Complex Langevin Equation

and Application to ChemicalPotential Problems

I.-O. Stamatescu (Heidelberg)

Report on common work with:

Gert Aarts, Frank James (Swansea),

Erhard Seiler (Munich), Denes Sexty (Heidelberg)

1

Items

1. Introduction: stochastic quantization.

2. Beyond Euclidean QFT.

3. CLE and FPE: Formal discussion, and a criterium for correctness.

4. Tests of the correctness criterium in simple models.

5. Lattice models: XY, SU3 spin model, lattice QCD.

6. Summary and Outlook.

Appologies for incomplete quotations!

2

1. Stochastic quantization

We here shall briefly review some features of the stochastic

quantization in QFT: LE, RW and FPE, relation to the path integral,

convergence.

Stochastic processes in Quantum Field Theory

Realize a sampling of field configurations by defining a supplementary

(noisy) dynamics in a 5-th “time”.

Example: Parisi Wu stochastic quantization

- proofs of equivalence with path integral formulation, proofs of

convergence relying on the definition of a probability distribution over

the space of field configurations via an associated Fokker-Planck eq.

- can define a “perturbation theory” without gauge fixing

- for numerical studies: comparable to MC (but: step-size dependence)

(see also Damgaard and Hueffel, Namiki, ...)

3

Usual realizations: Langevin Equation and Random Walk.

Here in discretized form, Ito calculus, ϑ: 5-th “time”, δϑ : “time” step;

for a field ϕ(x) (random variable), K[ϕ]: drift force,

Langevin equation:

δϕ(x;ϑ) ≡ ϕ(x;ϑ+ δϑ)− ϕ(x;ϑ) = K[ϕ(x;ϑ)] δϑ + η(x;ϑ)

〈η(x;ϑ)〉 = 0, 〈η(x;ϑ)η(x′;ϑ′)〉 = 2 δϑ δx,x′ δϑ,ϑ′

Random Walk:

δϕ(x;ϑ) = ±ω, with pbb : 12 (1± 1

2ωK[ϕ(x;ϑ)]) , ω =√δϑ

In the following we shall use t insead of ϑ.

4

Relation to path integral and MC

If the drift is the gradient of a real action, bounded from below

then there is a probability density P (ϕ, t) satisfying an associated

Fokker-Planck Equation in the limit δt −→ 0:

∂tP (ϕ, t) = ∂ϕ (∂ϕ −K) P (ϕ, t), K = −∂ϕS

and we have:

P (ϕ, t) = c0e−S(x) +

∑

En>0

cnφne−Ent → Pas(ϕ) = c0e

−S[ϕ], (t → ∞)

with En the eigenvalues of the Fokker–Planck Hamiltonian:

HFP = −∂2ϕ +

1

4(∂ϕS)

2 − 1

2(∂2

ϕS)

5

- expectation values 〈f(ϕ)〉 can be calculated as averages over the

noise, equivalently as t averages:

f(ϕ) =1

T

∫ T

0

dt f(ϕ(t)) = 〈f(ϕ)〉+O(1/√T ) ,

- the convergence is controlled by the properties of the FP

Hamiltonian,

- in practice δt 6= 0: ρas(ϕ) has O(δt) corrections (controllable).

6

0 2 4 6 8 10β

0

0.2

0.4

0.6

0.8

1

⟨Uµν

⟩

Langevin without g.f., 124, ε=0.001, 10

4/100/50

heatbath, 124, 500/100/20

0 2 4 6 8 10β

0

0.2

0.4

0.6

0.8

1

⟨Uµν

⟩

random walk without g.f., 124, η=0.03, 10

4/100/50

heatbath, 124, 500/100/20

Figure 1: Plaquette averages by LE and RW compared with MC

7

Langevin equation: special interest

Essential feature: uses a drift force to define the process (the 5-th time

dynamics). Does not need probabilities!

−→ versatility

- can be directly related to expectation values

- can be directly defined from the set up of the problem without

needing an action or a probability interpretation for the path

integral

This may be of interest in cases where other approaches (e.g., MC) do

not work.

8

2. Beyond Euclidean QFT

Here we shall briefly discuss extensions of the LE method, primarily to

complex action problems: real time and chemical potential

Developments based on the versatility of the method:

• redefining the drift force,

• redefining the process (kernel methods),

• redefining the noise (e.g., nonlinear processes: “active brownian

motion”), . . .

Very much used in modeling (statistical physics, complex systems, etc.)

In QM and QFT: open systems, continuous localization, special

simulation problems.

Interesting if Euclidean formulation is not possible or ambiguous.

In the following: The complex action case.

9

Complex action in Quantum Field Theory

Special problems with complex action

- real time (Minkowski) path integral,

- non-zero chemical potential in euclidean LGT

- θ− vacua

- ...

The task: Define a Complex Langevin Equation (CLE) to reproduce

the expectation values of the Path Integral with complex action.

(many papers since the 1980’s: Parisi, Klauder, Gausterer,

Guralnik, Salcedo, Namiki, Schultke, Nakazano, ... - )

10

Minkowski path integral

Example: Nonequilibrium dynamics

Various methods: PTh, classical dynamics, CLE

(Berges, Smit, Bodeker, Garcia Perez, Aarts, Borsanyi, Sexty,

Stamatescu , ...)

11

Non-zero chemical potential.

Various models:

- Heavy Dense model for QCD (Stamatescu, Feo, Aarts, Philipsen, ..)

- Various approximations using the fact that the chemical potential

selects the Polyakov loops in the fermionic determinant: Polyakov loop

models, Potts model, etc(Green and Karsch, Karsch and Wyld,

Karsch and Stickan, Alford et al, Aarts and James, ..)

- Relativistic Bose gas, XY model, etc (Aarts, James,..), ...

Difficulty: Sign problem, no direct MC possible, ..

Various methods:

- Canonical ensemble (Karsch, ..), series expansion (TARO), analytic

continuation (from imaginary µ) (de Forcrand, Philipsen, ..),

reweighting (Barbour, Feo, Stamatescu, ..), worm algorithm

(Banerjee, Chandrasekaran, ..), CLE (Aarts, Seiler, Stamatescu, ...)

12

QCD with chemical potential and the HDM model.

The lattice HDM - QCD (heavy dense matter) model relies on the

hopping parameter expansion of the fermionic determinant. In the limit

κ → 0, µ → ∞, ζ = κ eµ : fixed (1)

only the Polyakov loops survive and the determinant factorizes

(Stamatescu, Karsch .. 1992, Blum, Tousaint, .. 1994, ..) . .

In the temporal gauge the Polyakov loop contains only one link and

the determinant is local in the links.

This has been used, e.g. in a reweighting analysis for QCD at non-zero

chemical potential (De Pietri, Feo, Seiler, Stamatescu).

In the following we shall repeatedly refer to it, therefore here some

detail.

13

QCD grand canonical partition function (Wilson fermions):

Z =

∫

DU e−S , S = SYM − log detW (2)

W = 1− κ3∑

i=1

(

Γ+iUx,iTi + Γ−iU†x,iT−i

)

−κγ(

eµΓ+4Ux,4T4 + e−µΓ−4U†x,4T−4

)

(3)

Here T are lattice translations, Γ±µ = 1± γµ, and κ is the hopping

parameter ∼ 1/M . γ is a (bare) anisotropy parameter. For non-zero µ

detW and S are complex. We still have detW (µ) = [detW (−µ)]∗.

The temperature is introduced as

aT =γ

Nτ

(4)

14

Hoping parameter expansion for a systematic approximation

(Stamatescu, Smit, Kawamoto, .. 1982):

DetW = exp(Tr lnW ) (5)

= exp

−∞∑

l=1

∑

Cl

∞∑

s=1

(κlλgCl

)s

sTrD,CLs

Cl

=∞∏

l=1

∏

Cl

DetD,C

(

1 − (κλ)lgCl

LCl

)

with Cl a closed, non-self-repeating path, λ the links on Cl and

LsCl

=

(

∏

λ∈Cl

ΓλUλ

)s

, gCl=(

ǫ e±Nτµf)r

or 1 (6)

with non-trivial gClfor loops winding r times in the ±4 direction with

periodic(antiperiodic) b.c. (ǫ = +1(−1)) and κλ = κ or κ γ for

spatial/temporal links.15

Large mass, large chemical potential limit (Bender, Stamatescu,.. ):

