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Free probability, random matrices and disorder in organic semiconductors

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Random matrix theory has long been used to study the spectral
properties of physical systems, and has led to a rich interplay
between probability theory and physics [1]. Historically, random
matrices have been used to model physical systems with random
fluctuations, or systems whose eigenproblems were too difficult to
solve numerically. This talk explores applications of RMT to the
physics of disorder in organic semiconductors [2,3]. Revisiting the
old problem of Anderson localization [4] has shed new light on the
emerging field of free probability theory [5]. I will discuss the
implications of free probabilistic ideas for finite-dimensional random
matrices [6], as well as some hypotheses about eigenvector locality.
Algorithms are available in the RandomMatrices.jl package [7] written
for the Julia programming language.

[1] M. L. Mehta. Random matrices, 3/e, Academic Press, 2000.
[2] J. Chen, E. Hontz, J. Moix, M. Welborn, T. Van Voorhis, A. Suarez,
R. Movassagh, and A. Edelman. Error analysis of free probability
approximations to the density of states of disordered systems.
Phys. Rev. Lett. (2012) 109:36403.
[3] M. Welborn, J. Chen, and T. Van Voorhis. Densities of states for
disordered systems from free probability. Phys. Rev. B (2013) 88:205113.
[4] P. W. Anderson. Absence of diffusion in certain random lattices.
Phys. Rev. (1958) 109:1492--1505.
[5] D. Voiculescu. Addition of certain non-commuting random variables.
J. Functional Anal. (1986) 66:323--346.
[6] J. Chen, T. Van Voorhis, and A. Edelman. Partial freeness of random
matrices. arXiv:1204.2257
[7] https://github.com/jiahao/RandomMatrices.jl

Published in: Science

Free probability, random matrices and disorder in organic semiconductors

  1. 1. What is random matrix theory? linear algebra matrix properties: - eigenvalues/vectors - singular values/vectors - trace, determinant, etc. M = ⎛ ⎜ ⎝ 2.4 1 − 0.5i · · · 1 + 0.5i 33 · · · ... ... ... ⎞ ⎟ ⎠ A. Edelman, B. D. Sutton and Y. Wang. “Modern Aspects of Random Matrix Theory”. Proceedings of Symposia in Applied Mathematics 72, (2014)
  2. 2. What is random matrix theory? linear algebra matrix properties: - eigenvalues/vectors - singular values/vectors - trace, determinant, etc. M = ⎛ ⎜ ⎝ 2.4 1 − 0.5i · · · 1 + 0.5i 33 · · · ... ... ... ⎞ ⎟ ⎠ A. Edelman, B. D. Sutton and Y. Wang. “Modern Aspects of Random Matrix Theory”. Proceedings of Symposia in Applied Mathematics 72, (2014) random matrix theory ensemble of matrices M = ⎛ ⎜ ⎝ g g · · · g g · · · ... ... ... ⎞ ⎟ ⎠ ensemble of matrix properties Noteb
  3. 3. 1. The semicircle law M = ⎛ ⎜ ⎝ g g · · · g g · · · ... ... ... ⎞ ⎟ ⎠ n=500 M=randn(n, n) M=(M+M’)/√2n hist(eigvals(M)) E P Wigner, Ann. Math. 62 (1955), 548; 67 (1958), 325 −4 −3 −2 −1 0 1 2 3 4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 −4 −3 −2 −1 0 1 2 3 4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 −4 −3 −2 −1 0 1 2 3 4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 −4 −3 −2 −1 0 1 2 3 4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 −4 −3 −2 −1 0 1 2 3 4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 −4 −3 −2 −1 0 1 2 3 4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 −4 −3 −2 −1 0 1 2 3 4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 −4 −3 −2 −1 0 1 2 3 4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 −4 −3 −2 −1 0 1 2 3 4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 distribution of eigenvalues Notebook
  4. 4. histogram of level spacings 2. Level spacings: nuclear transitions M. L. Mehta, “Random Matrices” 3/e (2004), Ch. 1energy levels level spacings uncorrelated eigenvalues “randomly” correlated eigenvalues distribution of eigenvalue gaps = distribution of nuclear energy levels
