1. Diagrammatic Monte-Carlo algorithms for large-N quantum field theories from Schwinger-Dyson equations
Pavel Buividovich (Regensburg University)
eNlarge Horizons, IFT Madrid’2015

2. Motivation: Diagrammatic Monte-Carlo
Quantum field theory:
Sum over fields
Euclidean action:
Sum over interacting paths
Perturbative
expansions

3. 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)

4. 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]

5. 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!!!

6. 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

7. 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

8. Solution using the Metropolis algorithm:
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

9. 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

10. Illustration: ϕ4 matrix model (Running a bit ahead)
Large autocorrelation time and large fluctuations near the phase transition

11. 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)

12. 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 has smaller radius of convergence
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

13. 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
(*)

14. 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

15. 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

16. 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!)

17. SD equations in finite-N matrix model
Basic operations of “diagram drawing”:
Insert line
Merge singlet operators
Create vertex
Split singlet operators

18. SD equations in finite-N matrix model
Think of each as an n-segment boundary of a random surface

19. 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
Expansion in λ, N fixed:

20. 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!

21. Borel resummation in practice
In practice: 4-5 poles in Borel plane

22. Test: triviality of φ4 theory in D ≥ 4
Renormalized mass: Renormalized coupling:
CPU time:
several hrs/point
(2GHz core)
[Buividovich,
ArXiv: ]

23. Genus expansion Alternatively, let’s expand in 1/N: Insert line
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

24. 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 !!!

25. 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 leading order

26. Lessons from SU(N) sigma-model
SD equations in momentum space:

27. Lessons from SU(N) sigma-model <1/N tr g+x gy> vs λ
Deep in the SC regime, only few orders relevant…

28. 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 Laurent series
Can one get Exp(-8π/ λ) ???

29. Lessons from SU(N) sigma-model Matrix Lagrange Multiplier
Nonperturbative improvement!!! [Vicari, Rossi,...]

30. 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.

31. 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!!!

32. Infinite hierarchy of quadratic equations!
Migdal-Makeenko loop equations
Loop equations in the closed string sector:
Loop equations in the open string sector:
Infinite hierarchy of quadratic equations!

33. Loop equations illustrated
Quadratic term

34. 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
…if have collinear links
Remove staples
Probability ~ β
Create open string
Identical spin states

35. Loop equations: stochastic interpretation
Stack of strings (= open or closed loops)!
Possible transitions (open string sector):
Truncate open string
Probability ~ κ
Close by adding link
Probability ~ Nf /N κ
Close by removing link
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...

36. 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:
Strings oriented
in the time direction
are favoured
No signs
or phases!
κ -> κ exp(+/- μ)

37. Phase diagram of the theory: a sketch
High temperature
(small cylinder radius) OR
Large chemical potential
Numerous winding strings
Nonzero Polyakov loop
Deconfinement phase

38. 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)

39. Backup

40. 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!

41. 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: – Effective theory capturing center symmetry
P. de Forcrand, M. Fromm, ArXiv: – Infinitely strong coupling
W. Unger, P. de Forcrand, ArXiv: – Infinitely strong coupling, continuos time
K. Miura et al., ArXiv: – Explicit strong-coupling series …

42. 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])

43. Worm-like algorithms from Schwinger-Dyson 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)

44. Example: Schwinger-Dyson equations in φ4 theory

45. 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

46. 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:

47. 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

48. 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

49. 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

50. No need for resummation at large N!!!
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!!!

51. Resummation: fits by multiple exponents

52. Resummation: positions of poles
Connected truncated four-point function
Two-point function
2-3 poles can be extracted with reasonable accuracy

53. Test: triviality of φ4 theory in D ≥ 4
Renormalized mass: Renormalized coupling:
CPU time:
several hrs/point
(2GHz core)
[Buividovich,
ArXiv: ]

54. Phase diagram of the theory: a sketch
High temperature
(small cylinder radius) OR
Large chemical potential
Numerous winding strings
Nonzero Polyakov loop
Deconfinement phase

55. 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 strong-coupling” expansion (so far?)
Requires large statistics in IR region
QCD in terms of strings without explicit “stringy” action!!!

56. 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?

57. Thank you for your attention!!!
References:
ArXiv: (φ4 theory)
ArXiv: , (large-N theories)
Some sample codes are available at:
This work was supported by the S. Kowalewskaja
award from the Alexander von Humboldt Foundation

58. Back-up slides

59. Some historical remarks
“Genetic” algorithm vs. branching random process
Probability to find
some configuration
of branches obeys nonlinear
equation
Steady state due to creation
and merging
Recursive Markov Chains
[Etessami, Yannakakis, 2005]
Also some modification of
McKean-Vlasov-Kac models
[McKean, Vlasov, Kac, 196x]
“Extinction probability” obeys
nonlinear equation
[Galton, Watson, 1974]
“Extinction of peerage”
Attempts to solve QCD loop
equations
[Migdal, Marchesini, 1981]
“Loop extinction”:
No importance sampling

60. Nonlinear Random Processes
[Buividovich, ArXiv: ]
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