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)
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Free probability, random matrices and disorder in organic semiconductors
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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)
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
1. The semicircle lawM =
⎛
⎜⎝g g · · ·g g · · ·...
.... . .
⎞
⎟⎠
n=500M=randn(n, n)M=(M+M’)/√2nhist(eigvals(M))
E P Wigner, Ann. Math. 62 (1955), 548; 67 (1958), 325
−4 −3 −2 −1 0 1 2 3 40
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−4 −3 −2 −1 0 1 2 3 40
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−4 −3 −2 −1 0 1 2 3 40
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distribution of eigenvalues
Notebook
histogram of level spacings
2. Level spacings: nuclear transitions
M. L. Mehta, “Random Matrices” 3/e (2004), Ch. 1energy levels
levelspacings
uncorrelated eigenvalues
“randomly” correlatedeigenvalues
distribution of eigenvalue gaps= distribution of nuclear energy levels
0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
1.2bus spacing
P(s)
s
Figure 1. Bus interval distributionP (s) obtained for city line number four. The full curve representsthe random matrix prediction (4), the markers (+) represent the bus interval data and bars displaythe random matrix prediction (4) with 0.8% of the data rejected.
2. Level spacings: bus arrival times
Nextbus.com/MBTA real-time data12/6/2012 and 12/7/2012Picture: transitboston.com
bus intervals in Cuernavaca, Mexico
Krbálek and Šeba, J. Phys. A 33 (2000) L229
0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
1.2bus spacing
P(s)
s
Figure 1. Bus interval distributionP (s) obtained for city line number four. The full curve representsthe random matrix prediction (4), the markers (+) represent the bus interval data and bars displaythe 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
3. Growth & the Tracy-Widom Law
−5 −4 −3 −2 −1 0 1 20
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K A Takeuchi and M Sano J. Stat. Phys. 147 (2012) 853C 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
Physical consequences of disorder
❖ Electrical resistance in metals thermal fluctuations
lattice defectschemical impurities
❖ Spontaneous magnetization ergodicity breaking
spontaneous symmetry breaking
❖ Dynamical localization interference between paths suppresses
E P Wigner, Ann. Math. 62 (1955), 548; 67 (1958), 325
The expected trace of Mn is actually a long sum of expectations
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
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
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
Can we add eigenvalues?
In general, no. One must add eigenvalues vectorially.
eig(A) + eig(B) = eig(A+B) ?1 + 1 = 2
vector 1direction = eigenvector of A
magnitude = eigenvalue of A
vector 2direction = eigenvector of B
magnitude = eigenvalue of Bvector sumdirection = eigenvector of A+B
magnitude = eigenvalue of A+B
Special cases of “matrix sums”eigenvector of Aeigenvalue of A
eigenvector of Beigenvalue of B
eigenvector of A+Beigenvalue 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.
=*
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
Noisy electronic structureTight binding Anderson Hamiltonian in 1D
constant couplingGaussian disorder
interactionJ
+
random fluctuation of site energies
Avoiding diagonalizationIn 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 40
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�
Results of free convolution
Approximation
Exact
high noise moderate noise low noise
JC et al. Phys. Rev. Lett. 109 (2012), 036403
-10 0
10 0.1
1
10 0
0.1
0.2
ρ(x)
xσ/J
ρ(x)
-10 0
10 0.1
1
10 0
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0.2
ρ(x)
xσ/J
ρ(x)
2D square 2D honeycomb
-10 0
10 0.1
1
10 0
0.1
0.2
ρ(x)
xσ/J
ρ(x)
3D cube
-10 0
10 0.1
1
10 0
0.1
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ρ(x)
xσ/J
ρ(x)
1D next-nearest neighbors
-10 0
10 0.1
1
10 0
0.1
0.2
ρ(x)
xσ*/σ
ρ(x)
1D NN with fluctuating interactions
exactapprox.
Spectral signature of localizationSpectral 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)⟩
Excitation energy (eV)R
MS
leng
th (n
orm
aliz
ed)
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 localizationspectral compressibility
Strategy 1. Model Hamiltonians
atomic coordinates electronic structure
dynamicsobservable
disordered system
ensemble-averaged observable
sampling in
phase space
...
ensemble ofmodel
Hamiltonians
Outline
❖ Introduction: organic solar cellsBulk heterojunctions Disorder matters! Computing
❖ Disordered excitons ab initioThe sampling challenge Exciton band structures
❖ Models for disordered excitonsRandom matrix theory Quantum mechanics without wavefunctions
±
+
Excitation energy (eV)
Loca
lizat
ion
leng
th (n
orm
aliz
ed)
1.4 1.6 1.8 2 2.2 2.40
0.2
0.4
0.6
0.8
A standard protocol of computational chemistry
crystal atomic coordinateselectronic structure
dynamicsobservable
A standard protocol of computational chemistry
crystal atomic coordinateselectronic structure
dynamicsobservable
?Xdisordered system
disordered system
observable
?
Modeling disorder: explicit sampling
Modeling disorder: explicit samplingdisordered system
observable
sampling in
phase space
...
atomic coordinates electronic structure
dynamicsobservable
ensemble averaging
Q-Chem inputgeneration
Infrastructure for large-scale quantum chemical simulations