  5. 5. 0 0.5 1 1.5 2 2.5 3 3.5 0 0.2 0.4 0.6 0.8 1 1.2 bus spacing P(s) s Figure1. BusintervaldistributionP(s) obtainedforcitylinenumberfour. Thefullcurverepresents the random matrix prediction (4), the markers (+) represent the bus interval data and bars display the random matrix prediction (4) with 0.8% of the data rejected. 2. Level spacings: bus arrival times Nextbus.com/MBTA real-time data 12/6/2012 and 12/7/2012 Picture: transitboston.com bus intervals in Cuernavaca, Mexico Krbálek and Šeba, J. Phys. A 33 (2000) L229
  6. 6. 0 0.5 1 1.5 2 2.5 3 3.5 0 0.2 0.4 0.6 0.8 1 1.2 bus spacing P(s) s Figure1. BusintervaldistributionP(s) obtainedforcitylinenumberfour. Thefullcurverepresents the random matrix prediction (4), the markers (+) represent the bus interval data and bars display the random matrix prediction (4) with 0.8% of the data rejected. 2. Level spacings: bus arrival times bus intervals in Cuernavaca, Mexico Krbálek and Šeba, J. Phys. A 33 (2000) L229 mean
  7. 7. 3. Growth & the Tracy-Widom Law −5 −4 −3 −2 −1 0 1 2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 K A Takeuchi and M Sano J. Stat. Phys. 147 (2012) 853 C A Tracy and H Widom, Phys. Lett. B 305 (1993) 115; Commun. Math. Phys. 159 (1994), 151; 177 (1996), 727 experimental fluctuations of phase boundary = theoretical fluctuations in Gaussian ensembles phase interface in a liquid crystal statistics of fluctuations: skewness, kurtosis distribution of largest eigenvalue −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −0.1 0 0.1 0.2 0.3 0.4 M = ⎛ ⎜ ⎝ g g · · · g g · · · ... ... ... ⎞ ⎟ ⎠ largest eigenvalue of a random matrix
  8. 8. Physical consequences of disorder ❖ Electrical resistance in metals thermal fluctuations
 lattice defects
 chemical impurities ❖ Spontaneous magnetization ergodicity breaking
 spontaneous symmetry breaking ❖ Dynamical localization interference between paths suppresses transport Pictures: Wikipedia ahmedmater.com UPAA MIC Winners, Sep. 2011
  9. 9. Why is the semicircle law true? E P Wigner, Ann. Math. 62 (1955), 548; 67 (1958), 325 Notebook Wigner’s original proof: Compute all moments of the eigenvalue distribution Recall: For a matrix M, the nth moment of its spectral density is the expected trace of Mn. Denote this quantity as <Mn>.
  10. 10. Why is the semicircle law true? E P Wigner, Ann. Math. 62 (1955), 548; 67 (1958), 325 The expected trace of Mn is actually a long sum of expectations
  11. 11. Why is the semicircle law true? E P Wigner, Ann. Math. 62 (1955), 548; 67 (1958), 325 The expected trace of Mn is actually a long sum of expectations
  12. 12. Why is the semicircle law true? E P Wigner, Ann. Math. 62 (1955), 548; 67 (1958), 325 The expected trace of Mn is actually a long sum of expectations = N-1 paths of weight 1 + 1 path of weight 1 on average = N
  13. 13. Why is the semicircle law true? E P Wigner, Ann. Math. 62 (1955), 548; 67 (1958), 325 The only distribution with these moments is the semicircle law (using Carleman, 1923). = N-1 paths of weight 1 + 1 path of weight 1 on average = N
  14. 14. Can we add eigenvalues? In general, no. One must add eigenvalues vectorially. eig(A) + eig(B) = eig(A+B) ? 1 + 1 = 2 vector 1 direction = eigenvector of A magnitude = eigenvalue of A vector 2 direction = eigenvector of B magnitude = eigenvalue of Bvector sum direction = eigenvector of A+B magnitude = eigenvalue of A+B
  15. 15. Special cases of “matrix sums” eigenvector of A eigenvalue of A eigenvector of B eigenvalue of B eigenvector of A+B eigenvalue of A+B Case 1. A and B commute. A and B have the same basis, i.e. all their corresponding eigenvectors are parallel. Case 2. A and B are in general position. The bases of A and B are randomly oriented and have no preferred directions in common. No deterministic analogue! The eigenvalue distribution (density of states) of A + B is the free convolution of the separate densities of states. = A. Nica and R. Speicher, Lectures on the Combinatorics of Free Probability, 2006 JC and A Edelman, arXiv:1204.2257 Generalization to random matrices: The eigenvalue distribution (density of states) of A + B is the convolution of the separate densities of states. =*
