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

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

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−4 −3 −2 −1 0 1 2 3 40

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0.15

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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

0.05

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0.15

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0.25

0.3

0.35

0.4

0.45

−4 −3 −2 −1 0 1 2 3 40

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 40

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 40

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 40

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 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

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

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) 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

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

�

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