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 Euclidean action: Sum over interacting paths Perturbative expansions

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

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

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 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

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 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: 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

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

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!!!

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!

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

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...

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)

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 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)

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

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!!!

Resummation: fits by multiple exponents

Resummation: positions of poles Connected truncated four-point function Two-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 strong-coupling” 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 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

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