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Pavel Buividovich (ITEP, Moscow and JINR, Dubna) Lattice 2011, Squaw Valley, USA, 13.07.2011

Stochastic solution of Schwinger-Dyson equations: an alternative to Diagrammatic Monte-Carlo [ArXiv: 1009.4033 , 1104.3459 , 1011.2664 ]. Pavel Buividovich (ITEP, Moscow and JINR, Dubna) Lattice 2011, Squaw Valley, USA, 13.07.2011. Motivation.

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Pavel Buividovich (ITEP, Moscow and JINR, Dubna) Lattice 2011, Squaw Valley, USA, 13.07.2011

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  1. Stochastic solution of Schwinger-Dyson equations: an alternative to Diagrammatic Monte-Carlo[ArXiv:1009.4033, 1104.3459, 1011.2664] Pavel Buividovich (ITEP, Moscow and JINR, Dubna) Lattice 2011, Squaw Valley, USA, 13.07.2011

  2. Motivation Look for alternatives to thestandard Monte-Carlo to address the following problems: • Sign problem(finite chemical potential, fermions etc.) • Large-Nextrapolation(AdS/CFT, AdS/QCD) • SUSY on the lattice? • Elimination of finite-volume effects Diagrammatic Methods

  3. Motivation: Diagrammatic MC, Worm Algorithm, ... • Standard Monte-Carlo: directly evaluate the path integral • Diagrammatic Monte-Carlo:stochastically sum all the terms in the perturbative expansion

  4. Motivation: Diagrammatic MC, Worm Algorithm, ... • Worm Algorithm[Prokof’ev, Svistunov]: Directly sample Green functions, Dedicated simulations!!! Example: Ising model X, Y – head and tail of the worm Applications: • Discrete symmetry groups a-la Ising[Prokof’ev, Svistunov] • O(N)/CP(N) lattice theories [Wolff] – so far quite complicated

  5. Difficulties with “worm’’ DiagMC • Typical problems: • Nonconvergence of perturbative expansion (non-compact variables) [Prokof’ev et al., 1006.4519] • Explicit knowledge of the structure of perturbative series required (difficult for SU(N) see e.g. [Gattringer, 1104.2503]) • Finite convergence radius for strong coupling • Algorithm complexity grows with N • Weak-coupling expansion (=lattice perturbation theory): complicated, volume-dependent...

  6. DiagMC based on SD equations • Basic idea: • Schwinger-Dyson (SD) equations: infinite hierarchy of linear equations for correlators G(x1, …, xn) • Solve SD equations: interpret them as steady-state equations for some random process • Space of states: sequences of coordinates {x1, …, xn} • Extension of the “worm” algorithm: multiple “heads” and “tails” but no “bodies” • Main advantages: • No truncation of SD equations required • No explicit knowledge of perturbative series required • Easy to take large-N limit

  7. Example: SD equations in φ4 theory

  8. SD equations for φ4 theory: stochastic interpretation • Steady-state equations for Markov processes: • Space of states: • sequences of momenta {p1, …, pn} • Possible transitions: • Add pair of momenta {p, -p} at positions 1, A = 2 … n + 1 • Add up three first momenta (merge) • Start with {p, -p} • Probability for new momenta:

  9. Example: sunset diagram…

  10. Normalizing the transition probabilities • Problem:probability of “Add momenta” grows as (n+1), rescaling G(p1, … , pn) – does not help. Manifestation of series divergence!!! • Solution: explicitly count diagram order m. Transition probabilities depend on m • Extended state space: {p1, … , pn} and m – diagram order • Field correlators: • wm(p1, …, pn) – probability to encounter m-th order diagram with momenta {p1, …, pn} on external legs

  11. Normalizing the transition probabilities • Finite transition probabilities: • Factorial divergence of series is absorbed into the growth ofCn,m!!! • Probabilities (for optimal x, y): • Add momenta: • Sum up momenta + increase the order: • Otherwise restart

  12. Diagrammatic interpretation Histories between “Restarts”: unique Feynman diagrams Measurements of connected, 1PI, 2PI correlators are possible!!! In practice: label connected legs Kinematical factor for each diagram: qi are independent momenta, Qj – depend on qi Monte-Carlo integration over independent momenta

  13. Critical slowing down? Transition probabilities do not depend on bare mass or coupling!!! (Unlike in the standard MC) No free lunch: kinematical suppression of small-p region (~ ΛIRD)

  14. Resummation • Integral representation of Cn,m = Γ(n/2 + m + 1/2) x-(n-2) y-m: • Pade-Borel resummation. Borel image of correlators!!! • Poles of Borel image: exponentials in wn,m • Pade approximants are unstable • Poles can be found by fitting • Special fitting procedure using SVD of Hankel matrices

  15. Resummation: fits by multiple exponents

  16. Resummation: positions of poles Connected truncated four-point function Two-point function 2-3 poles can be extracted with reasonable accuracy

  17. Test: triviality of φ4 theory Renormalized mass: Renormalized coupling: CPU time: several hrs/point (2GHz core) Compare [Wolff 1101.3452] Several core-months (!!!)

  18. Conclusions: DiagMC from SD eq-s • Advantages: • Implicit construction of perturbation theory • No critical slow-down • Naturally treats divergent series • Easy to take large-N limit [Buividovich 1009.4033] • No truncation of SD eq-s • Disadvantages: • No “strong-coupling” expansions (so far?) • Large statistics in IR region • Requires some external resummation procedure • Extensions? • Spontaneous symmetry breaking (1/λ – terms???) • Non-Abelian LGT: loop equations [Migdal, Makeenko, 1980] • Strong-coupling expansion: seems quite easy • Weak-coupling expansion: more adequate, but not easy • Supersymmetry and M(atrix)-models

  19. Thank you for your attention!!! • References: • ArXiv:1104.3459 (this talk) • ArXiv:1009.4033, 1011.2664 (large-N theories) • Some sample codes are available at: • http://www.lattice.itep.ru/~pbaivid/codes.html

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