Stochastic solution of Schwinger-Stochastic solution of Schwinger-Dyson equations: an alternative to Dyson equations: an alternative to

Diagrammatic Monte-CarloDiagrammatic Monte-Carlo

[ArXiv:[ArXiv:1009.40331009.4033, , 1104.34591104.3459, , 1011.26641011.2664]]

Pavel BuividovichPavel Buividovich

(ITEP, Moscow and JINR, Dubna)(ITEP, Moscow and JINR, Dubna)

Lattice 2011, Squaw Valley, USA, 13.07.2011Lattice 2011, Squaw Valley, USA, 13.07.2011

MotivationMotivation

Look for alternatives to theLook for alternatives to the standard Monte-standard Monte-CarloCarlo

to address the following problems:to address the following problems:

• Sign problemSign problem (finite chemical potential, (finite chemical potential, fermions etc.)fermions etc.)

• Large-NLarge-N extrapolationextrapolation (AdS/CFT, AdS/QCD)(AdS/CFT, AdS/QCD)• SUSY on the lattice?SUSY on the lattice?• Elimination of finite-volume effectsElimination of finite-volume effects

Diagrammatic MethodsDiagrammatic Methods

Motivation: Motivation: Diagrammatic MC, Diagrammatic MC, Worm Algorithm, ...Worm Algorithm, ...

• Standard Monte-Carlo:Standard Monte-Carlo: directly evaluate the directly evaluate the path integralpath integral

• Diagrammatic Monte-Carlo:Diagrammatic Monte-Carlo: stochastically sumstochastically sum all the terms in the all the terms in the perturbative expansionperturbative expansion

Motivation: Motivation: Diagrammatic MC, Diagrammatic MC, Worm Algorithm, ...Worm Algorithm, ...

• Worm AlgorithmWorm Algorithm [Prokof’ev, Svistunov]:[Prokof’ev, Svistunov]: Directly sample Directly sample Green functions, Green functions, Dedicated simulations!!!Dedicated simulations!!!

Example: Example: Ising modelIsing model

X, YX, Y – – head and head and tail of the wormtail of the worm

Applications:Applications: • Discrete symmetry groups Discrete symmetry groups a-la Isinga-la Ising [Prokof’ev, Svistunov][Prokof’ev, Svistunov]• O(N)/CP(N) lattice theories O(N)/CP(N) lattice theories [Wolff][Wolff] – – so far quite complicatedso far quite complicated

Difficulties with “worm’’ Difficulties with “worm’’ DiagMCDiagMC

Typical problems:Typical problems:

• NonconvergenceNonconvergence of perturbative expansion of perturbative expansion (non-compact variables) (non-compact variables) [Prokof’ev et al., [Prokof’ev et al., 1006.4519]]• Explicit knowledgeExplicit knowledge of the structure of of the structure of perturbative seriesperturbative series required (difficult for required (difficult for SU(N) see e.g. SU(N) see e.g. [Gattringer, [Gattringer, 1104.2503]]))• Finite convergence radiusFinite convergence radius for for strong strong couplingcoupling• Algorithm complexity Algorithm complexity grows withgrows with N N• Weak-coupling expansionWeak-coupling expansion (=lattice (=lattice perturbation theory): perturbation theory): complicated, volume-complicated, volume-dependent...dependent...

DiagMC based on SD DiagMC based on SD equationsequations

Basic idea:Basic idea:• Schwinger-Dyson (SD) equations:Schwinger-Dyson (SD) equations: infinite infinite hierarchy of linear equations for correlators hierarchy of linear equations for correlators G(xG(x11, , …, x…, xnn))• Solve SD equations:Solve SD equations: interpret them as interpret them as steady-steady-state equationsstate equations for some random process for some random process• Space of states:Space of states: sequences of coordinates sequences of coordinates {x{x11, , …, x…, xnn}}• Extension of the “worm” algorithm:Extension of the “worm” algorithm: multiple multiple “heads”“heads” and and “tails”“tails” but no but no “bodies”“bodies”

Main advantages:Main advantages:• No truncation No truncation of SD equations requiredof SD equations required• No explicitNo explicit knowledge of knowledge of perturbative seriesperturbative series requiredrequired• Easy to take Easy to take large-N limitlarge-N limit

Example: Example: SD equations in SD equations in φφ44 theorytheory

SD equations for SD equations for φφ44 theory: theory: stochastic interpretationstochastic interpretation

• Steady-state equations for Markov processes:Steady-state equations for Markov processes:

• Space of states:Space of states: sequences of momenta sequences of momenta {p{p11, …, p, …, pnn}}

• Possible transitions:Possible transitions: Add pair of momentaAdd pair of momenta {p, -p} {p, -p} at positions at positions 1, A = 2 … n + 11, A = 2 … n + 1 Add up three first momentaAdd up three first momenta (merge)(merge)

• Start with {p, -p}Start with {p, -p}

• Probability for Probability for new momentanew momenta::

Example: sunset diagram…Example: sunset diagram…

Normalizing the transition probabilitiesNormalizing the transition probabilities• Problem:Problem: probabilityprobability of “Add momenta” of “Add momenta” grows as (n+1)grows as (n+1), rescaling , rescaling G(pG(p11, … , p, … , pnn) – does not help. ) – does not help. Manifestation of series Manifestation of series divergence!!!divergence!!!

