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

2. Motivation Look for alternatives to the standard Monte-Carlo
to address the following problems:
Sign problem (finite chemical potential, fermions etc.)
Large-N extrapolation (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., ]
Explicit knowledge of the structure of perturbative series required (difficult for SU(N) see e.g. [Gattringer, ])
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 of Cn,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 ]
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 ]
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: (this talk)
ArXiv: , (large-N theories)
Some sample codes are available at: