Schwinger-Dyson equations, Schwinger-Dyson equations, nonlinear random processes nonlinear random processes and diagrammatic and diagrammatic algorithms algorithms Pavel Buividovich Pavel Buividovich (Regensburg University) (Regensburg University) GGI Workshop “New Frontiers in Lattice Gauge Theory”, 27.08.2012 GGI Workshop “New Frontiers in Lattice Gauge Theory”, 27.08.2012
Motivation: Lattice Q QC CD D at finite at finite Motivation: Lattice baryon density baryon density • Lattice QCD Lattice QCD is one of the main tools is one of the main tools to study to study quark-gluon plasma quark-gluon plasma • Interpretation of heavy-ion collision experiments: Interpretation of heavy-ion collision experiments: RHIC, LHC, FAIR,… RHIC, LHC, FAIR,… But: baryon density is finite in experiment !!! But: baryon density is finite in experiment !!! Dirac operator is not Hermitean anymore Dirac operator is not Hermitean anymore exp(-S) is complex complex!!! !!! Sign problem Sign problem exp(-S) is Monte-Carlo methods are not applicable !!! Monte-Carlo methods are not applicable !!! Try to look for alternative numerical simulation strategies Try to look for alternative numerical simulation strategies
Lattice Q QC CD D at finite baryon density: at finite baryon density: Lattice some approaches some approaches • Taylor expansion in powers of Taylor expansion in powers of μ μ • Imaginary chemical potential Imaginary chemical potential SU(2) SU(2) or or G G 2 gauge theories • 2 gauge theories • Solution of truncated Solution of truncated Schwinger-Dyson Schwinger-Dyson equations in equations in a fixed gauge a fixed gauge • Complex Langevin dynamics Complex Langevin dynamics • Infinitely-strong coupling limit Infinitely-strong coupling limit • Chiral Matrix models ... Chiral Matrix models ... “Reasonable” approximations with Reasonable” approximations with unknown errors unknown errors, , “ BUT BUT No systematically improvable methods! No systematically improvable methods!
Path integrals: sum over paths vs. sum over fields Path integrals: sum over paths vs. sum over fields Quantum field theory: Quantum field theory: Sum over fields fields Sum over interacting paths interacting paths Sum over Sum over Perturbative expansions Euclidean action: Euclidean action:
Worm Algorithm [ [Prokof’ev, Svistunov Prokof’ev, Svistunov] ] Worm Algorithm • Monte-Carlo sampling of Monte-Carlo sampling of closed vacuum diagrams: closed vacuum diagrams: nonlocal updates, closure constraint nonlocal updates, closure constraint • Worm Algorithm: Worm Algorithm: sample closed diagrams sample closed diagrams + open diagram + open diagram • Local updates: open graphs Local updates: open graphs closed graphs closed graphs • Direct sampling of field correlators (dedicated simulations) Direct sampling of field correlators (dedicated simulations) x, y – head and x, y – head and tail of the worm tail of the worm Correlator = probability Correlator = probability distribution of head and distribution of head and tail tail • Applications: Applications: systems with “simple” and convergent systems with “simple” and convergent perturbative expansions (Ising, Hubbard, 2d fermions …) (Ising, Hubbard, 2d fermions …) perturbative expansions • Very fast and efficient algorithm!!! Very fast and efficient algorithm!!!
Worm algorithms for QC CD D? ? Worm algorithms for Q Attracted a lot of interest recently as a tool for Attracted a lot of interest recently as a tool for QCD at finite density: QCD at finite density: • Y. D. Mercado Y. D. Mercado, , H. G. Evertz H. G. Evertz, , C. Gattringer C. Gattringer, , ArXiv:1102.3096 1102.3096 – Effective theory capturing – Effective theory capturing ArXiv: center symmetry center symmetry • P. de P. de Forcrand Forcrand, , M. Fromm M. Fromm, , ArXiv:0907.1915 ArXiv:0907.1915 – Infinitely strong coupling – Infinitely strong coupling • W. Unger W. Unger, , P. de P. de Forcrand Forcrand, , ArXiv:1107.1553 ArXiv:1107.1553 – Infinitely strong coupling, continuos time – Infinitely strong coupling, continuos time • K. Miura et al. K. Miura et al., , ArXiv:0907.4245 ArXiv:0907.4245 – Explicit – Explicit strong-coupling series … strong-coupling series …
Worm algorithms for QC CD D? ? Worm algorithms for Q • Strong-coupling expansion for lattice gauge theory: Strong-coupling expansion for lattice gauge theory: confining strings [Wilson 1974] [Wilson 1974] confining strings • Intuitively: Intuitively: basic d.o.f.’s in gauge theories = basic d.o.f.’s in gauge theories = confining strings (also (also AdS/CFT AdS/CFT etc.) etc.) confining strings • Worm Worm something like “tube” “tube” something like • BUT: BUT: complicated group-theoretical factors!!! complicated group-theoretical factors!!! Not Not known explicitly Still no worm algorithm for no worm algorithm for known explicitly Still non-Abelian LGT (Abelian version: non-Abelian LGT (Abelian version: [Korzec, Wolff’ 2010] [Korzec, Wolff’ 2010]) )
Worm-like algorithms from Schwinger- Worm-like algorithms from Schwinger- Dyson equations Dyson equations Basic idea: Basic idea: • Schwinger-Dyson (SD) equations: Schwinger-Dyson (SD) equations: infinite hierarchy of infinite hierarchy of linear equations for field correlators G(x G(x 1 , …, x n ) linear equations for field correlators 1 , …, x n ) • Solve SD equations: Solve SD equations: interpret them as interpret them as steady-state steady-state equations for some random process for some random process equations G(x G(x 1 , ..., x n ): ~ probability probability to obtain {x to obtain {x 1 , ..., x n } • 1 , ..., x n ): ~ 1 , ..., x n } (Like in Worm algorithm, but for all correlators) (Like in Worm algorithm, but for all correlators)
Example: Schwinger-Dyson equations in Schwinger-Dyson equations in φ φ theory Example: 4 theory 4
Schwinger-Dyson equations for φ φ 4 Schwinger-Dyson equations for 4 theory: stochastic interpretation stochastic interpretation theory: • Steady-state equations for Markov processes: Steady-state equations for Markov processes: • Space of states: Space of states: sequences of coordinates sequences of coordinates {x {x 1 , …, x n } 1 , …, x n } • Possible transitions: Possible transitions: Add pair of points Add pair of points {x, x} {x, x} at random position at random position 1 … n + 1 1 … n + 1 Random walk Random walk for topmost for topmost • No truncation of SD No truncation of SD coordinate coordinate equations equations If If three points meet three points meet – – merge merge • No explicit form of No explicit form of perturbative series perturbative series Restart Restart with two points {x, x} with two points {x, x}
Stochastic interpretation in Stochastic interpretation in momentum space momentum space • 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 {p 1 , …, p n } 1 , …, p n } • Possible transitions: Possible transitions: Add pair of momenta Add pair of momenta {p, -p} {p, -p} at positions at positions 1, A = 2 … n + 1 1, A = 2 … n + 1 Add up Add up three first three first momenta momenta (merge) (merge) • Restart Restart with {p, -p} with {p, -p} • Probability for Probability for new momenta new momenta: :
Diagrammatic interpretation Diagrammatic interpretation History of such a random process: unique Feynman diagram unique Feynman diagram History of such a random process: BUT: no need to remember intermediate states no need to remember intermediate states BUT: Measurements of connected, 1PI, 2PI correlators connected, 1PI, 2PI correlators are are Measurements of possible!!! In practice: label connected legs label connected legs possible!!! In practice: Kinematical factor for each diagram: for each diagram: Kinematical factor q i are independent momenta independent momenta, Q , Q j – depend on q i q i are j – depend on q i Monte-Carlo integration over independent momenta Monte-Carlo integration over independent momenta
Normalizing the transition probabilities Normalizing the transition probabilities • Problem: Problem: probability probability of “Add momenta” of “Add momenta” grows as (n+1) grows as (n+1), , rescaling G(p 1 , … , p n ) – does not help. rescaling G(p 1 , … , p n ) – does not help. • Manifestation of Manifestation of series divergence!!! series divergence!!! • Solution: Solution: explicitly explicitly count diagram order count diagram order m. Transition m. Transition probabilities depend on m probabilities depend on m Extended state space: {p Extended state space: {p 1 , … , p n } and m – diagram order • 1 , … , p n } and m – diagram order • Field correlators: Field correlators: w w m (p 1 , …, p n ) – probability to encounter – probability to encounter m-th order diagram m-th order diagram • m (p 1 , …, p n ) with momenta {p {p 1 , …, p n } on external legs with momenta 1 , …, p n } on external legs
Recommend
More recommend