κ → 0, µ → ∞, κ eµ ≡ ζ : fixed (7)

0-th order:

Z [0]F (C, U) = exp

−2∑

~x

∞∑

s=1

(ǫC)s

sTr(P~x)

s

=∏

~x

Det (1I − ǫCP~x)2, C = (2 ζ)Nτ , (8)

(P~x: Polyakov loops).

NB: If one wants to ensure the symmetry detW (µ) = [detW (−µ)]∗

one can take also non-dominant terms in the determinant, thus

Z [0]F =

∏

~x

det(

1− ǫheµT P~x

)2

det(

1− ǫhe−µT P−1

~x

)2

, h = (2κ)Nτ

16

(κ

(κ

(κ

κ

e

e

e−µ

µ

µ )

)

)

N

N

N

n

τ

τ

τ

τ σ

2κ

T = 1 / N a τ τ

Figure 2: The HDM - QCD model.

17

2-nd order:

Z [2]F (κ, µ, U) = exp

−2∑

~x

∞∑

s=1

(ǫC)s

s×

× Tr

(P~x)s + κ2

∑

r,q,i,t,t′

(ǫC)s(r−1)(Pr,q~x,i,t,t′)

s

= Z [0]F (C, U)

∏

~x,r,q,i,t,t′

Det(

1I− (ǫC)r κ2 Pr,q~x,i,t,t′

)2

. (9)

Use temporal gauge for easy bookkeeping. The model is defined in

terms of the “Boltzmann factor” B = e−SY M ZF within various

approximation for ZF (Aarts et al 2002, Feo et al 2006, ...).

18

3. CLE and FPE: formal discussion and a criterium for correctness.

Here we shall introduce the CLE and its associated FPE, examine the

formal arguments for correctness, identify the loopholes which are

responsible for the occasional convergence to the wrong limit, and

establish a ”correctness criterium”.

19

The complex process.

Complex drift generates imaginary parts for the variables.

Hence the process is defined on a manifold Mc: the complex extension

of the original manifold Mr (typically Rn, Tn, SU(n).

The LE with complex drift K(z) for a complex variable

z(t) = x(t) + i y(t) , x ∈ Mr , z ∈ Mc (10)

amounts to two related real LE with independent noise terms

δz(t) = K(z, t) δt+ η(t) , η =√

NR ηR + i√

NI ηI (11)

i.e. : δx(t) = ReK(z, t) δt+√

NR ηR(t) (12)

δy(t) = ImK(z, t) δt +√

NI ηI(t) (13)

〈ηR〉 = 〈ηI〉 = 0 , 〈η2R〉 = 〈η2I 〉 = 2 δt , 〈ηRηI〉 = 0 , NR −NI = 1

20

We can also define a corresponding pair of real RW processes

δx(t) = ±ωx , Px,± = 12 (1±

ωx

2NR

ReK(z, t)) (14)

δy(t) = ±ωy , Py,± = 12 (1±

ωy

2NI

ImK(z, t)) (15)

ωx =√

2NRδt , ωy =√

2NIδt (16)

with P±: real transition probabilities (with appropriate definition of

the steps.

Both for the CLE and CRW the noise normalizations NR, NI are

arbitrary with the condition NR −NI = 1.

Here and in the following the drift is: K(z) = −∇zS

21

Observables and distributions.

Consider an “observable” O: an analytic function on Mc.

The expectation values are averages over the noise. We have:

∂t〈O(z(t))〉 = 〈LO(z(t))〉 = 〈LO(z(t))〉 (17)

L = [NR∇x +Kx]∇x + [NI∇y +Ky]∇y (18)

L = [∇z +K(z)]∇z (19)

(the second equality in Eq.(17) follows from the C-R conditions).Notice that L explicitely involves NR, NI while L does not.

