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

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)

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)

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

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

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

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 

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:

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

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  

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 …

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

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

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

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

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!

Frohlich polaron = single electron in ionic semiconductor              

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

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

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

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

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

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

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

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

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

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

Normalization using “desined bin”:

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:

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.

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

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

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

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

s-wave scattering length?