1 General Concept of Polaron and its Manifestations in Transport and Spectral Propertiesn A.S. Mishchenko RIKEN ( Institute of Physical and Chemical Research

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General Concept of Polaron and its Manifestations in Transport and

Spectral Propertiesn

A. S. MishchenkoRIKEN (Institute of Physical and Chemical Research), Japan

Electron-phonon interaction. Origin and Hamiltonian.Properties of ground state.Dynamical properties. What to look at? Why dificult? How to overcome?Frohlich 3D polaron: spectral function and optical conductivityPolaron mobility: 5 different regimes.Hole in t-J-Holstein model.

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ToTokyo500 m

FromTokyo



• Electron-phonon

interaction.

• Origin and Hamiltonian.

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Polaron: momentum space

Where this interaction comes from?

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H0= E0 Σi Ci+ Ci

+ t Σij Ci+ Cj

Band structure in rigid lattice.

E0

E0-αu

Hint= α Σi Ci+ Ci u u ~ (bi

+ + bi)

Hint= γ Σi Ci+ Ci (bi

+ + bi)

Brething

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H0= E0 Σi Ci+ Ci

+ t Σij Ci+ Cj


E0

E0-αu

Hint= α Σi Ci+ Ci u u ~ (bi

+ + bi)

Hint= γ Σi Ci+ Ci (bi

+ + bi)

Holstein

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H0= E0 Σi Ci+ Ci

+ t Σij Ci+ Cj


t-αu

Hint= α Σij Ci+ Cj u u ~ (bi

+ + bi)

Hint= γ Σi Ci+ Cj (bi

+ + bi) t

SSH

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Scattered in momentum space

k-qk

-q


Which physical characteristics may change?

Properties of

groundstate

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Which physical characteristics may change?

1. Energy of particle2. Dispersion of particle3. Effective mass of particle4. Z-factor of particle

Which physical characteristics are interesting to study?

1. Energy of particle2. Dispersion of particle3. Effective mass of particle4. Z-factor of particle5. Structure of phonon cloud6. ……

Frohlich polaron

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GS energy Effective mass

Frohlich polaron

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Polaron cloudZ-factor

Holstein polaron

16Number of phonons in the polaron cloud

V(q)=const

• Seems that weak and strong coupling regime are separated by crossover, not by real phase transition.

• This is true for V(q)

• However, this is not true for V(k,q). SSH model.

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SSH polaron(Su-Schrieffer-Heeger)

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phase diagram

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Physical properties under interest

Polaron: green function

No simple connection to measurable properties!

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Physical properties under interest: Lehman functionLehmann spectral function (LSF)

LSF has poles (sharp peaks)at the energies of stable(metastable) states. It is a measurable (in ARPES)quantity.

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Physical properties under interest: Lehmann function.Lehmann spectral function (LSF)

LSF has poles (sharp peaks)at the energies of stable(metastable) states. It is a measurable (in ARPES)quantity.

LSF can be determined from equation:

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Physical properties under interest: Z-factor and energy

Lehmann spectral function (LSF)

If the state with the lowest energy in the sector of given momentumis stable

The asymptotic behavior is

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Physical properties under interest: Z-factor and energy

The asymptotic behavior is

Dynamicalproperties.

What to look at?Why difficult?

How to overcome?

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Physical properties under interest: Lehmann function.Lehmann spectral function (LSF)

LSF can be determined from equation:

Solving of this equation is a notoriously difficult problem

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Physical properties under interest: absorption by polaron.

Such relation between imaginary-time function and spectralproperties is rather general:

Current–current corelation functionis related to optical absorption by polarons by the same expression:

ARPES

Optical absorption

μ = σ (ω→0)/en Mobility

There are a lot ofproblems where one has to solveFredholmintegral equationof the first kind

Many-particle Fermi/Boson system in imaginary times representation

Many-particle Fermi/Boson system in Matsubara representation

Optical conductivity at finite T in imaginary times representation

Image deblurring with e.g. known 2D noise K(m,ω)

K(m,ω) is a 2D x 2D noise distributon function

m and ω are2D vectors

Tomography image reconstruction (CT scan)

K(m,ω) is a 2D x 2Ddistribution function

m and ω are2D vectors

Aircraftstability

Nuclearreactor

operation

Image deblurring

A lot ofother…

What is dramaticin the problem?

