1. DIAGRAMMATIC MONTE CARLO:
N. Prokof’ev
Advancing Research in Basic Science and Mathematics
Kourovka 2018

2. Diagrammatic Monte Carlo (Diag.MC)
Unbiased sampling of configuration space with varying number of continuous variables
path integrals
impurity solvers & continuous-time methods
Feynman diagrammatic expansions
just any sum of multi-dimensional integrals
Frochlich polaron case in practice (with all algorithmic details)

3. MC approach in one slide:
Configuration weight
quantity of interest
Multi-dimensional sum/integral
(label all possible states=configuration space)
Average over configurations generated
with probability density
How general?
or go fancy
For arbitrary functions F, G, and W > 0
Ignore sign for the moment (efficiency, not principle issue)

4. Single-spin flip algorithm
The most familiar example – Ising model
; configuration space , and
MC Markov-chain cycle:
Contribute
suggest an update
Accept with probability
Config.
Select at random any site
and suggest to flip spin
For “current” config.
(same as before or
updated, if accepted)
Single-spin flip algorithm
Guarantees that contributing configurations are generated with probability W

5. Contribution to the answer or weight (with differential measures!)
More general classical case:
the number of variables N is constant
General “quantum” case:
Integration variables
term order
different terms of
of the same order
(say, topologies)
Contribution to the answer
or weight (with differential measures!)
Diagrammatic Monte Carlo (DiagMC) = unbiased sampling of configuration space
with varying number of continuous variables
+ Estimators for thermodynamic properties and functions; data processing
+ Normalization, if there is no denominator

6. Configuration space = (diagram order, topology and types of lines, internal & external variables)
Diagram weight =

7. MC Markov-chain cycle:
Configuration space:
Config. weight:
MC Markov-chain cycle:
Contribute
suggest an update
Accept with probability
Config.
Updates of type A for same-order diagrams:
Business “as usual”
Updates of type B changing the diagram order:
Ooops …
Not a problem, because there are other factors 

8. Detailed Balance: solve equation for each pair of updates separately
Balance Equation:
If the desired probability density distribution
of configurations in the stochastic sum is
then the updating process has to be stationary
with respect to (equilibrium condition). Often
Flux out of
Flux to
Is the probability of proposing an update transforming to
Detailed Balance: solve equation for each pair of updates separately
Acceptance ratio:

9. Detailed Balance equation:
Updates of type A for same-order diagrams (the number of possibilities is unlimited):
1. Choose to perform a particular type A update with propability
2. Select variable in the set , e.g. at random. Probability is
where is the number of variables to choose from in this update. Let it be variable .
3. Propose new value for the selected variable. Probability is for discrete variables or
for continuous ones.
4. Accept the change with probability
Detailed Balance equation:
Solution for acceptance ratio
is normalized

10. Updates of type B changing the diagram order form complementary pairs:
2. Propose values for new variables from the probability distribution
3. Accept the change with probability
2. Accept the change with probability

11. Detailed Balance equation:
Updates of type B changing the diagram order form complementary pairs:
Detailed Balance equation:
Solution for acceptance ratio
All differential measures are gone!
Efficiency rules:
ENTER
Who has the biggest button …

12. Configuration space = (diagram order, types of lines, topology (if not summed over), internal variables)
Diagram order
Diagram topology
MC update
Cont. variables
This is NOT: write/enumerate diagram after diagram,
compute its value, and then sum

13. Polaron problem: quasiparticle
Electrons in semiconducting crystals (electron-phonon polarons) e e
electron
phonons
el.-ph.
interaction

14. + + + + + + … Green function: phonons el.-ph. interaction electron
Sum of all Feynman diagrams
Positive definite series in the representation + + + + +
+ …

15. Feynman
digrams
Graph-to-math correspondence:
is a product of

16. Doing MC in the Feynman diagram configuration space is an endless fun!
Diagrams for:
there are also diagrams for optical conductivity, etc.
Doing MC in the Feynman diagram
configuration space is an endless fun!

17. Frohlich polaron = single electron in ionic semiconductor

18. Always accepted, New time:
Type A: changing ”external” time (the simplest version)
New time:
Exponential probability density
Transformation method:
Always accepted,

19. Type A: changing “external” time (another, more fancy, version)
Acceptance ratio

20. Always accepted, Type A: changing internal time
1. Select any electron interval except the last one at random
Always accepted,

21. Type A: changing internal momentum angle
1. Select any phonon line out of n interval at random
Momenta for q=0

22. Type A: changing internal momentum modulus
1. Select any phonon line out of n interval at random

23. Type A: changing local “topology”
1. Select any electron interval except the first and last one at random

24. Type B: changing diagram order (one possible example)
Insert/Delete a phonon line (increasing/decreasing the diagram order by one)

25. Detailed Balance equation:
Type B: changing diagram order (one possible example)
Detailed Balance equation:
Solution for acceptance ratio in Insert
Solution for acceptance ratio in Delete

26. Possible distribution
Type B: changing diagram order (one possible example)
Recall that
Possible distribution
Not perfect, but FAPP “good enough”

27. Normalization: histogram special “bin” where is known exactly
Normalized histogram

28. Normalization using “desined bin”:

29. Normalization example (statistics is that for G diagrams)
Define “normalization” subspace and compute some physical answer in it
Do it analytically (if possible) or numerically (to high accuracy)
Collect statistics for your answers as usual
and record the number of configurations in the normalization subspace
Properly normalized physical answer:

30. This is it! Collect statistics for , Monte Carlo estimators for energy,
group velocity, effective mass, number and distribution of phonons in the
cloud, or some corr. function.
Analyze it.

31. Analysing data: Quasiparticle energy
probability of getting a bare electron
(Lehman expansion)
probability of getting
two phonons in the
polaron cloud
Slope

32. Analyzing data: polaron energy estimator
[ In the limit ]
[ In the limit ]

33. Not Landau Pekar limit yet: broad superposition of phonon-number states, while
A. Mishchenko, B. Svistunov, A. Sakamoto, NP, ‘98, ‘00

34. The simplest (still physical) “bold” Diag.MC example
S-wave scattering by spherically symmetric potential
B. Svistunov, NP ‘07

35. s-wave scattering length?