STOCHASTIC QUANTIZATION AT FINITE …...Bilic, Gausterer, Sanielevici ’88 Lattice08, July 2008 – p.4 FINITE CHEMICAL POTENTIAL WHAT WE DID three models of the form Z = Z DUe−SB

STOCHASTIC QUANTIZATION AT FINITE

CHEMICAL POTENTIAL

Gert Aarts

with I.-O. Stamatescu

arXiv:0807.1597 [hep-lat]

Swansea University

Lattice08, July 2008 – p.1

INTRODUCTIONQCD AT NONZERO BARYON DENSITY

QCD at finite µ: complex fermion determinant

det M(µ) = [det M(−µ)]∗

partition function: Z =

∫

DU e−SB(U) det M

importance sampling not possible

reweighting

Taylor expansion

analytical continuation

density of states

canonical ensemble

...

here: stochastic quantization


STOCHASTIC QUANTIZATIONLANGEVIN DYNAMICS

alternative nonperturbative numerical approach

weight = equilibrium distribution of stochastic process

think: Brownian motion

particle in a fluid: friction (γ) and kicks (η)Langevin equation:

d

dt~v(t) = −γ~v(t) + ~η(t) 〈ηi(t)ηj(t

′)〉 = 2kTγδijδ(t − t′)

equilibrium solution/noise average:

limt→∞

1

2〈vi(t)vj(t)〉 =

1

2δijkT



apply to field theory (Parisi and Wu ’81)

∂φ(x, θ)

∂θ= − δS[φ]

δφ(x, θ)+ η(x, θ)

Gaussian noise

〈η(x, θ)〉 = 0 〈η(x, θ)η(x′, θ′)〉 = 2δ(x − x′)δ(θ − θ′)

corresponding Fokker-Planck equation

∂P [φ, θ]

∂θ=

∫

ddxδ

δφ(x, θ)

(

δ

δφ(x, θ)+

δS[φ]

δφ(x, θ)

)

P [φ, θ]

stationary solution: P [φ] ∼ e−S



real action: formal proofs of convergence(but can also use importance sampling)

complex action: no formal proofs available(but other methods in serious trouble)

force δS/δφ complex: complex Langevin dynamics

example: real scalar field φ → Re φ + iIm φ

∂Re φ

∂θ= −Re

δS

δφ+ η

∂Im φ

∂θ= −Im

δS

δφ

observables: analytic extension

〈O(φ)〉 → 〈O(Re φ + iIm φ)〉Lattice08, July 2008 – p.3

(PRE)HISTORY

Parisi and Wu ’81

Damgaard and Hüffel, Physics Reports ’87

application to finite µ:

effective three-dimensional spin models

Karsch and Wyld ’85

Ilgenfritz ’86

Bilic, Gausterer, Sanielevici ’88


FINITE CHEMICAL POTENTIALWHAT WE DID

three models of the form

Z =

∫

DUe−SB det M det M(µ) = [det M(−µ)]∗

QCD in hopping expansion

SU(3) one link model

U(1) one link model

observables:

(conjugate) Polyakov loops

density

phase of determinant


THREE MODELSI: QCD IN HOPPING EXPANSION

fermion matrix:

M = 1 − κ

3∑

i=1

space − κ(

eµΓ+4Ux,4T4 + e−µΓ−4U−1x,4T−4

)

hopping expansion:

det M ≈ det[

1 − κ(

eµΓ+4Ux,4T4 + e−µΓ−4U−1x,4T−4

)]

=∏

x

det(

1 + heµ/TPx

)2det

(

1 + he−µ/TP−1x

)2

with h = (2κ)Nτ and the (conjugate) Polyakov loops P(−1)x

full gauge dynamics included


THREE MODELSII: SU(3) ONE LINK MODEL

Z =

∫

dUe−SB det M link U ∈ SU(3)

SB = −β

6

(

Tr U + Tr U−1)

determinant:

det M = det[

1 + κ(

eµσ+U + e−µσ−U−1)]

= det (1 + κeµU) det(

1 + κe−µU−1)

with σ± = (11 ± σ3)/2

det in colour space remaining

exact evaluation by integrating over the Haar measureLattice08, July 2008 – p.6

THREE MODELSIII: U(1) ONE LINK MODEL

U(1) model: link U = eix with −π < x ≤ π

SB = −β

2

(

U + U−1)

= −β cos x

determinant:

det M = 1 +1

2κ

[

eµU + e−µU−1]

= 1 + κ cos(x − iµ)

partition function:

Z =

∫ π

−π

dx

2πeβ cos x [1 + κ cos(x − iµ)]

all observables can be computed analyticallyLattice08, July 2008 – p.6

COMPLEX LANGEVIN DYNAMICS

Langevin update:

U(θ + ε) = R(θ) U(θ) R = exp[

iλa

(

εKa +√

εηa

)]

drift term

Ka = −DaSeff Seff = SB+SF SF = − ln det M

noise〈ηa〉 = 0 〈ηaηb〉 = 2δab

real action: ⇒ K† = K ⇔ U ∈ SU(3)

complex action: ⇒ K† 6= K ⇔ U ∈ SL(3, C)


