Schwinger-Dyson equations, Schwinger-Dyson equations, nonlinear - - PowerPoint PPT Presentation
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
GGI Workshop “New Frontiers in Lattice Gauge Theory”, 27.08.2012 GGI Workshop “New Frontiers in Lattice Gauge Theory”, 27.08.2012
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: 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 exp(-S) is complex complex!!! !!! Sign problem Sign problem 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
Taylor expansion in powers of μ μ
Imaginary chemical potential
SU(2) or
G2
2 gauge theories
gauge theories
Solution of truncated Schwinger-Dyson Schwinger-Dyson equations in equations in a fixed gauge a fixed gauge
Complex Langevin dynamics
Infinitely-strong coupling limit
Chiral Matrix models ... “ “Reasonable” approximations with Reasonable” approximations with unknown errors unknown errors, , BUT BUT
Quantum field theory: Quantum field theory:
Sum over Sum over fields fields Euclidean action: Euclidean action: Sum over Sum over interacting paths interacting paths Perturbative expansions
Monte-Carlo sampling of closed vacuum diagrams: closed vacuum diagrams: nonlocal updates, closure constraint nonlocal updates, closure constraint
Worm Algorithm: sample closed diagrams sample closed diagrams + open diagram + open diagram
Local updates: open graphs closed graphs closed graphs
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: systems with “simple” and convergent systems with “simple” and convergent perturbative expansions perturbative expansions (Ising, Hubbard, 2d fermions …) (Ising, Hubbard, 2d fermions …)
Very fast and efficient algorithm!!!
Strong-coupling expansion for lattice gauge theory: confining strings confining strings [Wilson 1974] [Wilson 1974]
Intuitively: basic d.o.f.’s in gauge theories = basic d.o.f.’s in gauge theories = confining strings confining strings (also (also AdS/CFT AdS/CFT etc.) etc.)
Worm something like something like “tube” “tube”
BUT: complicated group-theoretical factors!!! complicated group-theoretical factors!!! Not Not known explicitly known explicitly Still Still no worm algorithm for no worm algorithm for non-Abelian LGT non-Abelian LGT (Abelian version:
(Abelian version: [Korzec, Wolff’ 2010] [Korzec, Wolff’ 2010]) )
Basic idea: Basic idea:
Schwinger-Dyson (SD) equations: infinite hierarchy of infinite hierarchy of linear equations for field correlators linear equations for field correlators G(x G(x1
1, …, x
, …, xn
n)
)
Solve SD equations: interpret them as interpret them as steady-state steady-state equations equations for some random process for some random process
G(x1
1, ..., x
, ..., xn
n): ~
): ~ probability probability to obtain {x to obtain {x1
1, ..., x
, ..., xn
n}
} (Like in Worm algorithm, but for all correlators) (Like in Worm algorithm, but for all correlators)
4 4 theory
4
Steady-state equations for Markov processes:
Space of states: sequences of coordinates sequences of coordinates {x {x1
1, …, x
, …, xn
n}
}
Possible transitions:
Add pair of points {x, x} {x, x} at random position at random position 1 … n + 1 1 … n + 1
Random walk for topmost for topmost coordinate coordinate
If three points meet three points meet – – merge merge
Restart with two points {x, x} with two points {x, x}
No truncation of SD equations equations
No explicit form of perturbative series perturbative series
Steady-state equations for Markov processes:
Space of states: sequences of momenta sequences of momenta {p {p1
1, …, p
, …, pn
n}
}
Possible transitions:
Add pair of momenta {p, -p} {p, -p} at positions at positions 1, A = 2 … n + 1 1, A = 2 … n + 1
Add up three first three first momenta momenta (merge) (merge)
Restart with {p, -p} with {p, -p}
Probability for new momenta new momenta: :
History of such a random process: History of such a random process: unique Feynman diagram unique Feynman diagram BUT: BUT: no need to remember intermediate states no need to remember intermediate states Measurements of Measurements of connected, 1PI, 2PI correlators connected, 1PI, 2PI correlators are are possible!!! In practice: possible!!! In practice: label connected legs label connected legs Kinematical factor Kinematical factor for each diagram: for each diagram: q qi
i are
are independent momenta independent momenta, Q , Qj
j – depend on q
– depend on qi
i
Monte-Carlo integration over independent momenta Monte-Carlo integration over independent momenta
Problem: probability probability of “Add momenta”
grows as (n+1), , rescaling G(p rescaling G(p1
1, … , p
, … , pn
n) – does not help.
) – does not help.
Manifestation of series divergence!!! series divergence!!!
Solution: explicitly explicitly count diagram order count diagram order m. Transition
probabilities depend on m probabilities depend on m
Extended state space: {p1
1, … , p
, … , pn
n} and m – diagram order
} and m – diagram order
Field correlators:
wm
m(p
(p1
1, …, p
, …, pn
n)
) – probability to encounter – probability to encounter m-th order diagram m-th order diagram with momenta with momenta {p {p1
1, …, p
, …, pn
n} on external legs
} on external legs
Finite transition probabilities:
Factorial divergence of series is absorbed into the growth of
C Cn,m
n,m
!!! !!!
