Diagrammatic Monte-Carlo
algorithms for large-N
quantum field theories from
Schwinger-Dyson equations
Pavel Buividovich (Regensburg University)
ECT* workshop on DiagMC
Motivation: QCD side
• DiagMC in strong-coupling lattice QCD at finite density [de
Forcrand,Vairinhos,Gattringer,Chandrasekharan,…]
• Few orders of SC expansion NOT SYSTEMATIC, BUT EFFICIENT
SYSTEMATIC WAY: DUAL REPRESENTATIONS/SC EXPANSIONS
• Abelian Lattice Gauge Theories (also with fermions)
• Scalar field theories
• O(N) / CPN sigma models
OFTEN, DUAL REPRESENTATIONS are very PHYSICAL
(In QCD, already the lowest order gives CONFINEMENT)
• BUT: no convenient duality for non-Abelian fields (ClebschGordan coefficients are cumbersome+sign-alternating)
• Weak-coupling expansions are also cumbersome and difficult to
re-sum
• Systematic DiagMC in the non-Abelian case?
• Avoid manual construction of duality transformations?
• Avoid Borel resummations?
Outline
DiagMC algorithms can be constructed in a
universal way from Schwinger-Dyson equations
• General structure of SD equations
• Solution of SD equations by Metropolis algorithm
• Practical implementation of the algorithm
• Sign problem and reweighting
• Simplifications in the large-N limit
• Strong-coupling QCD at finite chemical potential
(how it could work in principle)
• Closer look at SU(N) sigma model
(possible ways out of the sign problem)
General structure of SD equations
(everywhere we assume lattice discretization)
Choose some closed set of observables φ(X)
X is a collection of all labels, e.g. for scalar field theory
SD equations (with disconnected correlators) are linear:
A(X | Y): infinite-dimensional, but sparse linear operator
b(X): source term, typically only 1-2 elements nonzero
Stochastic solution of linear equations
Assume: A(X|Y), b(X) are positive, |eigenvalues| < 1
Solution with Metropolis algorithm:
Sample sequences {Xn, …, X0} with the weight
Two basic transitions:
• Add new index Xn+1 ,
•
Remove index
•
Restart
Stochastic solution of linear equations
Compare [Prokof’ev,Svistunov, PRL 81 (1998) 2514]
• With probability p+: Add index step
• With probability (1-p+):
Remove index/Restart
Ergodicity: any sequence can be reached
(unless A(X|Y) has some block-diagonal structure)
Acceptance probabilities
(no detailed balance, Metropolis-Hastings)
• Parameter p+ can be tuned to reach optimal acceptance
• Probability distribution of N(X) is crucial to asses convergence
Finally: make histogram of the last element Xn in the sequence
Solution φ(X) , normalization factor
Illustration: ϕ4 matrix model
(Running a bit ahead)
Large autocorrelation time and large
fluctuations near the phase transition
Practical implementation for QFT
• Every transition is a summand in a symbolic representation of SD
equations
• Every transition is a “drawing” of some element of diagrammatic
expansion (either weak- or strong-coupling one)
• Diagrams with any number of legs are sampled
(extension of the worm algorithm)
Keep in computer memory:
• current diagram
• history of drawing
Need DO and UNDO operations
for every diagram element
Almost automatic algorithmization:
• We do not have to remember what’s inside the diagrams (but we
can, if necessary !!!)
• We do not have to know diagrammatic rules – only the SD
equations, quite easy to derive
Sign problem and reweighting
• Now lift the assumptions A(X | Y) > 0 , b(X)>0
• Use the absolute value of weight for the Metropolis sampling
• Sign of each configuration:
• Define
• Effectively, we solve the system
• The expansion
convergence
has typically smaller radius of
• Reweighting fails if the system
approaches the critical point (one of eigenvalues approach 1)
One can only be saved by a suitable reformulation of equations which
makes the sign problem milder
Diagrammatic MC in the large-N limit
for matrix-valued fields
In the above scheme, only series of the form
In real QFTs/non-Abelian LGT’s:
• No. of diagrams ~ n! +UV renormalon
• IR divergences ~ n! (IR renormalon)
Things become simpler in the large-N limit
• UV renormalons are absent in the planar limit
• Number of planar diagrams grows exponentially
• IR renormalons are interesting, but can be removed by
IR regularization (finite volume, twisted BC etc.)