(CONJUGATE) POLYAKOV LOOPS

U(1) ONE LINK MODEL

0 2 4 6 8µ

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Re

< eix

>

β=1β=2β=3

0 2 4 6 8µ

0.5

1

1.5

2

2.5

Re

< e-i

x >

β=1β=2β=3

data points: complex Langevinstepsize ε = 5 × 10−5, 5 × 107 time steps

lines: exact results

excellent agreement for all µ



SU(3) ONE LINK MODEL

0 1 2 3 4

µ

0

0.1

0.2

0.3

0.4

0.5

Re

< P

>

β=1β=2β=3

0 1 2 3 4

µ

0

0.1

0.2

0.3

0.4

0.5

Re

< P

-1>

β=1β=2β=3

data points: complex Langevinstepsize ε = 5 × 10−5, 5 × 107 time steps

lines: exact results




SU(3) ONE LINK MODEL

scatter plot of P during Langevin evolution



QCD IN HOPPING EXPANSION

first results on 44 lattice at β = 5.6, κ = 0.12, Nf = 3

0.4 0.5 0.6 0.7 0.8 0.9 1

µ

0

0.05

0.1

0.15

Pol

yako

v lo

op<P>

<P-1

>

low-density “confining” phase ⇒ high-density “deconfining” phase


DENSITYU(1) ONE LINK MODEL SU(3) ONE LINK MODEL

0 2 4 6 8

µ

0

0.2

0.4

0.6

0.8

1

Re

<n>

β=1β=2β=3

0 1 2 3 4

µ

0

0.5

1

1.5

2

2.5

3

Re

<n>

β=1β=2β=3

linear increase at small µ

saturation at large µ



DENSITYQCD IN HOPPING EXPANSION

0.4 0.5 0.6 0.7 0.8 0.9 1

µ

0

2

4

6

8

10

12

dens

ity

first results on 44 lattice at β = 5.6, κ = 0.12, Nf = 3

low-density phase ⇒ high-density phase


REAL VS. COMPLEX LANGEVINU(1) ONE LINK MODEL

-60 -40 -20 0 20 40 60

µ2

0

0.5

1

1.5

Re

<co

s x>

β=1β=2β=3

real Langevin complex Langevin

plaquette as a function of µ2

µ2 < 0: imaginary chemical potential ⇔ real actionLattice08, July 2008 – p.10

SIGN PROBLEMU(1) ONE LINK MODEL

det M(µ) = [det M(−µ)]∗ = | det M(µ)|eiφ

average phase factor: 〈e2iφ〉 =

⟨

det M(µ)

det M(−µ)

⟩

0 2 4 6 8

µ

0

0.2

0.4

0.6

0.8

1

Re

< e2i

φ >

β=1β=2β=3




Re e2iφ

scatter plot of e2iφ during Langevin evolutionLattice08, July 2008 – p.11



Re e2iφ





SIGN PROBLEMQCD IN HOPPING EXPANSION

average phase factor: 〈e2iφ〉 =

⟨

det M(µ)

det M(−µ)

⟩


SU(3) → SL(3,C)QCD IN HOPPING EXPANSION

1

3Tr U †U ≥ 1 = 1 if U ∈ SU(3)


WHY DOES IT (APPARENTLY) WORK?

one link models: excellent

precise agreement with exact results

sign problem not a problem

well defined distributions

field theory encouraging

why?

classical flow

Fokker-Planck equationin U(1) model


CLASSICAL FLOWU(1) ONE LINK MODEL

link U = eix complexification x → z = x + iy

Langevin dynamics: x = Kx + η y = Ky

classical forces: Kx = −Re∂S

∂x

∣

∣

∣

x→zKy = −Im

∂S

∂x

∣

∣

∣

x→z

classical fixed points: Kx = Ky = 0

one stable fixed point at x = 0, y = ys(µ)

unstable fixed points at x = π, y = yu(µ)

structure is independent of µ!


CLASSICAL FLOWU(1) ONE LINK MODEL

flow diagrams and Langevin evolution

-2 -1 0 1 2 3 4 5

x

-2

0

2

y

µ=0.1

-2 -1 0 1 2 3 4 5

x

-2

0

2

4

y

µ=2

black dots: classical fixed points

µ = 0: dynamics only in x direction

µ > 0: spread in y directionLattice08, July 2008 – p.15

COMPLEX FOKKER-PLANCK EQUATIONU(1) ONE LINK MODEL

complex Fokker-Planck equation:

∂P (x, θ)

∂θ=

∂

∂x

(

∂

∂x+

∂S

∂x

)

P (x, θ)

all eigenvalues are real ⇔ det M(µ) = [det M(−µ)]∗

0 0.5 1 1.5 2 2.5 3

β

0

5

10

15

20

λ

µ=0µ=1µ=2µ=3µ=4

smallest nonzero eigenvalue

all eigenvalues ≥ 0

open question: real Fokker-Planck equation for ρ(x, y, θ)Lattice08, July 2008 – p.16

SUMMARY

finite chemical potential: complex actionstochastic quantization and complex Langevin dynamics

one link models: excellent

field theory: encouraging

detailed study of

(sign problem and) phase of the determinant

why? partly understood in simple models

classical flow qualitatively unchanged

complex FP equation: eigenvalues ≥ 0

to do: more field theoryLattice08, July 2008 – p.17

Documents

STOCHASTIC QUANTIZATION AT FINITE …...Bilic, Gausterer, Sanielevici ’88 Lattice08, July 2008 – p.4 FINITE CHEMICAL POTENTIAL WHAT WE DID three models of the form Z = Z DUe−SB