Felipe Attanasio - INDICO-FNAL (Indico) · 2018-07-23 · Complex Langevin Equation CLE: Complexiﬁcation I Complexiﬁcation [Aarts, Stamatescu, hep-lat/0807.1597] Allow gauge links

Stabilising complex Langevin simulations

Felipe Attanasio

In collaboration with B. Jager

Felipe Attanasio Stabilising complex Langevin simulations 1 / 24

Complex Langevin Equation

CLE: Motivation

Description of QCD under different thermodynamical conditions

Reweighting, Taylor expansion

µ

T

Hadrons

Quark-GluonPlasma

Nuclear matter

ColourSuperconductor?

Critical point?

Experimental investigationsin progress (LHC, RHIC)and planned (FAIR)Perturbation theory onlyapplicable at hightemperature/density(asymptotic freedom)Full exploration requiresnon-perturbative (e.g.lattice) methods



CLE: Motivation

Description of QCD under different thermodynamical conditions

Sign problem: chemical potential in Euclidean path integral ⇒ complexactionExpectation values ⇒ precise cancellations of oscillating quantitiesIn QCD: fermion determinant

[detM(U, µ)]∗ = detM(U,−µ∗)

is complex for real chemical potential µTraditional Monte-Carlo methods unreliable for severe sign problem



CLE: Stochastic quantization

On the lattice [Damgaard and Huffel, Physics Reports]

Evolve gauge links according to the Langevin equation

Uxµ(θ + ε) = exp [Xxµ]Uxµ(θ) ,

with the Langevin driftXxµ = iλa(−εDa

xµS [U(θ)] +√ε ηaxµ(θ)) ,

λa are the Gell-Mann matrices, ε is the stepsize, ηaxµ are white noise fieldssatisfying

〈ηaxµ〉 = 0 , 〈ηaxµηbyν〉 = 2δabδxyδµν ,

S is the QCD action and Daxµ is defined as

Daxµf(U) = ∂

∂αf(eiαλ

a

Uxµ)∣∣∣∣α=0



CLE: Complexification I

Complexification [Aarts, Stamatescu, hep-lat/0807.1597]

Allow gauge links to be non-unitary: SU(3) 3 Uxµ → Uxµ ∈ SL(3,C)

Use U−1xµ instead of U†xµ as

keeps the action/observables holomorphic;coincide on SU(3), but on SL(3,C) it is U−1 that represents thebackwards-pointing link.

Circumvents the sign problem by doubling the degrees of freedom

similar to∫ ∞−∞

dx e−x2→

√∫r drdθ e−r2



CLE: Complexification II – Gauge cooling

Gauge cooling [Seiler, Sexty, Stamatescu, hep-lat/1211.3709]

SL(3,C) is not compact ⇒ gauge links can get arbitrarily far from SU(3)

During simulations monitor the distance from the unitary manifold with

d = 1N3sNτ

∑x,µ

Tr[UxµU

†xµ − 1

]2 ≥ 0

Use gauge transformations to decrease d

Uxµ → ΛxUxµΛ−1x+µ

necessary, but not always sufficient



CLE: Complexification II – Gauge cooling

Gauge cooling (mild sign problem)

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

50 100 150 200 250 300 350 400 450 500

Pol

yako

vlo

op

θ

Gauge coolingReweighting

0

0.2

0.4

0.6

0.8

1

1.2

50 100 150 200 250 300 350 400 450 500U

nita

rity

norm

θ

Gauge cooling

Left: Langevin time history of Polyakov loopRight: Langevin time history of unitarity norm



CLE: Complexification III – Dynamic stabilisation

Dynamic stabilisation [Attanasio, Jager, hep-lat/1607.05642]

New term in the drift to reduce the non-unitarity of Ux,ν

Xxν = iλa(−εDa

x,νS − εαDSMax +√ε ηax,ν

).

with αDS being a control coefficient

Max : constructed to be irrelevant in the continuum limit

Max only depends on Ux,νU†x,ν (non-unitary part)




Dynamic stabilisation (mild sign problem)

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

50 100 150 200 250 300 350 400 450 500

Pol

yako

vlo

op

θ

Gauge coolingDyn. stab.

Reweighting

1e-08

1e-07

1e-06

1e-05

1e-04

1e-03

1e-02

1e-01

1e+00

1e+01

50 100 150 200 250 300 350 400 450 500U

nita

rity

norm

θ

Gauge coolingDyn. stab.

Left: Langevin time history of Polyakov loopRight: Langevin time history of unitarity norm (notice log scale!)