  16. 16. Free convolutions Case 2. A and B are in general position. The bases of A and B are randomly oriented and have no preferred directions in common. The eigenvalue distribution (density of states) of A + B is the free convolution of the separate densities of states. = D. Voiculescu, Inventiones Mathematicae, 1991, 201-220. function eigvals_free(A, B) n = size(A, 1) Q = qr(randn(n, n)) M = A + Q*B*Q’ eigvals(M) end The spectral density of M can be given by free probability theory
  17. 17. Noisy electronic structure Tight binding Anderson Hamiltonian in 1D constant coupling Gaussian disorder interaction J + random fluctuation of site energies
  18. 18. Avoiding diagonalization In general, exact diagonalization is expensive. Strategy: split H into pieces with known eigenvalues then recombine using free convolution. How accurate is it? −4 −3 −2 −1 0 1 2 3 4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
  19. 19. Results of free convolution Approximation Exact high noise moderate noise low noise JC et al. Phys. Rev. Lett. 109 (2012), 036403
  20. 20. -10 0 10 0.1 1 10 0 0.1 0.2 ρ(x) x σ/J ρ(x) -10 0 10 0.1 1 10 0 0.1 0.2 ρ(x) x σ/J ρ(x) 2D square 2D honeycomb -10 0 10 0.1 1 10 0 0.1 0.2 ρ(x) x σ/J ρ(x) 3D cube -10 0 10 0.1 1 10 0 0.1 0.2 0.3 ρ(x) x σ/J ρ(x) 1D next-nearest neighbors -10 0 10 0.1 1 10 0 0.1 0.2 ρ(x) x σ*/σ ρ(x) 1D NN with fluctuating interactions exact approx.
  21. 21. Spectral signature of localization Spectral compressibility 0 for Wigner statistics (maximally delocalized states) 1 for Poisson statistics (localized states) measures fine-scale fluctuations in the level density Can tell something about eigenvectors from the eigenvalues?! B. L. Altshuler, I. K. Zharekeshev, S. A. Kotochigova, and B. I. Shklovskii, Sov. Phys. JETP 67 (1988) 625. Relationships between c and localization length of eigenvectors are conjectured to hold for certain random matrix ensembles χ(E) = lim ⟨N(E)⟩→∞ d ∆N2 (E) d ⟨N(E)⟩ ∼ ∆N2 (E) ⟨N(E)⟩
  22. 22. Excitation energy (eV) RMSlength(normalized) 1.6 1.8 2 2.2 2.4 0.2 0.4 0.6 0.8 Can tell something about eigenvectors from the eigenvalues?! Spectral signature of localization spectral compressibility
  23. 23. Strategy 1. Model Hamiltonians atomic coordinates electronic structure dynamics observable disordered system ensemble-averaged observable sampling in phase space ... ensemble of model Hamiltonians
  24. 24. Outline ❖ Introduction: organic solar cells Bulk heterojunctions
 Disorder matters!
 Computing ❖ Disordered excitons ab initio The sampling challenge
 Exciton band structures ❖ Models for disordered excitons Random matrix theory
 Quantum mechanics without wavefunctions ± + Excitation energy (eV) Localizationlength(normalized) 1.4 1.6 1.8 2 2.2 2.4 0 0.2 0.4 0.6 0.8
  25. 25. A standard protocol of computational chemistry crystal atomic coordinates electronic structure dynamics observable
  26. 26. A standard protocol of computational chemistry crystal atomic coordinates electronic structure dynamics observable ?X disordered system
  27. 27. disordered system observable ? Modeling disorder: explicit sampling
  28. 28. Modeling disorder: explicit sampling disordered system observable sampling in phase space ... atomic coordinates electronic structure dynamics observable ensemble averaging
  29. 29. Q-Chem input generation Infrastructure for large-scale quantum chemical simulations HDF5 Database Q-Chem output file parsing Q-Chem QM/MM Grid Engine CHARMM error handling convergence failures system failures ... job dispatcher queue monitor error handling for cluster-wide failures post-analysis scripts sampling electronic structure observables
  30. 30. thermalizedperfect crystal Step 1. Sample thermalized states using molecular dynamics NVT dynamics of 8x8x8 supercell in CHARMM
  31. 31. Cost: ~4 CPU-hours Step 2. Calculate absorption frequencies (energies) of each molecule using ab initio electronic structure theory TD-PBE0/6-31G* with electrostatic embedding in Q-Chem + 0.2 eV shift *L Edwards and M. Gouterman, J. Mol. Spect. 33 (1970), 292 Qx Qy B
  32. 32. Step 3. Collect statistics to recover averaged spectra 100 snapshots x 128 molecules, ~6 CPU-years Absorption spectrum (gas) 1.2 1.4 1.6 1.8 2 2.2 (condensed) Energy (eV) Qx Qy Qx Qy tail states?