Probabilities (for optimal x, y): (for optimal x, y):
Add momenta:
Sum up momenta + increase the order: increase the order:
Otherwise restart restart
Transition probabilities do not depend on bare mass or Transition probabilities do not depend on bare mass or coupling!!! coupling!!! (Unlike in the standard MC) (Unlike in the standard MC) No free lunch: No free lunch: kinematical suppression of small-p region kinematical suppression of small-p region ( (~ ~ Λ ΛIR
IR D D)
)
Integral representation of
Cn,m
n,m =
= Γ Γ(n/2 + m + 1/2) x (n/2 + m + 1/2) x-(n-2)
y-m
: Pade-Borel resummation. Pade-Borel resummation. Borel image of correlators!!! Borel image of correlators!!!
Poles of Borel image:
exponentials in w in wn,m
n,m
Pade approximants are unstable unstable
Poles can be found by fitting fitting
Special fitting procedure using SVD of Hankel matrices using SVD of Hankel matrices
Two-point function Two-point function Connected truncated Connected truncated four-point function four-point function
2-3 poles can be extracted with reasonable accuracy 2-3 poles can be extracted with reasonable accuracy
4 theory in D
Renormalized mass: Renormalized coupling: Renormalized mass: Renormalized coupling: CPU time: several hrs/point (2GHz core) [Buividovich, ArXiv:1104.3459]
Gauge theory with the action
t-Hooft-Veneziano limit: N -> N -> ∞, N ∞, Nf
f ->
∞, λ λ fixed, N fixed, Nf
f/N fixed
/N fixed
Only planar diagrams planar diagrams contribute! contribute! connection with strings connection with strings
Factorization of Wilson loops
W(C) = 1/N tr P exp(i ∫dx W(C) = 1/N tr P exp(i ∫dxμ
μ A
Aμ
μ):
):
Better approximation for real Q QC CD D than than pure large-N gauge pure large-N gauge theory: meson decays, deconfinement phase etc. theory: meson decays, deconfinement phase etc.
Lattice action:
No EK reduction No EK reduction in the large-N limit! in the large-N limit! Center symmetry broken by fermions. Center symmetry broken by fermions. Naive Dirac fermions: Naive Dirac fermions: N Nf
f is infinite, no need to care about doublers!!!.
is infinite, no need to care about doublers!!!.
Basic observables:
Wilson loops = closed string closed string amplitudes amplitudes
Wilson lines with quarks at the ends = open string
amplitudes
Zigzag symmetry for QC CD D strings!!! strings!!!
Loop equations in the Loop equations in the closed string closed string sector: sector: Loop equations in the Loop equations in the open string
sector:
Quadratic term Quadratic term
[Buividovich, ArXiv:1009.4033] [Buividovich, ArXiv:1009.4033] Also: Also: Recursive Markov Chain Recursive Markov Chain [ [Etessami,Yannakakis, 2005 Etessami,Yannakakis, 2005] ]
Let X X be some be some discrete set discrete set
Consider stack stack of the
elements of X
At each process step process step: :
Create: with probability P with probability Pc
c(x)
(x) create new x create new x and and push it to stack push it to stack
Evolve: with probability P with probability Pe
e(x|y)
(x|y) replace y replace y on
the top of the stack the top of the stack with x with x
Merge: with probability P with probability Pm
m(x|y
(x|y1
1,y
,y2
2)
) pop pop two two elements elements y y1
1, y
, y2
2 from the stack and
from the stack and push x push x into the into the stack stack
Otherwise restart
Probability to find to find n elements x n elements x1
1 ... x
... xn
n in the stack:
in the stack: W(x W(x1
1, ..., x
, ..., xn
n)
)
Propagation of chaos [McKean, 1966] [McKean, 1966] ( = ( = factorization at large-N factorization at large-N [tHooft, Witten, 197x]): [tHooft, Witten, 197x]): W(x W(x1
1, ..., x
, ..., xn
n) = w
) = w0
0(x
(x1
1) w(x
) w(x2
2) ... w(x
) ... w(xn
n)
)
Steady-state equation (sum over y, z): (sum over y, z): w(x) = P w(x) = Pc
c(x) + P
(x) + Pe
e(x|y) w(y) + P
(x|y) w(y) + Pm
m(x|y,z) w(y) w(z)
(x|y,z) w(y) w(z)
Stack of strings Stack of strings (= open or closed loops)! (= open or closed loops)! Wilson loop W[C] ~ Probabilty of generating loop C Wilson loop W[C] ~ Probabilty of generating loop C Possible transitions ( Possible transitions (closed string sector closed string sector): ):
Create new string Create new string Append links to string Append links to string Join strings with links Join strings with links Join strings Join strings
…if have collinear links
Remove staples Remove staples
Probability ~ Probability ~ β β
Create open string Create open string
Identical spin states
Stack of strings Stack of strings (= open or closed loops)! (= open or closed loops)! Possible transitions ( Possible transitions (open string sector
):
Truncate open string Truncate open string
Probability ~ Probability ~ κ κ
Close by adding link Close by adding link
Probability ~ N Probability ~ Nf
f /N
/N κ κ
Close by removing link Close by removing link
Probability ~ N Probability ~ Nf
f /N
/N κ κ
Disclaimer: Disclaimer: this work is in progress, so the algorithm this work is in progress, so the algorithm is far from optimal... is far from optimal...
