1 general concept of polaron and its manifestations in transport and spectral propertiesn a.s....
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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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A. S. MishchenkoRIKEN (Institute of Physical and Chemical Research), Japan
General Concept of Polaron and its Manifestations in Transport and
Spectral Propertiesn
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A. S. MishchenkoRIKEN (Institute of Physical and Chemical Research), Japan
ToTokyo500 m
FromTokyo
General Concept of Polaron and its Manifestations in Transport and
Spectral Propertiesn
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• 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
Band structure in rigid lattice.
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
Band structure in rigid lattice.
t-αu
Hint= α Σij Ci+ Cj u u ~ (bi
+ + bi)
Hint= γ Σi Ci+ Cj (bi
+ + bi) t
SSH
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Polaron: momentum space
Scattered in momentum space
k-qk
-q
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Polaron: momentum space
Which physical characteristics may change?
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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
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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. ……
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Frohlich polaron
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GS energy Effective mass
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Frohlich polaron
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Polaron cloudZ-factor
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Holstein polaron
16Number of phonons in the polaron cloud
V(q)=const
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• 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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SSH polaron(Su-Schrieffer-Heeger)
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SSH polaron(Su-Schrieffer-Heeger)
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SSH polaron(Su-Schrieffer-Heeger)
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
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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
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There are a lot ofproblems where one has to solveFredholmintegral equationof the first kind
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Many-particle Fermi/Boson system in imaginary times representation
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Many-particle Fermi/Boson system in Matsubara representation
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Optical conductivity at finite T in imaginary times representation
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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
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Tomography image reconstruction (CT scan)
K(m,ω) is a 2D x 2Ddistribution function
m and ω are2D vectors
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Aircraftstability
Nuclearreactor
operation
Image deblurring
A lot ofother…
What is dramaticin the problem?
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Aircraftstability
Nuclearreactor
operation
Image deblurring
A lot ofother…
What is dramaticin the problem?
Ill-posed!
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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!
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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.
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Ill-posed!
Physics department:Max Ent.
Engineering department:Tikhonov Regularization
Statistical department:ridge regression
Next player: stochastic methods
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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:
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Ill-posed!
Explicit expression:
Saw tooth noise instabilitydue to small singular values.
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Ill-posed!
General formulation of methods to solve ill-posed problems in terms ofBayesian statistical inference.
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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
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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
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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
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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:
All methods to solve the above problemcan be formulated in terms of this relation
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Historically first method to solve the problem of Fredholm kind I integral equation.
Tikhonov regularization method (1943)
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Tikhonov regularization method (1943)
If Г is unit matrix:
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Ill-posed!
Tikhonov regularizationto fight with the
saw tooth noise instability.
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P[A|G] ~ P[G|A] P[A]Maximum entropy method
Likelihood(objective)function
Prior knowledge function
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P[A|G] ~ P[G|A] P[A]Maximum entropy method
Prior knowledge function
D(ω) is default model
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P[A|G] ~ P[G|A] P[A]Maximum entropy method
Prior knowledge function
1. One has escaped extra smoothening.
2. But one has got default model as an extra price.
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P[A|G] ~ P[G|A] P[A]
Prior knowledge function
1. We want to avoid extra smoothening.
2. We want to avoid default model as an extra price.
Maximum entropy method
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P[A|G] ~ P[G|A] P[A]
Stochastic methods
Both items (extra smoothening and arbitrary default model) can be somehow circumvented by the group of stochastic methods.
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P[A|G] ~ P[G|A] P[A]
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.
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P[A|G] ~ P[G|A] P[A]
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.
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P[A|G] ~ P[G|A] P[A]
Stochastic methods
Change the likelihood function to the likelihood functional. Avoids sawtooth noise.
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Stochastic methodsLikelihood functional. Avoids sawtooth noise.
Sandvik, Phys. Rev. B 1998, isthe first practical attempt to think stochastically.
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Stochastic methodsLikelihood functional. Avoids sawtooth noise.
SOM was suggested in 2000.Mishchenko et al, Appendix B in Phys. Rev. B.
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Stochastic methodsLikelihood functional. Avoids sawtooth noise.
What is the special need for the stochastic sampling methods?
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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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Self-averaging of the saw-tooth noise.
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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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Frohlich polaron: spectral function and optical 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