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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−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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−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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0.5
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
transport
Pictures: Wikipediaahmedmater.com
UPAA MIC Winners, Sep. 2011
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>.
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
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
0.1
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
0.2
0.3
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
HDF5Database
Q-Chem output file parsingQ-Chem QM/MM
Grid Engine
CHARMMerror handlingconvergence failuressystem failures...
job dispatcher
queue monitorerror handling for cluster-wide failures
post-analysis scripts
sampling electronic structure observables
thermalizedperfect crystal
Step 1. Sample thermalized states using molecular dynamicsNVT dynamics of 8x8x8 supercell in CHARMM
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
Step 3. Collect statistics to recover averaged spectra100 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?
• 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
###
incoherentdiffusive
coherentballistic
Localization of statesExcitation energy (eV)
RM
S le
ngth
(nor
mal
ized
)
1.6 1.8 2 2.2 2.4
0.2
0.4
0.6
0.8
Localization of statesExcitation energy (eV)
RM
S le
ngth
(nor
mal
ized
)
1.6 1.8 2 2.2 2.4
0.2
0.4
0.6
0.8
delocalized
Localization of statesExcitation energy (eV)
RM
S le
ngth
(nor
mal
ized
)
1.6 1.8 2 2.2 2.4
0.2
0.4
0.6
0.8
0 50 100 20406060
80
100
120
140
160
180
sample 61, state 5, energy = 2.126573
- delocalized along herringbone axis only- antiferromagnetic order in transition dipoles
Localization of statesExcitation energy (eV)
RM
S le
ngth
(nor
mal
ized
)
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
Localization of statesExcitation energy (eV)
RM
S le
ngth
(nor
mal
ized
)
1.6 1.8 2 2.2 2.4
0.2
0.4
0.6
0.8
0 50 100204060
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
Excitation energy (eV)
RM
S le
ngth
(nor
mal
ized
)
1.6 1.8 2 2.2 2.4
0.2
0.4
0.6
0.8
Asymmetry from neighbor shells
Excitation energy (eV)
RM
S le
ngth
(nor
mal
ized
)
1.6 1.8 2 2.2 2.4
0.2
0.4
0.6
0.8
1 2 allnumber of neighbor
shells
Neighbor shells in crystal Excitation energy (eV)
RM
S le
ngth
(nor
mal
ized
)
1.7 1.75 1.8 1.85 1.90.6
0.7
0.8
0.9
1
Excitation energy (eV)
RM
S le
ngth
(nor
mal
ized
)
1.5 1.6 1.7 1.8 1.9 2
0.8
0.85
0.9
0.95
1
Excitation energy (eV)
RM
S le
ngth
(nor
mal
ized
)
1.5 1.6 1.7 1.8 1.9 2
0.85
0.9
0.95
1
Excitation energy (eV)
RM
S le
ngth
(nor
mal
ized
)
1.4 1.6 1.80.7
0.75
0.8
0.85
0.9
0.95
1
1 2 3 allnumber of neighbor
shells
SummaryExcitons 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)
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
Modeling disorderdisordered system
observable
sampling in
phase space
...
atomic coordinates electronic structure
dynamicsobservable
ensemble averaging
Modeling disorderdisordered system
observable
sampling in
phase space
...
atomic coordinates electronic structure
dynamicsobservable
ensemble averaging
4 CPU-hours
Absorption spectrum (gas)
1.2 1.4 1.6 1.8 2 2.2
(condensed)
Energy (eV)
6 CPU-years1 “CPU-PhD”
Modeling disorder
atomic coordinates electronic structure
dynamicsobservable
disordered system
ensemble-averaged observable
sampling in
phase spacerandom matrix theory?
...spatial disorder
spectral disorder
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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