Complex Langevin dynamics for nonabelian gauge...

Complex Langevin dynamicsfor nonabelian gauge theories

Gert Aarts

XQCD 14, June 2014 – p. 1

QCD phase diagram

QCD partition function

Z =

∫

DUDψDψ e−SYM−SF =

∫

DU detD e−SYM

at nonzero quark chemical potential

[detD(µ)]∗ = detD(−µ∗)

fermion determinant is complex

straightforward importance sampling not possible

sign problem

⇒ phase diagram has not yet been determinednon-perturbatively

XQCD 14, June 2014 – p. 2

Outline

complex Langevin: exploring a complexified field space

gauge theories: from SU(N ) to SL(N,C)

gauge cooling

status report:

SU(3) + θ-term

heavy dense QCD

full QCD

summary

XQCD 14, June 2014 – p. 3

Complex Langevin dynamics

various scattered results since mid 1980s

here: recent results obtained with

Erhard Seiler, Dénes Sexty, Nucu Stamatescu

Benjamin Jaeger

Lorenzo Bongiovanni, Felipe Attanasio

. . .

some refs:0807.1597 [GA, IOS]

0912.3360 [GA, ES, IOS], 0912.0617, 1101.3270 [GA, FJ, ES, I OS]

1211.3709 [ES, DS, IOS], 1307.7748 [DS]

1306.3075 [GA, PG, ES], 1308.4811 [GA]

1311.1056 [GA, LB, IOS, ES, DS]

reviews: 1302.3028 [GA], 1303.6425 [GA, LB, IOS, ES, DS]

XQCD 14, June 2014 – p. 4

Complexified field space

partition function Z =∫

dx ρ(x) with complex weight ρ(x)

dominant configurations in the path integral?

x

Re

ρ(x)

⇒

y

x

real and positive distribution P (x, y): how to obtain it?

⇒ solution of stochastic process

complex Langevin dynamicsParisi 83, Klauder 83

XQCD 14, June 2014 – p. 5

Complex Langevin dynamics

does it work?

for real actions: stochastic quantization Parisi & Wu 81

equivalent to path integral quantization

Damgaard & Huffel, Phys Rep 87

for complex actions: formal proof was notably absent

recent progress:

theoretical foundation given

practical criteria for correctness formulated

severe sign and Silver Blaze problems explicitly solved

first results for gauge theories and even full QCD

XQCD 14, June 2014 – p. 6

Localised distributions

if

the action is holomorphic

[log dets are an additional worry, see Kim Splittorff ]

and

the distribution is localised, i.e.

P (x, y) = 0 for |y| > ymax [or P (x, y) → 0 fast enough]

then

the correct result is obtained GA, ES, IOS 09 + FJ 11

applications to gauge theories ...

XQCD 14, June 2014 – p. 7

Gauge theories

SU(N ) gauge theory: complexification to SL(N,C)

links U ∈ SU(N ): CL update

U(n+1) = R(n)U(n) R = exp[

iλa(

ǫKa +√ǫηa

)]

Gell-mann matrices λa (a = 1, . . . N2 − 1)

drift: Ka = −Da(SB + SF ) SF = − ln detM

XQCD 14, June 2014 – p. 8

Gauge theories

SU(N ) gauge theory: complexification to SL(N,C)

links U ∈ SU(N ): CL update

U(n+1) = R(n)U(n) R = exp[

iλa(

ǫKa +√ǫηa

)]

Gell-mann matrices λa (a = 1, . . . N2 − 1)

drift: Ka = −Da(SB + SF ) SF = − ln detM

complex action: K† 6= K ⇔ U ∈ SL(N,C)

deviation from SU(N ): unitarity norms

1

NTr

(

UU † − 11)

≥ 01

NTr

(

UU † − 11)2 ≥ 0

XQCD 14, June 2014 – p. 8

Gauge theories

deviation from SU(3): unitarity norm GA & IOS 08

1

3TrUU † ≥ 1

heavy dense QCD, 44 lattice with β = 5.6, κ = 0.12, Nf = 3XQCD 14, June 2014 – p. 9

Gauge theories

controlled evolution: stay close to SU(N ) submanifold when

small chemical potential µ

small non-unitary initial conditions

in presence of roundoff errors

XQCD 14, June 2014 – p. 10

Gauge theories

controlled evolution: stay close to SU(N ) submanifold when

small chemical potential µ

small non-unitary initial conditions

in presence of roundoff errors

in practice this is not the case

⇒ unitary submanifold is unstable!

process will not stay close to SU(N )

wrong results in practice, e.g. jumps when µ2 crosses 0

also seen in abelian XY model

XQCD 14, June 2014 – p. 10

Gauge cooling

what is the origin? can this be fixed?