Different terms in loop equations have different signs signs
Configurations should be additionally reweighted reweighted
Each loop comes with a complex-valued phase complex-valued phase ( (+/-1 +/-1 in in pure gauge pure gauge, , exp(i exp(i π π k/4 k/4) ) with with Dirac fermions Dirac fermions ) )
Sign problem is very mild (strong-coupling only)? (strong-coupling only)? for for 1x1 Wilson loops 1x1 Wilson loops
For large β β ( (close to the continuum close to the continuum): ): sign problem sign problem should be important should be important
Large terms ~ ~β β sum up to ~1 sum up to ~1
Chemical potential: Chemical potential: κ κ ->
κ exp(± exp(±μ μ) ) No additional phases No additional phases
Interacting fermions: Interacting fermions:
Extremely severe sign problem in configuration space severe sign problem in configuration space [U. [U. Wolff, ArXiv:0812.0677] Wolff, ArXiv:0812.0677]
BUT: most time is spent on generating most time is spent on generating “free” random walks “free” random walks
All worldlines can be summed up analytically summed up analytically
Manageable sign in momentum space [Prokof’ev, [Prokof’ev, Svistunov] Svistunov]
Momentum space loops Momentum space loops for for Q QC CD D? ?
Easy to construct in the continuum in the continuum [Migdal, Makeenko, 198x] [Migdal, Makeenko, 198x]
BUT no obvious discretization suitable for numerics suitable for numerics
Measurement of string tension string tension: probability to get a : probability to get a rectangular R x T Wilson loop rectangular R x T Wilson loop - almost
ZERO
Physical observables = Mesonic correlators Mesonic correlators = sums over all = sums over all loops loops
Mesonic correlators = Loops in momentum space Loops in momentum space [Makeenko, [Makeenko, Olesen, ArXiv: Olesen, ArXiv:0810.4778 0810.4778] ]
Finite temperature: strings on cylinder R ~1/T cylinder R ~1/T
Winding strings = Polyakov loops ~ = Polyakov loops ~ quark free energy quark free energy
No way to create winding string in pure gauge theory pure gauge theory at large-N at large-N EK reduction EK reduction
Veneziano limit:
Chemical potential:
Strings oriented in the time direction in the time direction are favoured are favoured κ κ ->
κ exp(+/- exp(+/- μ μ) )
No signs No signs
High temperature High temperature (small cylinder radius) (small cylinder radius) OR OR Large Large chemical potential chemical potential Numerous Numerous winding strings winding strings Nonzero Nonzero Polyakov loop Polyakov loop Deconfinement phase Deconfinement phase
useful strategies complimentary to standard Monte-Carlo
a novel way to stochastically sum up perturbative series Advantages: Advantages:
Implicit construction of perturbation theory perturbation theory
No truncation of SD eq-s
Large-N limit is very easy
Naturally treats divergent series series
No sign problem at μ μ≠0 ≠0 Disadvantages: Disadvantages:
Limited to the “very strong- coupling” expansion (so far?) coupling” expansion (so far?)
Requires large statistics in IR region IR region
References: References:
ArXiv:1104.3459 1104.3459 ( (φ φ4
4 theory
theory) )
ArXiv:1009.4033 1009.4033, , 1011.2664 1011.2664 (large-N theories) (large-N theories)
Some sample codes sample codes are available at: are available at: http://www.lattice.itep.ru/~pbaivid/codes.html http://www.lattice.itep.ru/~pbaivid/codes.html This work was supported by the S. Kowalewskaja This work was supported by the S. Kowalewskaja award from the Alexander von Humboldt Foundation award from the Alexander von Humboldt Foundation
“ “Genetic” algorithm Genetic” algorithm vs.
branching random process
Probability to find Probability to find some configuration some configuration
equation equation Steady state due to creation Steady state due to creation and merging and merging Recursive Markov Chains Recursive Markov Chains [ [Etessami, Yannakakis, 2005 Etessami, Yannakakis, 2005] ] Also some modification of Also some modification of McKean-Vlasov-Kac models McKean-Vlasov-Kac models [McKean, Vlasov, Kac, 196x] [McKean, Vlasov, Kac, 196x] “ “Extinction probability” Extinction probability” obeys
nonlinear equation nonlinear equation [Galton, Watson, 1974] [Galton, Watson, 1974] “ “Extinction of peerage” Extinction of peerage” Attempts to solve Attempts to solve QCD loop QCD loop equations equations [Migdal, Marchesini, 1981] [Migdal, Marchesini, 1981] “ “Loop extinction”: Loop extinction”: No importance sampling No importance sampling