Aircraftstability

Nuclearreactor

operation

Image deblurring

A lot ofother…

What is dramaticin the problem?

Ill-posed!

Ill-posed!

We cannot obtain an exact solution not becauseof some approximations of our approaches.

Instead, we have to admit that the exact solution does not exist at all!

Ill-posed!

1. No unique solution in mathematical sense No function A to satisfy the equation

2. Some additional information is required which specifies which kind of solution is expected. In orderto chose among many approximate solutions.

Ill-posed!

Physics department:Max Ent.

Engineering department:Tikhonov Regularization

Statistical department:ridge regression

Next player: stochastic methods

Ill-posed!

Because of noise present in the input data G(m)there is no unique A(ωn)=A(n) which exactly satisfies the equation. Hence, one can search for the least-square fittedsolution A(n) which minimizes:

Ill-posed!

Explicit expression:

Saw tooth noise instabilitydue to small singular values.

Ill-posed!

General formulation of methods to solve ill-posed problems in terms ofBayesian statistical inference.

Bayes theorem:

P[A|G] P[G] = P[G|A] P[A]

P[A|G] – conditional probability that the spectral function is A provided the correlation function is G

Bayes theorem:

P[A|G] P[G] = P[G|A] P[A]

P[A|G] – conditional probability that the spectral function is A provided the correlation function is G

To find it is just the analytic continuation

P[A|G] ~ P[G|A] P[A]

P[G|A] is easier problem of finding G given A: likelihood function

P[A] is prior knowledge about A:

Analytic continuation


P[G|A] is easier problem of finding G given A: likelihood function

P[A] is prior knowledge about A:

All methods to solve the above problemcan be formulated in terms of this relation

Historically first method to solve the problem of Fredholm kind I integral equation.

Tikhonov regularization method (1943)

Tikhonov regularization method (1943)

If Г is unit matrix:

Ill-posed!

Tikhonov regularizationto fight with the

saw tooth noise instability.

P[A|G] ~ P[G|A] P[A]Maximum entropy method

Likelihood(objective)function

Prior knowledge function



D(ω) is default model



1. One has escaped extra smoothening.

2. But one has got default model as an extra price.



1. We want to avoid extra smoothening.

2. We want to avoid default model as an extra price.

Maximum entropy method


Stochastic methods

Both items (extra smoothening and arbitrary default model) can be somehow circumvented by the group of stochastic methods.


Stochastic methods

The main idea of the stochastic methods is:

1.Restrict the prior knowledge to the minimal possible level (positive, normalized, etc…).

2. Change the likelihood function to the likelihood functional.


Stochastic methods

The main idea of the stochastic methods is:

1.Restrict the prior knowledge to the minimal possible level (positive, normalized, etc…). Avoids default model. 2. Change the likelihood function to the likelihood functional. Avoids saw-tooth noise.


Stochastic methods

Change the likelihood function to the likelihood functional. Avoids sawtooth noise.

Stochastic methodsLikelihood functional. Avoids sawtooth noise.

Sandvik, Phys. Rev. B 1998, isthe first practical attempt to think stochastically.


SOM was suggested in 2000.Mishchenko et al, Appendix B in Phys. Rev. B.


What is the special need for the stochastic sampling methods?

Stochastic optimization method.

1. One has to sample through solutions A(ω) which fit the correlation function G well.

2. One has to make some weighted sum of these well solutions A(ω).

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Stochastic Optimization method.

Particular solution L(i)(ω) for LSF is presented as a sum of a number K of rectangles with some width, height and center. Initial configuration of rectangles is created by random number generator (i.e. number K and all parameters of of rectangles are randomly generated). Each particular solution L(i)(ω) is obtained by a naïve method without regularization (though, varying number K).

Final solution is obtained after M steps of such procedure L(ω) = M-1 ∑i

L(i)(ω)

Each particular solution has saw tooth noise Final averaged solution L(ω) has no saw tooth noise though not regularized with sharp peaks/edges!!!!

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Self-averaging of the saw-tooth noise.

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Frohlich 3D polaron:spectral function

andoptical conductivity

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Frohlich polaron: spectral function and optical conductivity.

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Polaron mobility:

5 different regimes

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Mobility of 1D Holstein polaron

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Band conductionregime

Hopping activationregime

Documents

1 General Concept of Polaron and its Manifestations in Transport and Spectral Propertiesn A.S. Mishchenko RIKEN ( Institute of Physical and Chemical Research