General Concept of Polaron and

1
General Concept of Polaron and its
Manifestations in Transport and Spectral
Propertiesn

• S. Mishchenko
• RIKEN (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 conductivity Polaro
n mobility 5 different regimes. Hole in
t-J-Holstein model.
2
General Concept of Polaron and its
Manifestations in Transport and Spectral
Propertiesn

• S. Mishchenko
• RIKEN (Institute of Physical and Chemical
  Research), Japan

3
General Concept of Polaron and its
Manifestations in Transport and Spectral
Propertiesn

• S. Mishchenko
• RIKEN (Institute of Physical and Chemical
  Research), Japan

From Tokyo
To Tokyo 500 m
4

• Electron-phonon
• interaction.
• Origin and Hamiltonian.

5
Polaron momentum space
Where this interaction comes from?
6
Band structure in rigid lattice.
H0 E0 Si Ci Ci t Sij Ci Cj
u (bi bi)
Hint a Si Ci Ci u
Hint ? Si Ci Ci (bi bi)
E0
Brething
E0-au
7
Band structure in rigid lattice.
H0 E0 Si Ci Ci t Sij Ci Cj
u (bi bi)
Hint a Si Ci Ci u
Hint ? Si Ci Ci (bi bi)
E0
Holstein
E0-au
8
Band structure in rigid lattice.
H0 E0 Si Ci Ci t Sij Ci Cj
u (bi bi)
Hint a Sij Ci Cj u
Hint ? Si Ci Cj (bi bi)
t
SSH
t-au
9
Polaron momentum space
Scattered in momentum space
-q
k
k-q
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Polaron momentum space
Which physical characteristics may change?
11
Properties of groundstate
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Which physical characteristics may change?

1. Energy of particle
2. Dispersion of particle
3. Effective mass of particle
4. Z-factor of particle

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Which physical characteristics are interesting to
study?

1. Energy of particle
2. Dispersion of particle
3. Effective mass of particle
4. Z-factor of particle
5. Structure of phonon cloud

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Frohlich polaron
GS energy
Effective mass
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Frohlich polaron
Polaron cloud
Z-factor
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Holstein polaron
V(q)const
Number of phonons in the polaron cloud
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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!
23
Physical properties under interest Lehman
function
Lehmann 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 momentum is 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
spectral properties is rather general
ARPES
Currentcurrent corelation function is related to
optical absorption by polarons by the same
expression
Optical absorption
µ s (??0)/en
Mobility
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There are a lot of problems where one has to
solve Fredholm integral equation of 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,?)
m and ? are 2D vectors
K(m,?) is a 2D x 2D noise distributon function
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Tomography image reconstruction (CT scan)
m and ? are 2D vectors
K(m,?) is a 2D x 2D distribution function
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Aircraft stability
Nuclear reactor operation
Image deblurring
What is dramatic in the problem?
A lot of other
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Ill-posed!
Aircraft stability
Nuclear reactor operation
Image deblurring
What is dramatic in the problem?
A lot of other
38
Ill-posed!
We cannot obtain an exact solution not because of
some approximations of our approaches. Instead,
we have to admit that the exact solution does
not exist at all!
39
Ill-posed!

• 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
  order
• to chose among many approximate solutions.

40
Ill-posed!
Next player stochastic methods
Physics department Max Ent.
Engineering department Tikhonov Regularization
Statistical department ridge regression
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Ill-posed!
Because of noise present in the input data
G(m) there is no unique A(?n)A(n) which exactly
satisfies the equation.
Hence, one can search for the least-square
fitted solution A(n) which minimizes
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Ill-posed!
Saw tooth noise instability due to small
singular values.
Explicit expression
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Ill-posed!
General formulation of methods to solve
ill-posed problems in terms of Bayesian
statistical inference.
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Bayes theorem PAG PG PGA PA PAG
conditional probability that the
spectral function is A provided
the correlation function is G
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Bayes theorem PAG PG PGA PA PAG
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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PAG PGA PA PGA is easier
problem of finding G given A
likelihood function PA is prior knowledge
about A
Analytic continuation
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PAG PGA PA PGA is easier
problem of finding G given A
likelihood function PA is prior knowledge
about A
All methods to solve the above problem can 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 ? is unit matrix
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Ill-posed!
Tikhonov regularization to fight with the saw
tooth noise instability.
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Maximum entropy method
PAG PGA PA
Likelihood (objective) function
Prior knowledge function
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Maximum entropy method
PAG PGA PA
D(?) is default model
Prior knowledge function
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Maximum entropy method
PAG PGA PA

1. One has escaped extra smoothening.
2. But one has got default model as an extra price.

Prior knowledge function
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Maximum entropy method
PAG PGA PA

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

Prior knowledge function
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Stochastic methods
PAG PGA PA
Both items (extra smoothening and arbitrary
default model) can be somehow circumvented by
the group of stochastic methods.
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Stochastic methods
PAG PGA PA

• The main idea of the stochastic methods is
• 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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Stochastic methods
PAG PGA PA

• The main idea of the stochastic methods is
• 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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Stochastic methods
PAG PGA PA
Change the likelihood function to the
likelihood functional. Avoids sawtooth noise.
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Stochastic methods
Likelihood functional. Avoids sawtooth noise.
Sandvik, Phys. Rev. B 1998, is the first
practical attempt to think stochastically.
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Stochastic methods
Likelihood functional. Avoids sawtooth noise.
SOM was suggested in 2000. Mishchenko et al,
Appendix B in Phys. Rev. B.
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Stochastic methods
Likelihood 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(?)
which fit the correlation function G well. 2.
One has to make some weighted sum of these well
solutions A(?).
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Stochastic Optimization method.

• Particular solution L(i)(?) 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)(?) is obtained by
  a naïve
• method without regularization (though,
  varying number K).
• Final solution is obtained after M steps of such
  procedure
• L(?) M-1 ?i L(i)(?)
• Each particular solution has saw tooth noise
• Final averaged solution L(?) 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 polaronspectral functionandoptical
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 mobility5 different regimes
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Mobility of 1D Holstein polaron
Band conduction regime
Hopping activation regime