Diagrammatic Monte-Carlo
algorithms for large-N
quantum field theories from
Schwinger-Dyson equations
Pavel Buividovich (Regensburg University)
eNlarge Horizons, IFT Madrid’2015
Motivation: Diagrammatic Monte-Carlo
Quantum field theory:
Sum over fields
Sum over interacting paths
Perturbative
expansions
Euclidean action:
Motivation: QCD side
• Diagrammatic Monte-Carlo algorithms: often better than standard
Monte-Carlo (Simple cond-mat systems)
• Recent attempts to apply DiagMC to lattice QCD at finite density and
reduce sign problem [de Forcrand, Gattringer, Chandrasekharan]
• Typically, DiagMC relies on “Duality transformations”, well-known for
Abelian lattice (gauge and scalar) field theories
• BUT: no convenient duality transformations are known for non-Abelian
fields (Clebsch-Gordan coefficients are cumbersome and difficult and have
sign problem)
• Weak-coupling expansions are also cumbersome and difficult to re-sum
• How to construct DiagMC in the non-Abelian case?
• How to avoid manual search for duality transformations?
• How to avoid Borel resummations?
Additional motivation: Large-N quantum field theory
Can DiagMC take advantage of the large-N limit?
(Exponentially vs. factorially growing number of diagrams)
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!!!
Outline
DiagMC algorithms can be constructed in a
universal way from Schwinger-Dyson equations
• General structure of SD equations
• Stochastic solution of SD equations by Metropolis algorithm
• Practical implementation of the algorithm
• Sign problem and reweighting
• Simplifications in the large-N limit
• Resummation: expansion in coupling vs expansion in 1/N
• Lessons from SU(N) sigma model
• Strong-coupling QCD at finite chemical potential
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 using the 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
• 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
• Keeping the whole sequence{Xn, …, X0} in memory is not practical
(size of X can be quite large)
• Use the sparseness of A(X | Y) , remember the sequence of
transitions Xn→ Xn+1
• 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)
Save:
• current diagram
• history of drawing
Need DO and UNDO operations for every diagram
element
Construction of algorithms is almost automatic and can be nicely
combined with symbolic calculus software (e.g. Mathematica)
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 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
SD equations in the large-N limit
• SD equations simplify (factorization), BUT are quadratic
• Still cast into the linear form, one step back before factorization
(*)
• Note we are now working in a much larger linear space
• Factorized solution
• Solve linear equation (*), make histograms of Xm only
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
Resummation/Rescaling
Growth of field correlators
with n/ order:
• Exponential in the large-N limit
• Factorial at finite N
How to interprete
as a probability distribution?
Exponential growth? Introduce “renormalization constants” :
is now finite ,
can be interpreted as probability
In the Metropolis algorithm: all the transition weights should be
finite, otherwise unstable behavior
How to deal with factorial growth?
Borel resummation
Resummation: finite-N matrix model
Consider 0D matrix field theory (finite-N matrix model)
Full set of observables: multi-trace correlators
Schwinger-Dyson equations
(Remember the topological recursion formula!)
SD equations in finite-N matrix model
Basic operations of “diagram drawing”:
• Insert line
• Merge singlet operators
• Create vertex
• Split singlet operators
SD equations in finite-N matrix model
Think of each
as an n-segment boundary
of a random surface
SD equations in finite-N matrix model
• Probability of “split” action grows as
Obviously, cannot be removed by rescaling of the form N cn
Introduce rescaling factors which depend on
number of vertices OR genus
1) Expansion in λ, N fixed:
Standard perturbative expansion
Upon rescaling, weights of transitions are:
• Insert line
• Merge singlet operators
• Create vertex
• Split singlet operators
()
Require all of them to be finite
Factorial growth of the number of diagrams with the
number of vertices!
)
Borel resummation in practice
In practice: 4-5 poles in Borel plane
Test: triviality of φ4 theory in D ≥ 4
Renormalized mass:
Renormalized coupling:
CPU time:
several hrs/point
(2GHz core)
[Buividovich,
ArXiv:1104.3459]
Genus expansion
Alternatively, let’s expand in 1/N:
Upon rescaling, weights of transitions are:
• Insert line
• Merge singlet operators
• Create vertex
• Split singlet operators
At fixed genus, exponential growth with n, m
Ansatz
Genus expansion: factorial divergence
Now the weights of transitions are:
• Insert line
• Merge singlet operators
• Create vertex
• Split singlet
operators
• Finiteness of cg implies
• Then,
• Finite if
is finite
cg should
be finite
for any g !!!
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
Lessons from SU(N) sigma-model
SD equations in momentum space:
Lessons from SU(N) sigma-model
<1/N tr g+x gy> vs λ
Deep in the SC regime, only few orders relevant…
Lessons from SU(N) sigma-model
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
• Singularity can be avoided by compactification + twist
Standard diagrammatic expansion
Can one get Exp(-8π/ λ) ???
Laurent series
Lessons from SU(N) sigma-model
Matrix Lagrange Multiplier
Nonperturbative improvement!!! [Vicari, Rossi,...]
Large-N gauge theory in the Veneziano limit
• Gauge theory with the action
• t-Hooft-Veneziano limit:
N -> ∞, Nf -> ∞, λ fixed, Nf/N fixed
• Only planar diagrams contribute!
connection with strings
• Factorization of Wilson loops W(C) = 1/N tr P exp(i ∫dxμ Aμ):
• Better approximation for real QCD than pure large-N gauge
theory: meson decays, deconfinement phase etc.
Large-N gauge theory in the Veneziano limit
• Lattice action:
No EK reduction in the large-N limit! Center symmetry broken by fermions.
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
• Zigzag symmetry for QCD strings!!!
Migdal-Makeenko loop equations
Loop equations in the closed string sector:
Loop equations in the open string sector:
Infinite hierarchy of quadratic equations!
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
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...
Temperature and chemical potential
• Finite temperature: strings on cylinder R ~1/T
• Winding strings = Polyakov loops ~ quark free energy
• No way to create winding string in pure gauge theory
at large-N
EK reduction
• Veneziano limit:
open strings wrap and close
• Chemical potential:
κ -> κ exp(+/- μ)
No signs
or phases!
• Strings oriented
in the time direction
are favoured
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]
Nonlinear Random Processes
[Buividovich, ArXiv:1009.4033]
Also: Recursive Markov Chain
[Etessami,Yannakakis, 2005]
• Let X be some discrete set
• Consider stack of the elements of X
• At each process step:
 Create: with probability Pc(x) create new x and
push it to stack
 Evolve: with probability Pe(x|y) replace y on
the top of the stack with x
 Merge: with probability Pm(x|y1,y2) pop two
elements y1, y2 from the stack and push x into the
stack
 Otherwise restart