Transcript Large-N Quantum Field Theories and Nonlinear Random Processes

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
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
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
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
Difficulties with “worm’’ DiagMC
Typical problems:
• Nonconvergence of perturbative expansion (noncompact 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...
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
Example: SD equations in φ4 theory
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:
Example: sunset diagram…
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
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
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
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)
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
Resummation: fits by multiple
exponents
Resummation: positions of poles
Two-point function
Connected truncated
four-point function
2-3 poles can be extracted with reasonable accuracy
Test: triviality of φ4 theory
Renormalized mass:
Renormalized coupling:
CPU time:
several hrs/point
(2GHz core)
Compare
[Wolff 1101.3452]
Several coremonths (!!!)
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
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