gauge freedom: link at site k

Uk → ΩkUkΩ−1k+1 Ωk = eiω

k

aλa → e−αfk

aλa

in SU(N ): ωka ∈ R ⇒ in SL(N,C): ωk

a ∈ C

choose ωka purely imaginary, orthogonal to SU(N )

control unitarity norm D =1

NTr

(

UU † − 11)

≥ 0

gauge cooling ES, DS & IOS 12

after one gauge cooling step at site k, D → D′, linearise

D′ − D = − α

N(fka )

2 +O(α2) ≤ 0XQCD 14, June 2014 – p. 11

Gauge cooling

what is fka? Ωk = e−αfk

aλa D′ − D = −α/N(fka )2 + . . .

choose fka as the gradient of the unitarity norm

fka = 2Trλa

(

UkU†k − U †

k−1Uk−1

)

if U ∈ SU(N ): fka = 0, D = 0, no effect

cooling brings the links asclose as possible to SU(N )

SU( )

NSL( ,C)

N

XQCD 14, June 2014 – p. 12

Gauge cooling

simple example: one-link model GA, LB, ES, DS & IOS 13

S =1

NTrU U → ΩUΩ−1

D =1

NTr

(

UU † − 11)

fa = 2Trλa(

UU † − U †U)

note: c = TrU/N, c∗ = TrU †/N invariant under cooling

cooling dynamics:

D′ − D ≡ ˙D = − α

Nf2a = −16α

NTrUU †[U,U †]

in SU(2)/SL(2,C):

˙D = −8α(D2 + 2

(

1− |c|2)D+ c2 + c∗2 − 2|c|2

)

XQCD 14, June 2014 – p. 13

Gauge cooling

SU(2)/SL(2,C) one-link model

˙D = −8α(D2 + 2

(

1− |c|2)D+ c2 + c∗2 − 2|c|2

)

c = 12TrU, c∗ = 1

2TrU† invariant under cooling

if c = c∗: U gauge equivalent to SU(2) matrix

˙D = 8α(D+ 2− 2c2)D D(t) ∼ e−16α(1−c2)t → 0

if c 6= c∗: U not gauge equivalent to SU(2) matrixD(t) → D0 = |c|2 − 1 +√

1− c2 − c∗2 + |c|4 > 0

minimal distance from SU(2)reached exponentially fast

XQCD 14, June 2014 – p. 14

Langevin with gauge cooling

complex Langevin dynamics with gauge cooling:

alternate CL updates with gauge cooling updates

monitor unitarity norm

stay fairly close to SU(N )

under investigation:

SU(3) with a θ-term Lorenzo Bongiovanni, GA, ES, DS & IOS

13 + in prep

heavy dense QCD GA & IOS 08, ES, DS & IOS 12

Benjamin Jaeger, Felipe Attanasio, GA, ES, DS & IOS in prep

full QCD Denes Sexty 13 + in prep

XQCD 14, June 2014 – p. 15

SU(3) Yang-Mills theory with a θ-term

XQCD 14, June 2014 – p. 16

SU(3) with aθ-term

pure SU(3) Yang-Mills theory (no fermions)

S = SYM − iθQ Q =g2

64π2

∫

d4xF aµνF

aµν

on the lattice:

S = SW − iθL∑

x

qL(x) qL(x) = discretised lattice version

θL bare parameter, requires renormalisation

lattice QL =∑

x qL is not topological (top. cooling)

complex action for real θL, real action for imaginary θL

imaginary θL: real Langevin and hybrid Monte Carlo (HMC)

real θL: use complex LangevinXQCD 14, June 2014 – p. 17

Analytical continuation

Z(θL) =

∫

DU e−SYM+iθLQ = e−Ωf(θL)

partition function and free energy even functions of θL

real θL:

−i〈Q〉θL =∂ lnZ

∂θL= ΩχLθL

(

1 + 2b2θ2L + 3b4θ

4L + . . .

)

imaginary θL = iθI :

〈Q〉θI = −∂ lnZ∂θI

= ΩχLθI(

1− 2b2θ2I + 3b4θ

4I + . . .

)

determine topological charge susceptibility χL andhigher-order coefficients bk

XQCD 14, June 2014 – p. 18


−i〈Q〉θL = −∂ lnZ∂θL

= ΩχLθL(

1 + 2b2θ2L + 3b4θ

4L + . . .

)

β = 6.1

Ω = 104

lines are fits:

y = b0θL(

1± 2b2θ2L

)

with b0 = 0.026for both andb2 ∼ 1 · 10−5

no renormalisation of θL yet!XQCD 14, June 2014 – p. 19


−i〈Q〉θL = −∂ lnZ∂θL

= ΩχLθL(

1 + 2b2θ2L + . . .