• SD equations are significantly simpler!!!
SD equations in the large-N limit
• SD equations simplify (factorization), BUT are
quadratic
• Still use the linear form without 1/N terms (step back)
• Factorized solution
• Solve linear equation (*), make histograms of Xm only
• Factorization is a property of random process
SD equations@large N: stack structure
The larger space of “indices”
labeling
has a natural stack structure
(Nonlinear random processes/ Recursive Markov chains in Math)
Now three basic transitions instead of X → Y (prob. ~ A(X | Y))
• Create: with probability ~b(X) create new X and push it to stack
• Evolve: with probability ~A(X|Y) replace topmostY with X
• Merge: with probability ~C(X|Y, Z) pop two elements Y, Z from
the stack and push X
Note that now Xi are strongly correlated (we update only topmost)
Makes sense to use only topmost element for statistical sampling
Minimal working example: finite-N matrix model
Consider 0D matrix field theory (finite-N matrix model)
Weights of all diagrams are positive!!!
Full set of observables: multi-trace correlators
Schwinger-Dyson equations
(Akin to the topological recursion formula [B. Eynard])
SD equations in finite-N matrix model
Basic operations of “diagram drawing”:
• Insert line
• Merge singlet operators
• Create vertex
• Split singlet operators (absent in the planar limit)
SD equations in finite-N matrix model
• Probability of the split action grows as
• Separate diagrams of different genus
• All transition probabilities are finite if
Borel non-summability
(or UV renormalon)
pops up in the 1/N expansion!!!
Genus expansion: ϕ4 matrix model
[From Marino,Schiappa,Weiss 0711.1954]
Genus expansion: ϕ4 matrix model
Heavy-tailed distributions
near criticality, P(x) ~ x-2
Large-N gauge theory in the Veneziano limit
(how it can work in principle)
• Lattice action:
Naive Dirac fermions: Nf is infinite,
no need to care about doublers!!!.
• Basic observables:
 Wilson loops = closed string amplitudes
 Wilson lines with quarks at the ends = open string amplitudes
Loop equations illustrated
Quadratic term
Loop equations: stochastic interpretation
Stack of strings (= open or closed loops)!
Wilson loop W[C] ~ Probabilty of generating loop C
Possible transitions (closed string sector):
Create new string
Append links to string
Join strings with links
Join strings
Remove staples
Create open string
…if have collinear links
Probability ~ β
Identical spin states
Temperature and chemical potential
• Finite temperature: strings on cylinder R ~1/T
• Winding strings = Polyakov loops ~ quark free energy
• Veneziano limit:
open strings wrap and close
(No EK reduction)
• Chemical potential:
κ -> κ exp(+/- μ) No signs
or phases!
• Strings stretch in time
Re-emergence of the sign problem
and the physics of QCD strings?
• At large ß, sign problem re-emerges…
• Consider e.g. strong-coupling expansion of Wilson
loop W[L x L] of fixed lattice size L
• At large ß, W[C] -> 1
• In SC expansion,
• Large positive powers of ß should cancel to give 1
(even though analiticity in ß in any finite lattice volume)
Sign problem is necessary!!!
• In contrast, for lattice Nambu-Goto strings
No sign problem
[Weingarten model]
No continuum limit
• Remember the idea on fermionic d.o.f.’s/extra dimension
for QCD strings [Polyakov, Migdal and Co]
Lessons from SU(N) sigma-model
Nontrivial playground similar to QCD!!!
Action:
Observables:
Schwinger-Dyson equations:
Stochastic solution naturally leads to strong-coupling series!
Alternating sign already @ leading order
Making sign problem milder: momentum space
SD equations in momentum space:
Making sign problem milder: momentum space
<1/N tr g+x gy> vs λ
Deep in the SC regime, only few orders relevant…
Sign problem becomes important towards WC
Making sign problem milder:
Matrix Lagrange Multiplier
Nonperturbative improvement!!! [Vicari, Rossi,...]
SD equations at the edge of convergence
The above SD equations are at the edge of convergence
for any λ
Matrix of SD equations has unit eigenvalue
In the SC limit, we reduce to 0D matrix model
Two COMPLETE sets of observables:
• Convergence for G+
• Edge of convergence for GShall we use positive powers of
Lagrange multiplier for SD equations?