In the standard way we can define a real FPE for the probability

distribution P (x, y, t) as realized in the process (12,13) or (14,15):

∂tP (x, y, t) = LT P (x, y, t) (20)

LT = [∇x (NR ∇x −ReK(z)) +∇y (NI ∇y − ImK(z))] (21)

〈P,LO〉 = 〈LTP,O〉, 〈P,O〉 =∫

O(x, y)P (x, y)dxdy (22)

22

We can also define a FPE for a complex “distribution” ρ(x) on Mr:

∂tρ(x, t) = LT0 ρ(x, t), LT

0 = ∇x (∇x −K) (23)

The complex density ρ(x;∞) ∝ exp [−S(x)] is a stationary solution of

Eq. (23), expected to be unique. Numerical studies (where feasible) of

Eq. (23) confirm this and the convergence appears exponentially fast.

We can thus define two types of expectation values

〈O〉P (t) ≡∫

O(x+ iy)P (x, y; t)dxdy∫

P (x, y; t)dxdy, 〈O〉ρ(t) ≡

∫

O(x)ρ(x; t)dx∫

ρ(x; t)dx.(24)

and what we should like to show is

〈O〉P (t) = 〈O〉ρ(t), (25)

23

Set up for the correctness question.

Instead of evolving P (x, y, t) we can evolve the observables.

For analytic functions we can use interchangeably L and L:

∂tO(z; t) = LO(z; t) (t ≥ 0) → O(z; t) = exp[t L]O(z). (26)

or equivalently

∂tO(x; t) = L0O(x; t) (t ≥ 0) → O(x; t) = exp[t L0]O(x). (27)

We consider the interpolation

FO(t, τ) ≡∫

P (x, y; t− τ)O(x+ iy; τ)dxdy, (28)

FO(t, 0) = 〈O〉P (t), FO(t, t) = 〈O〉ρ(t). (29)

24

Proof:

FO(t, t) =

∫

P (x, y; 0)(

etLO)

(x+ iy; 0)dxdy

=

∫

ρ(x; 0)(

etL0O)

(x; 0)dx

=

∫

O(x; 0)(

etLT0 ρ)

(x; 0)dx

= 〈O〉ρ(t), (30)

25

Formal proof of correctness and its loopholes.

Equality 〈O〉P (t) = 〈O〉ρ(t) is ensured if FO(t, τ) is independent of τ :

∂

∂τFO(t, τ) =−

∫

(

LTP (x, y; t − τ))

O(x+ iy; τ)dxdy

+

∫

P (x, y; t − τ)LO(x+ iy; τ)dxdy. (31)

The RHS =0 after integration by parts.

This proof relies on neglecting boundary terms from the y-integration.

Loophole: The proof fails if the fall off of the probability distribution is

not sufficient for the boundary terms to vanish.

26

A criterium for correctness.

As a sufficient condition for correctness we require

∂τFO(t, τ) = 0 . (32)

A weaker condition (suggested by numerical studies) is

limt→∞

∂τFO(t, τ)∣

∣

τ=0= 0 , (33)

For the equilibrium measure (LTP (x, y;∞) = 0) it is equivalent with

EO ≡∫

P (x, y;∞)LO(x+ iy; 0)dxdy = 〈LO〉 = 0 , (34)

Strong statement: If the ”Consistency Conditions” (CC) Eq.(34) are

satisfied for “a sufficient number of observables” the results are

correct. (Conjecture; needs further conditions.)

Weak statement: Violation of the CC’s indicate false convergence.

The set of CC’s are related to the Dyson-Schwinger Equations.

27

4. Tests of the correctness criterium in simple models.

We here shall discuss the problem of correctness and test the

consistency conditinons in simple models where the exact results can

be directly calculated:

- one link U(1),

- Guralnik-Pehlevan (GP ) PT action,

- SU(3) one link effective model for HDM-QCD.