• Solution:Solution: explicitly count diagram order m. explicitly count diagram order m. Transition Transition probabilities depend on mprobabilities depend on m

• Extended state space: {pExtended state space: {p11, … , p, … , pnn} and m – diagram order} and m – diagram order

• Field correlators:Field correlators:

• wwmm(p(p11, …, p, …, pnn)) – – probabilityprobability to encounter to encounter m-th order diagramm-th order diagram with with momenta momenta {p{p11, …, p, …, pnn} on external legs} on external legs

Normalizing the transition probabilitiesNormalizing the transition probabilities

• Finite transition probabilities:Finite transition probabilities:

• Factorial divergence Factorial divergence of series is absorbed into the of series is absorbed into the growth ofgrowth of CCn,mn,m !!!!!!

• Probabilities Probabilities (for optimal x, y):(for optimal x, y): Add momenta:Add momenta:

Sum up momenta +Sum up momenta + increase the order:increase the order:

• Otherwise Otherwise restartrestart

Diagrammatic Diagrammatic interpretationinterpretation

Histories between “Restarts”: Histories between “Restarts”: unique Feynman unique Feynman diagramsdiagrams

Measurements of Measurements of connected, 1PI, 2PI correlatorsconnected, 1PI, 2PI correlators are are possible!!! possible!!! In practice: label connected legsIn practice: label connected legs

Kinematical factorKinematical factor for each diagram: for each diagram:

qqii are are independent momentaindependent momenta, Q, Qjj – depend on q – depend on qii

Monte-Carlo integration over independent Monte-Carlo integration over independent momentamomenta

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

DD))

ResummationResummation• Integral representationIntegral representation of of CCn,m n,m = = ΓΓ(n/2 + m + 1/2) x(n/2 + m + 1/2) x-(n--(n-

2)2) y y-m-m::

Pade-Borel resummation.Pade-Borel resummation. Borel image of Borel image of correlators!!!correlators!!!

• PolesPoles of Borel image: of Borel image: exponentialsexponentials in w in wn,mn,m

• Pade approximants are Pade approximants are unstableunstable• Poles can be found byPoles can be found by fitting fitting• Special fitting procedureSpecial fitting procedure using SVD of Hankel using SVD of Hankel matricesmatrices

Resummation: Resummation: fits by multiple fits by multiple exponentsexponents

Resummation: Resummation: positions of positions of polespoles

Two-point functionTwo-point function Connected truncated Connected truncated four-point functionfour-point function

2-3 poles can be extracted with reasonable accuracy2-3 poles can be extracted with reasonable accuracy

Test: triviality of Test: triviality of φφ44 theory theoryRenormalized mass: Renormalized coupling:

CPU time:CPU time: several several hrs/pointhrs/point(2GHz core)(2GHz core)

CompareCompare[Wolff [Wolff 1101.3452]1101.3452]Several core-Several core-months (!!!)months (!!!)

Conclusions: DiagMC from SD Conclusions: DiagMC from SD eq-seq-s

Advantages:Advantages:• Implicit construction of Implicit construction of perturbation theoryperturbation theory• No critical slow-downNo critical slow-down• Naturally treats divergent Naturally treats divergent seriesseries• Easy to take large-N limit Easy to take large-N limit [Buividovich [Buividovich 1009.4033]]• No truncation of SD eq-sNo truncation of SD eq-s

Disadvantages:Disadvantages:• No “strong-coupling” No “strong-coupling” expansions (so far?)expansions (so far?)• Large statistics in IR Large statistics in IR regionregion• Requires some external Requires some external resummation procedureresummation procedure

Extensions?Extensions?• Spontaneous symmetry breaking Spontaneous symmetry breaking (1/(1/λλ – terms??? – terms???))• Non-Abelian LGT: loop equations Non-Abelian LGT: loop equations [Migdal, Makeenko, [Migdal, Makeenko, 1980]1980]

Strong-coupling expansion:Strong-coupling expansion: seems quite easy seems quite easy Weak-coupling expansion:Weak-coupling expansion: more adequate, but not more adequate, but not easyeasy

• SupersymmetrySupersymmetry and M(atrix)-models and M(atrix)-models

Thank you for your Thank you for your attention!!!attention!!!References:References:• ArXiv:ArXiv:1104.3459 (this talk) (this talk)• ArXiv:ArXiv:1009.4033, , 1011.2664 (large-N (large-N theories) theories)

• Some Some sample codessample codes are available at: are available at:

http://www.lattice.itep.ru/~pbaivid/http://www.lattice.itep.ru/~pbaivid/codes.htmlcodes.html