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DIAGRAMMATIC MONTE CARLO: From polarons to path-integrals to skeleton technique

DIAGRAMMATIC MONTE CARLO: From polarons to path-integrals to skeleton technique. N. Prokof’ev. AFOSR MURI. Advancing Research in Basic Science and Mathematics. KITPC 5/13/14. Classical MC. the number of variables N is constant. Quantum MC (often). Integration variables. term order.

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DIAGRAMMATIC MONTE CARLO: From polarons to path-integrals to skeleton technique

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  1. DIAGRAMMATIC MONTE CARLO: From polarons to path-integrals to skeleton technique N. Prokof’ev AFOSR MURI Advancing Research in Basic Science and Mathematics KITPC 5/13/14

  2. Classical MC the number of variables N is constant Quantum MC (often) Integration variables term order Contribution to the answer or weight (with differential measures!) different terms of of the same order

  3. Monte Carlo (Metropolis) cycle: Accept with probability Diagram suggest a change Same order diagrams: Business as usual Updating the diagram order: Ooops

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

  5. Let us be more specific. Equation to solve: new variables are proposed from the normalized probability distribution Solution: All differential measures are gone! Efficiency rules: - try to keep - simple analytic function ENTER

  6. Standard Monte Carlo setup: (depends on the model and it’s representation) - configuration space - each cnf. has a weight factor - quantity of interest

  7. Statistics: Monte Carlo states generated from probability distribution Anything: Monte Carlo states generated from probability distribution Connected Feynman Anything = diagrams, e.g. for the proper self-energy diagram order topology internal variables Answer to S. Weinberg’s question Monte Carlo

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

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

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

  11. Feynman digrams Graph-to-math correspondence: is a product of Positive definite series in the representation

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

  13. The simplest algorithm has three updates: Insert/Delete pair (increasing/decreasing the diagram order) In Delete select the phonon line to be deleted at random

  14. The optimal choice of depends on the model Frohlich polaron: 1. Select anywhere on the interval from uniform prob. density 2. Select anywhere to the left of from prob. density (if reject the update) 3. Select from Gaussian prob. density , i.e.

  15. New : Standard “heat bath” probability density Always accepted,

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

  17. Normalization using “desined bin”: bin “0”

  18. This is it! Collect statistics for or some other corr. function. Analyze it.

  19. Analysing data: dispersion relation probability of getting a bare electron (Lehman expansion) probability of getting two phonons in the polaron cloud Slope

  20. Standard model Hubbard model Coulomb gas Heisenberg model Periodic Anderson & Kondo lattice models … High-energy physics High-Tc superconductors Quantum chemistry & band structure Quantum magnetism Heavy fermion materials … - Introduced in mid 1960s or earlier - Still not solved (just a reminder, today is 01/13/2012) - Admit description in terms of Feynman diagrams

  21. Feynman Diagrams & Physics of strongly correlated many-body systems In the absence of small parameters, are they useful in higher orders? “Divergent series are the devil's invention...” Yes, with sign-blessing for regularized skeleton graphs! N.Abel, 1828: And if they are, how to handle millions and billions of skeleton graphs? Sample them with Diagrammatic Monte Carlo techniques (teach computers rules of quantum field theory) Steven Weinberg, Physics Today, Aug. 2011 : “Also, it was easy to imagine any number of quantum field theories of strong interactions but what could anyone do with them?” Unbiased solutions beased on millions of graphs with extrapolation to the infinite diagram order From current strong-coupling theories based on one lowest order skeleton graph (MF, RPA, GW, SCBA, GG0, GG, …

  22. Skeleton diagrams up to high-order: do they make sense for ? NO YES Divergent series outside of finite convergence radius can be re-summed. Series diverge for large even if they converge for Skeleton series are not based on Many systems remain well-defined for Lattice fermions, quantum magnetism, resonant fermions … _+ regularization Dyson “Expansion in is asymptotic if for some the system becomes pathological”: Continuous space bosons and fermions collapse to infinite density for Math. Statement: number of skeleton graphs asymptotic series with zero conv. radius Number of graphs is but because they alternate in sign they may all cancel each other to near zero ! “Sign blessing” Asymptotic series for with zero convergence radius are useless! Start computing high-order diagrams!

  23. Re-summation of divergent series with finite convergence radius. Example: бред какойто Define a function such that: (Gauss) (Lindeloef) Construct sums and extrapolate to get

  24. Conventional Sign-problem vs Sign-blessing Sign-problem: Computational complexity is exponential in system volume and error bars explode before a reliable exptrapolation to can be made (diagrams for ) Feynamn diagrams: No limit to take, selfconsistent formulation, admit analytic results and partial resummations. (for ) Sign-blessing: Number of diagram of order is factorial thus the only hope for good series convergence properties is sign alter- nation of diagrams leading to their cancellation. Still, i.e. Smaller and smaller error bars are likely to come at exponential price (unless convergence is exponential). (diagrams for )

  25. Diagrammatic Monte Carlo in the generic many-body setup Feynman diagrams for free energy density x x x x

  26. Bold (self-consistent) Diagrammatic Monte Carlo Diagrammatic technique for ln(Z ) diagrams: admits partial summation and self-consistent formulation No need to compute all diagrams for and : Dyson Equation: Screening: Calculate irreducible diagrams for , , … to get , , …. from Dyson equations

  27. . . . . . . . . In terms of “exact” propagators Dyson Equation: Screening:

  28. More tools: Build diagrams using ladders: (contact potential) In terms of “exact” propagators Dyson Equations:

  29. Fully dressed skeleton graphs (Heidin): . . Irreducible 3-point vertex: all accounted for already!

  30. Unpolarized system at unitarity:BCS-BEC crossover Unitary gas: when and are the only length/energy scales

  31. Answering Weinberg’s question: Equation of State for ultracold fermions & neutron matter at unitarity MIT group: Mrtin Zwierlein, Mark Ku, Ariel Sommer, Lawrence Cheuk, Andre Schirotzek Uncertainty due to location of the resonance BDMC results Kris Van Houcke, Felix Werner, Evgeny Kozik, Boris Svistunov, NP virial expansion (3d order) Ideal Fermi gas QMC for connected Feynman diagrams NOT particles! Sign blessing Sign problem

  32. Lattice path-integrals for bosons and spins are “diagrams” of closed loops! + + imaginary time

  33. Diagrams for Diagrams for imaginary time imaginary time lattice site lattice site The rest is conventional worm algorithm in continuous time

  34. Simulating Bose-Hubbard model “as is” and comparing to experiments: (in this example, ) Nature Physics, 6, 998–1004, (2010)

  35. It is realistic to do about 2,000,000 or more particles at temperatures relevant for the experiment. Phys. Rev. Lett. 103, 085701 (2009)

  36. Path-integrals in continuous space are “diagrams” of closed loops too! P 2 1 P

  37. Diagrams for the attractive tail in : Ifand N>>1 all the effort is for something small ! statistical interpretation ignore : stat.weight 1 Account for : stat. weight p Faster than conventional scheme for N>30, scalable (size independent) updates with exact account of interactions between all particles

  38. 3D @ s.v.p. 64 experiment 2048

  39. Other applications: Continuous-time QMC solves (impurity solvers) are standard DMC schemes Fermions with contact interaction 1 3 2 9 Rubtsov (2003) Most efficient solvers for DMFT and DCA are based on this approach

  40. Maier et al.

  41. Critical point from pair distribution function Criticality: from zero of

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