28

Example 1: One link U(1) model

Z =

∫

dU e−SY M+i p x detW =

∫ π

−π

dx

2πe−S(x), (35)

SYM = −β

2

(

U + U−1)

= −β cosx , U = ei x

detW = 1 +κ

2

[

eµU + e−µU−1]

= 1 + κ cos(x− iµ).

Observables (exact expressions in terms of Bessel functions):

〈Un〉 = 〈einx〉 , 〈U−n〉 = 〈e−inx〉 (Polyakov loop and its inverse)

〈cos x〉 = ∂∂β

lnZ (plaquette), 〈n〉 = ∂∂µ

lnZ =⟨

iκ sin(x−iµ)1+κ cos(x−iµ)

⟩

Notice that in complexifying the variable x → z = x+ i y we must use

U−1 instead of U∗ everywhere. p may be used for “reweighting”.

For κ = 0, β : imaginary we mock Minkowski quantization,

For β : real and κ, µ 6= 0 we mock non-zero chemical potential.

29

M

x

y

M

x

y

r

cM

rMc

Figure 3: Complexification of the U(1) one link model (left).

Compactification of Mc (right). NB: This exlicitly affects the model!

30

NI dependence

Since the expectation values of analytic observable should not depend

on NI we shall use NI to check the validity of formal arguments.

We also introduce a periodic cut off Y by compactifying the y

direction. This represents an explicit modification of the model.

We performed both CLE and FPE analyses (for the latter a small

NI > 0 is necessary to stabilize the DE).

We find that the results depend both on NI (formally not expected)

and on Y (expected if the probability measures are wide).

We first calculate explicitely FO(t, τ) for various observables using the

FPE solution.

31

0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

0 1 2 3 4 5 6 7

f(t,tau) vs tau, NI=0.1, Y=3.162, taumax =< t =< 10 (from up)

0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

0 1 2 3 4 5 6 7

f(t,tau) vs tau, NI=0.1, Y=0.474, taumax =< t =< 10 (from up)

Figure 4: U(1) one-link model: FU (t, τ) vs. τ for several values of t,

with 0 < τ < t, NI = 0.1 and cutoff Y = 3.162 (left) and 0.362 (right).

We observe that for NI = 0.1 FO(t, τ) does depend on τ and this

dependence changes with Y such that for an intermediate value of Y it

disappears.

32

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0 0.2 0.4 0.6 0.8 1

kappa=0.25, mu=0.5, NI=0.1 vs tanh(Y/1.581): 0FPE O(1)/ex-1CLE O(1)/ex-1FPE O(2)/ex-1

FPE E(1)CLE E(1)FPE E(2)

Figure 5: Cutoff (Y ) dependence of various quantities in the U(1) one-

link model, for β = 1, κ = 0.25, µ = 0.5 and NI = 0.1.

We observe that at Y ≃ 0.362 (which makes FO(t, τ) flat) the CC’s

are satisfied and the results are correct.

33

Example 2: The Guralnik − Pehlevan model

The model is defined through the PT-symmetric action

S = −iβ

(

z +1

3z3)

. (36)

The complexification of the model is the complex plane. We find a

similar situation: Fulfilment of the CC’s is correlated to the correctness

of the results. Notice that fulfilment of only one CC is not enough!

The equilibrium measure shows a weaker decay for NI > 0, this should

explain the (incorrect) NI > 0 dependence of the result without cut

off.

34

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8

GP, beta=1, NI=1, X=3.16, vs Y: 0FPE, Imz/ex-1CLE, Imz/ex-1

FPE, Rez2/ex-1CLE, Rez2/ex-1FPE, Imz3/ex-1CLE, Imz3/ex-1FPE, Rez4/ex-1CLE, Imz4/ex-1

FPE, E2CLE, E2FPE, E3CLE, E3

Figure 6: GP model: Cutoff dependence for β = 1, NI = 1 (E1 is trivial

in this model).

35

Figure 7: GP model: Fall off of of the equilibrium measures (log-log

scale).

36

Example 3: Effective SU(3) one link model model.

For an effective model for the HDM-QCD problem we concentrate on

one link in the field of its neighbors. This can be used as a probe for

the convergence questions concerning the full lattice model.