Making sign problem milder:
weak-coupling expansion + stereo projection
Representation in terms of Hermitian matrices:
Bare propagator:
General pattern: bare mass term ~λ
(Reminder of the compactness of the group manifold)
• Infinite number of higher-order vertices
• No double-trace terms
• Bare mass is again the bare coupling…
What do we get if we perform
the standard perturbative expansion?
Simpler case: O(N) sigma-model
Stereographic projection: maps whole RN to whole SN
Let’s blindly denote m02 = λ/2 and do expansion
We get series of the form
(something like transseries)
Convergence of double series
Mass gap vs. order
Z vs order
λ=4
λ=2
X axis: 1/(max order)
λ=2 (m=0.2)
DiagMC for O(N) σ-model: sign problem
At every order in λ, many diagrams with different
signs contribute
<sign> vs order in λ
Reweighting seems feasible!!!
(here λ=2, m = 0.24)
Resume
• Schwinger-Dyson equations can be used to construct
DiagMC algorithms, when one does not have any
convenient dual representation
• Works perfectly in the absence of sign problem
• Provides a way to think about systematic SC expansion
• Sign problem for models with SU(N) degrees of
freedom
• Origin in sign-alternating SC expansion series
• Reformulation of SD equations might help
• Interesting structure of perturbative expansion from
stereographic projection:
• Convergent double series in λ and log(λ)
BACKUP SLIDES
Loop equations: stochastic interpretation
Stack of strings (= open or closed loops)!
Possible transitions (open string sector):
Truncate open string
Close by adding link
Close by removing link
Probability ~ κ
Probability ~ Nf /N κ
Probability ~ Nf /N κ
• Hopping expansion for fermions (~20 orders)
• Strong-coupling expansion (series in β) for
gauge fields (~ 5 orders)
Disclaimer: this work is in progress, so the algorithm
is far from optimal...
Phase diagram of the theory: a sketch
High temperature
(small cylinder radius)
OR
Large chemical potential
Numerous winding strings
Nonzero Polyakov loop
Deconfinement phase
Conclusions
• Schwinger-Dyson equations provide a convenient
framework for constructing DiagMC algorithms
• 1/N expansion is quite natural (other algorithms
cannot do it AUTOMATICALLY)
• Good news: it is easy to construct DiagMC
algorithms for non-Abelian field theories
• Then, chemical potential does not introduce
additional sign problem
• Bad news: sign problem already for higher-order
terms of SC expansions
• Can be cured to some extent by choosing proper
observables (e.g. momentum space)
Backup
Lattice QCD at finite baryon density:
some approaches
• Taylor expansion in powers of μ
• Imaginary chemical potential
• SU(2) or G2 gauge theories
• Solution of truncated Schwinger-Dyson equations in
a fixed gauge
• Complex Langevin dynamics
• Infinitely-strong coupling limit
• Chiral Matrix models ...
“Reasonable” approximations with unknown errors,
BUT
No systematically improvable methods!
Worm algorithms for QCD?
Attracted a lot of interest recently as a tool for
QCD at finite density:
• Y. D. Mercado, H. G. Evertz, C. Gattringer,
ArXiv:1102.3096 – Effective theory capturing
center symmetry
• P. de Forcrand, M. Fromm, ArXiv:0907.1915
– Infinitely strong coupling
• W. Unger, P. de Forcrand, ArXiv:1107.1553
– Infinitely strong coupling, continuos time
• K. Miura et al., ArXiv:0907.4245 – Explicit
strong-coupling series …
Worm algorithms for QCD?
• Strong-coupling expansion for lattice gauge theory:
•
•
confining strings [Wilson 1974]
Intuitively: basic d.o.f.’s in gauge theories =
confining strings (also AdS/CFT etc.)