)

expected β dependence: drop in χL

no renormalisation of θL yet!XQCD 14, June 2014 – p. 20

Heavy dense QCD (and beyond)

XQCD 14, June 2014 – p. 21

Heavy dense QCD

consider static quarks: fermion determinant simplifies

detM =∏

x

det(

1 + heµ/TPx

)2det

(

1 + he−µ/TP−1x

)2

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

full Wilson gauge action is included

nontrivial phase diagram:

thermal deconfinement transition (as in pure glue)µ-driven transition at µc ∼ − ln(2κ)

test case for full QCD

XQCD 14, June 2014 – p. 22

Heavy dense QCD

consider static quarks: fermion determinant simplifies

detM =∏

x

det(

1 + heµ/TPx

)2det

(

1 + he−µ/TP−1x

)2

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

preliminary results for Polyakov loop and density for

β = 5.8 (a ∼ 0.15 fm)

κ = 0.12 (µc ∼ − ln(2κ) = 1.43)

volume 83 ×Nτ

Nτ = 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 22 24 26 28

Benjamin Jaeger

XQCD 14, June 2014 – p. 23

Heavy dense QCD (in progress)

Polyakov loop

XQCD 14, June 2014 – p. 24


density

XQCD 14, June 2014 – p. 24


density µc = − ln(2κ) = 1.43 nsat = 12

first order transition at T = 0 (expected)see poster of Ben for more results

XQCD 14, June 2014 – p. 24

Beyond heavy dense QCD

include higher-order κ corrections DS, IOS et al, in prep

closer to real physicse.g. mB/3 6= mπ/2, no immediate saturation, . . .

0

0.05

0.1

0.15

0.2

14 14.5 15 15.5 16 16.5 17 17.5

0103.16, nf=2, NLO, beta=5.8, kappa=0.12 vs mu/T; denstot

dens0dens2

〈n〉/T 3 vs µ at NLO (including O(κ2) corrections)see poster of Nucu

XQCD 14, June 2014 – p. 25

Beyond heavy dense QCD

include higher-order κ corrections DS, IOS et al, in prep

closer to real physicse.g. mB/3 6= mπ/2, no immediate saturation, . . .

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 5 10 15 20 25 30

dens

ity

order of NLO corrections

43*4β=5.9κ=0.12 NF=2µ=0.9

fullqcdκ expansion

convergence in systematic expansion in κ2 (DS)XQCD 14, June 2014 – p. 25

Full QCD

XQCD 14, June 2014 – p. 26

Full QCD

first application to full QCD: Denes Sexty arXiv:1307.7748

fermion determinant: additional drift term in CLE

requires inversion of fermion matrix

stochastic inversion using conjugate gradient

staggered fermions with 4 flavours(Wilson fermions in progress)

monitor unitarity norm, log det, distributions, . . .

compare with HDQCD for heavy quarks andreweighting for light quarks

XQCD 14, June 2014 – p. 27

Full QCD

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25

n/n s

at

µ/T

83*6 latticeNf=4β=5.8

HQCD ma=1staggered ma=1


density / densitysat vs µ/T

for heavier quarks and lighter quarks

comparison with HDQCD Sexty 13

XQCD 14, June 2014 – p. 28

Full QCD

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 5 10 15 20 25

Pol

yako

v lo

op

µ/T

83*6 latticeNf=4β=5.8



Polyakov loop vs µ/T

for heavier quarks and lighter quarks

comparison with HDQCD Sexty 13

XQCD 14, June 2014 – p. 28

Full QCD

comparison with reweighting

0.27

0.28

0.29

0.3

0.31

0.32

0.33

0.34

0.35

0.36

0.37

0 0.5 1 1.5 2 2.5 3 3.5 4

µ/T

83*4 latticeβ=5.4mass=0.05NF=4

Polyakov loop CLEinverse Polyakov CLEPolyakov reweighting

inv. Polyakov reweighting

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4

Re

<ex

p(i 2

φ)>

µ/T

83*4 latticeβ=5.4mass=0.05NF=4

rewCLE

Polyakov loop vs µ/T average sign

for light quarks

agreement until reweighting breaks down

Fodor, Katz & Sexty in prep

XQCD 14, June 2014 – p. 28

Summary and outlook

complex Langevin dynamics:

recent progress for gauge theories

better mathematical and practical understanding

gauge cooling for SU(N ) gauge theories

work in progress forSU(3) + θ-termheavy dense QCDfull QCD

many things to do!

XQCD 14, June 2014 – p. 29

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