−S =β

2

(

trUA+ trA−1U−1)

+ lnD + ln D (37)

D = 1 + CtrU + C2trU−1 + C3, C = 2κeµ (38)

D = 1 + CtrU−1 + C2trU + C3, C = 2κe−µ (39)

Here the matrices A ∈ GL(3, C) stand for the staples.

I-st step: We diagonalize A = V −1DV and obtain for the Y-M action

−SY =β

2

(

trUD + trD−1U−1)

, D = diag(α1, α2, α3), U = V UV −1

Using the invariance of the Haar measure we can make the integration

over the U ∈ SL(3, C) matrices. The determinant is invariant under

this transformation.

37

II-nd step: reduce then integration to the 3 complex angles in the

diagonal of U and simulate the integral over the remaining angles by

the reduced Haar measure.

→ Effective model:

−S =β

2

3∑

i=1

(

αieiwi + α−1

i e−iwi)

+ lnD + ln D + f lnH (40)

H = sin2w2 − w3

2sin2

w3 − w1

2sin2

w1 − w2

2, w1 + w2 + w3 = 0

(we introduced a parameterf to test the effect of the Haar measure).

The observables and the CC conditions are:

On = tr(Un) = ei nw1 + ei nw2 + ei nw3 (41)

En =(

∇2 +K∇)

On, K = −∇S (42)

There follow some tests. Solid lines represent the exact results. A

generic set of α’s is (1.0, 0.2), (0.8, 0.0), (0.5,−0.5) .

38

-0.5

0

0.5

1

0 0.5 1 1.5 2

bt=(1,0), k=0.25, NI=0, nz=3, vs mu,*: O1 O2 O3

O-1 O-2 O-3 E1 E2 E3

E-1 E-2 E-3

Figure 8: Effective model: Observables and CC’s vs µ for generic αi.

39

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

Beta=(1,0), k=0.25, mu=0.5, NI=0, nz=3, alpha*, vs fhm: O1 O2

O-1 O-2 E1 E2

E-1 E-2 E-3

Figure 9: Effective model: Observables and CC’s vs f , the Haar measure

parameter, for generic αi.

40

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.5 1 1.5 2

ReBeta=1, k=0.25, mu=0.5, NI=0, nz=3, vs ImBeta: O1 O2

O-1 O-2 E1 E2

E-1 E-2

Figure 10: Effective model: Observables and CC’s vs Imβ, for αi = 1.

The violation of the CC’s signalizes discrepant results.

41

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1

Beta=(1,0), k=0.25, mu=0.5, NI=0, nz=3, vs alpha3*: O1 O2

O-1 O-2 E1 E2

E-1 E-2 E-3

Figure 11: Effective model: Dependence of the observables and the CC’s

on the αi’s. The violation of the CC’s signalizes discrepant results.

42

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

-0.1 -0.05 0 0.05 0.1

Y=infty, p(y) vs y: alpha3*=0.4, y(1)y(2)

alpha3*=0.3, y(1)y(2)

alpha3*=0.2, y(1)y(2)

alpha3*=0.1, y(1)y(2)

alpha3*=0.0, y(1)y(2)

Figure 12: Effective model: Histograms of the equilibrium measure for

different αi’s. The wide structures correlate with the violation of the

CC’s and the wrong results.

43

5. Lattice models: XY, SU3 spin model, QCD.

We here shall briefly describe recent work with the CLE for some

lattice models.

Further work in progress pertains the Thirring model (Pawlowski,

Spielmann, Zielinski, Stamatescu, Hands, Aarts, ..).

Also mentioned should be earlier work concerning real time,

non-equilibrium studies, both for φ4 and QCD models (Berges,

Borsanyi, Sexty, Stamatescu).

44

XY model at non-zero chemical potential

(Frank James and Gert Aarts)

The model is implemented on a Ω = NτN2s pbc lattice with the

action:

S = −β∑

x

2∑

ν=0

cos(φx − φx+ν − iµδν,0), 0 ≤ φx < 2π (43)

The authors use adaptive step size for the simulation. They use a

worm algorithm to test the CLE results.