Worm
something like “tube”
• BUT: complicated group-theoretical factors!!! Not
known explicitly
Still no worm algorithm for
non-Abelian LGT (Abelian version: [Korzec, Wolff’ 2010])
Worm-like algorithms from SchwingerDyson equations
Basic idea:
• Schwinger-Dyson (SD) equations: infinite hierarchy of
linear equations for field correlators G(x1, …, xn)
• Solve SD equations: interpret them as steady-state
equations for some random process
• G(x1, ..., xn): ~ probability to obtain {x1, ..., xn}
(Like in Worm algorithm, but for all correlators)
Example: Schwinger-Dyson equations in φ4 theory
Schwinger-Dyson equations for φ4
theory: stochastic interpretation
• Steady-state equations for Markov processes:
• Space of states:
sequences of coordinates {x1, …, xn}
• Possible transitions:
 Add pair of points {x, x}
at random position
1…n+1
 Random walk for topmost
coordinate
 If three points meet – merge
 Restart with two points {x, x}
• No truncation of SD
equations
• No explicit form of
perturbative series
Stochastic interpretation in
momentum space
• 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)
• Restart with {p, -p}
• Probability for new momenta:
Diagrammatic interpretation
History of such a random process: unique Feynman diagram
BUT: no need to remember intermediate states
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
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
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
No need for resummation at large N!!!
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 in D ≥ 4
Renormalized mass:
Renormalized coupling:
CPU time:
several hrs/point
(2GHz core)
[Buividovich,
ArXiv:1104.3459]
Phase diagram of the theory: a sketch
High temperature
(small cylinder radius)
OR
Large chemical potential
Numerous winding strings
Nonzero Polyakov loop
Deconfinement phase
Summary and outlook
• Diagrammatic Monte-Carlo and Worm algorithm:
useful strategies complimentary to standard Monte-Carlo
• Stochastic interpretation of Schwinger-Dyson equations:
a novel way to stochastically sum up perturbative series
Advantages:
• Implicit construction of
perturbation theory
• No truncation of SD eq-s
• Large-N limit is very easy
• Naturally treats divergent
series
• No sign problem at μ≠0
Disadvantages:
• Limited to the “very strongcoupling” expansion (so far?)
• Requires large statistics in
IR region
QCD in terms of strings without
explicit “stringy” action!!!
Summary and outlook
Possible extensions:
• Weak-coupling theory: Wilson loops in
momentum space?
 Relation to meson scattering amplitudes
 Possible reduction of the sign problem
• Introduction of condensates?
 Long perturbative series ~ Short
perturbative series + Condensates
[Vainshtein, Zakharov]
 Combination with Renormalization-Group
techniques?
Thank you for your attention!!!
References:
• ArXiv:1104.3459 (φ4 theory)
• ArXiv:1009.4033, 1011.2664 (large-N theories)
• Some sample codes are available at:
http://www.lattice.itep.ru/~pbaivid/codes.html
This work was supported by the S. Kowalewskaja
award from the Alexander von Humboldt Foundation
Back-up slides
Some historical remarks
“Genetic” algorithm
vs. branching random process
Probability to find
some configuration
of branches obeys nonlinear
equation
“Extinction probability” obeys
nonlinear equation
[Galton, Watson, 1974]
“Extinction of peerage”
Steady state due to creation
and merging
Attempts to solve QCD loop
equations
[Migdal, Marchesini, 1981]
Recursive Markov Chains
[Etessami, Yannakakis, 2005] “Loop extinction”:
No importance sampling
Also some modification of
McKean-Vlasov-Kac models
[McKean, Vlasov, Kac, 196x]
Motivation: Large-N QFT side
Can DiagMC take advantage of the large-N limit?
• Only exponentially (vs. factorially) growing number of diagrams
• At least part of non-summabilities are absent
• Large-N QFTs are believed to contain most important physics of real QFT
Theoretical aspects of large-N expansion
• Surface counting problems: calculation of coefficients of 1/N
expansions (topological recursion)
[Formula by B. Eynard]
• Instantons via divergences of 1/N expansion
• Resurgent structure of 1/N expansions [Marino, Schiappa, Vaz]
Worm Algorithm [Prokof’ev, Svistunov]
• Monte-Carlo sampling of closed vacuum diagrams:
nonlocal updates, closure constraint
• Worm Algorithm: sample closed diagrams + open diagram
• Local updates: open graphs
closed graphs
• Direct sampling of field correlators (dedicated simulations)
x, y – head and
tail of the worm
Correlator = probability
distribution of head and
tail
• Applications: systems with “simple” and convergent
perturbative expansions (Ising, Hubbard, 2d fermions …)
• Very fast and efficient algorithm!!!
Stochastic solution of SD equations
• Sampling the correlators of the theory –
momenta or coordinates
• Histograms of X are correlators (later about
positivity)
• Extension of the WORM algorithm: Diagrams
with any number of open legs!!!