  33. 33. • Normalized root mean square spread
 of the exciton wavefunction • Localization determines nature of transport Localization of states 1 N 1 l = 1 Lmax ψ |r| 2 ψ − ⟨ψ |r| ψ⟩ 2 incoherent diffusive coherent ballistic
  34. 34. Localization of states Excitation energy (eV) RMSlength(normalized) 1.6 1.8 2 2.2 2.4 0.2 0.4 0.6 0.8
  35. 35. Localization of states Excitation energy (eV) RMSlength(normalized) 1.6 1.8 2 2.2 2.4 0.2 0.4 0.6 0.8 delocalized
  36. 36. Localization of states Excitation energy (eV) RMSlength(normalized) 1.6 1.8 2 2.2 2.4 0.2 0.4 0.6 0.8 0 50 100 20 40 60 60 80 100 120 140 160 180 sample 61, state 5, energy = 2.126573 - delocalized along herringbone axis only - antiferromagnetic order in transition dipoles
  37. 37. Localization of states Excitation energy (eV) RMSlength(normalized) 1.6 1.8 2 2.2 2.4 0.2 0.4 0.6 0.8 - mostly delocalized along herringbone axis - polaron-like - antiferromagnetic order in transition dipoles
  38. 38. Localization of states Excitation energy (eV) RMSlength(normalized) 1.6 1.8 2 2.2 2.4 0.2 0.4 0.6 0.8 0 50 100 20 40 60 60 80 100 120 140 160 180 sample 1, state 51, energy = 1.715239 - mostly delocalized along herringbone axis - polaron-like - ferromagnetic order in transition dipoles
  39. 39. Excitation energy (eV) RMSlength(normalized) 1.6 1.8 2 2.2 2.4 0.2 0.4 0.6 0.8 Asymmetry from neighbor shells Excitation energy (eV) RMSlength(normalized) 1.6 1.8 2 2.2 2.4 0.2 0.4 0.6 0.8 1 2 all number of neighbor shells
  40. 40. Neighbor shells in crystal Excitation energy (eV) RMSlength(normalized) 1.7 1.75 1.8 1.85 1.9 0.6 0.7 0.8 0.9 1 Excitation energy (eV) RMSlength(normalized) 1.5 1.6 1.7 1.8 1.9 2 0.8 0.85 0.9 0.95 1 Excitation energy (eV) RMSlength(normalized) 1.5 1.6 1.7 1.8 1.9 2 0.85 0.9 0.95 1 Excitation energy (eV) RMSlength(normalized) 1.4 1.6 1.8 0.7 0.75 0.8 0.85 0.9 0.95 1 1 2 3 all number of neighbor shells
  41. 41. Summary Excitons in H2Pc come in three distinct flavors high energy:
 localized in 2D, delocalized along herringbone axis • medium energy:
 delocalized • low energy:
 dressed states localized predominantly in 2D Asymmetric density of states is a nonlocal effect J.C. et al., Phys. Rev. Lett. 2012 Excitation energy (eV) Localizationlength(normalized) 1.4 1.6 1.8 2 2.2 2.4 0 0.2 0.4 0.6 0.8
  42. 42. Modeling disorder disordered system observable sampling in phase space ... atomic coordinates electronic structure dynamics observable ensemble averaging
  43. 43. Modeling disorder disordered system observable sampling in phase space ... atomic coordinates electronic structure dynamics observable ensemble averaging 4 CPU-hours Absorption spectrum (gas) 1.2 1.4 1.6 1.8 2 2.2 (condensed) Energy (eV) 6 CPU-years 1 “CPU-PhD”
  44. 44. Modeling disorder atomic coordinates electronic structure dynamics observable disordered system ensemble-averaged observable sampling in phase space random matrix theory? ... spatial disorder spectral disorder
  45. 45. Summary •Free probability allows us to construct accurate approximations to analytic model Hamiltonians •An error analysis of this phenomenon is known •Statistics of eigenvalues may be able to tell us information about experimental observations J.C. et al., Phys. Rev. Lett. 2012

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