They find that in the ordered phase (large β) the CLE analysis

reproduces the correct results while in the disordered phase (small β) it

fails. This behaviour clearly correlates with the width of the various

distributions.

45

-0.2 -0.1 0 0.1 0.2

µ2

-1.6

-1.55

-1.5

-1.45

-1.4

<S>

/Ω

complex Langevinreal Langevinworld line

β=0.7, 83

-0.2 -0.1 0 0.1 0.2

µ2

-0.2

-0.18

-0.16

-0.14

<S>

/Ω

complex Langevinreal Langevinworld line

β=0.3, 83

Figure 13: XY model for large (left) and small (right) β, CLE and worm

algorithm: Real part of the action density around µ2 = 0 (negative

values imply imaginary µ, i.e. a euclidean simulation.

46

1 10 100 1000

Kmax

/6β

0

5

10

15

20

25

30P[

Km

ax/6

β]hotcold

β = 0.7, µ = 0.1, 83

1 10 100 1000

Kmax

/6β

0

5

10

15

20

P[K

max

/6β]

hotcold

β = 0.3, µ = 0.1, 83

Figure 14: XY model for large (left) and small (right) β, µ = 0.1, CLE:

Histogram of the maximal value of the drift force, Kmax.

47

3-d SU(3) spin model at non-zero chemical potential

(Karsch and Wyld revisited).(Frank James and Gert Aarts)

SB = −β∑

x,i

[

TrUxTrU†x+i + TrU †

xTrUx+i

]

SF = −h∑

x

[

eµTrUx + e−µTrU †x

]

This is work in progress. The authors test the CLE simulation and

higher order algorithms. They find good description of the phase

structure and that the higher order algorithms allow a much larger step

size and therefore a significant speed up.

48

-1 -0.5 0 0.5 1

µ2

0

0.5

1

1.5

2

< T

r(U

+U

-1)/

2 >

β=0.135β=0.134β=0.132β=0.130

-1 -0.5 0 0.5 10

0.5

1

1.5

2

β=0.128β=0.126β=0.124β=0.120

h=0.02, 103

Figure 15: SU(3) spin model: action density vs µ.

49

0 2000 4000 6000 8000 10000Langevin time

0

0.5

1

1.5

2

<T

r(U

+U

-1)/

2>

β=0.134, µ2=−0.65, h=0.02, 103

Figure 16: SU(3) spin model: first order behaviour.

50

1.695

1.7

1.705

1.71

<T

rU>

lowest orderimproved

1.735

1.74

1.745

1.75<

TrU

-1>

0 0.0002 0.0004 0.0006 0.0008 0.001ε

0.68

0.682

0.684

<n>

β=0.125, µ=3, h=0.02, 103

Figure 17: SU(3) spin model: test of higher order algorithms.

51

The lattice QCD problem.

This is more work in progress. We attack the problem in the HDM

approximation. The updating:

dXi = −∇iS dτ + ηi (44)

U → U ′ = ei∑

8

i=1dXiλi U (45)

proceeds in the algebra and is followed by exponentiation. The latter

is trivial if we chose to update one direction at a time.

Run-aways are eliminated by using adaptive step size.

52

Figure 18: SU(3) updating.

53

6. Summary and outlook.

- In earlier work we have shown that one of the problems of CLE

(run-aways) can be eliminated by using adaptive step size.

- We think now of having identified also the reason for the other

problem (occasional convergence to the wrong limit), namely slow

fall off of the distributions in the imaginary direction.

- Most important: we could set up a criterium for the correctness of

the result, namely the fulfilment of the Consistency Conditions

(CC’s), which can easily be checked. In principle one needs the full

set of CC’s (and further bounds), but we conjectured that already

the first few CC’s can provide a reliable test.

- We have confirmed the above findings and conjectures in a

number of models, including lattice models.

54

- We must extend the analysis to the most interesting problems:

non-equilibrium QFT and full QCD with chemical potential.

- We should try to go beyond a criterium for correctness toward a

mechanism to “repair” the method in the cases where the

criterium is not fulfilled.

55