that Scheme I in the infinite limit reduces to the coherent potential approximation, a self-consistent mean-field 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 excellent approximation universally regardless of the strength of noise. But why?
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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.
Transcript of Eigenvalues of sums from sums of eigenvalues: how accurate is free probability in calculating the...
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?