Dynamic stabilisation

1e-07

1e-06

1e-05

1e-04

1e-03

1e-02

1e-01

1e+00

1e-04 1e+00 1e+04 1e+08 1e+12 1e+16

A

v

e

r

a

g

e

u

n

i

t

a

r

i

t

y

n

o

r

m

αDS

103 × 4, µ = 0.7083 × 20, µ = 2.45

Unitarity norm as a function of αDS for HDQCD at β = 5.8 and κ = 0.04



Observables considered

“Real part” of the Polyakov loop

P s = 12(〈P 〉+ 〈P−1〉

),

Chiral condensate (for dynamical quarks)

〈ψψ〉 = T

V

∂ lnZ∂m

Quark density〈n〉 = T

V

∂ lnZ∂µ


HDQCD

Heavy-dense QCD

Heavy-dense approximation [Aarts, Stamatescu, hep-lat/0807.1597]

Heavy quarks → quarks evolve only in Euclidean time direction:

detM(U, µ) =∏~x

{det[1 + (2κeµ)Nτ P~x

]2det[1 +

(2κe−µ

)Nτ P−1~x

]2}

Polyakov loopP~x =

∏τ

U4(~x, τ)

Exhibits the sign problem: [detM(U, µ)]∗ = detM(U,−µ∗)Transition to higher densities (at T = 0 it happens at µ = µ∗c ≡ − ln(2κ))


HDQCD

Comparison of DS and GC

αDS scan of the Polyakov loop, with results from gauge cooling

0.195

0.200

0.205

0.210

0.215

0.220

-0.065

-0.060

-0.055

-0.050

-0.045

-0.040

100 105 1010 1015

Rea

lPol

yako

vlo

op

β = 5.8, µ/µ0c = 0.96, κ = 0.04, V = 83 × 20

Imag

.Pol

yako

vlo

op

αDS

GC d < 0.03GC d > 0.03

Wide region of αDS where DS agrees with GC with low unitarity norm


HDQCD

Histograms I (HDQCD)

DS reduces only imaginary part of the Langevin drift

100

102

104

106

108

1010

1012

10−4 10−3 10−2

Cou

nt

ε|Im[Kax,µ]|

HDQCD, 83 × 20, β = 5.8, κ = 0.04, µ/µ0c = 0.96

αDS = 1.0αDS = 105

αDS = 107

αDS = 1010

100

102

104

106

108

1010

1012

10−4 10−3 10−2

Cou

nt

ε|Re[Kax,µ]|

HDQCD, 83 × 20, β = 5.8, κ = 0.04, µ/µ0c = 0.96

αDS = 1.0αDS = 105

αDS = 107

αDS = 1010

Histograms of the Langevin driftLeft: Imaginary part is clearly affected by larger αDSRight: For αDS large enough, the real part of the drift is essentially unchanged


HDQCD

Histograms II (HDQCD)

DS decreases with the lattice spacing

100

102

104

106

108

1010

10−5 10−4 10−3 10−2 10−1

Cou

nt

ε|Re[Kax,µ]|

HDQCD, 83 × 20, β = 5.8, κ = 0.04, µ/µ0c = 0.96

β = 5.8β = 6.0β = 6.2

10−2

100

102

104

106

108

1010

1012

10−4 10−3 10−2

Cou

nt

αDSε|Max |

HDQCD, 83 × 20, β = 5.8, κ = 0.04, µ/µ0c = 0.96

β = 5.8β = 6.0β = 6.2

Histograms for different values of βLeft: Real part of the drift changes very little (a is changing)Right: DS term decreases with a (higher β)


HDQCD

Deconfinement in HDQCD

Good agreement with reweighting across the deconfinement region for fixed µ

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

5.4 5.6 5.8 6.0 6.2

Pol

yako

vlo

op

β

HDQCD 64, κ = 0.12, µ/µ0c = 0.6

RWDS

Polyakov loop as a function of the inverse coupling β for HDQCD


HDQCD

Deconfinement in HDQCD

Good agreement with reweighting – even when GC converges to the wrong limit

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

0.62

5.4 5.6 5.8 6.0 6.2

Spat

ialp

laqu

ette

β

HDQCD 64, κ = 0.12, µ/µ0c = 0.6

ReweightingDyn. stab. αDS = 1000Gauge cooling, d < 0.03

Gauge cooling

Spatial plaquette as a function of the inverse coupling β for HDQCD


Staggered quarks

Dynamical fermions

Staggered quarks

The Langevin drift for Nf flavours of staggered quarks

Dax,νSF ≡ Da

x,ν ln detM(U, µ)

= NF4 Tr

[M−1(U, µ)Da

x,νM(U, µ)]

Inversion is done with conjugate gradient method

Trace is evaluated by bilinear noise scheme – introduces imaginarycomponent even for µ = 0!


Staggered quarks

Staggered quarks (β = 5.6, m = 0.025, NF = 4)

Comparison between CLE + DS runs and HMC (results by P. de Forcrand)

0.1194

0.1196

0.1198

0.1200

0.1202

0.1204

0.1206

0.1208

10−4 10−2 100 102 104 106 108

〈ψψ〉 µ

=0

αDS

HMCDS

Grey band represents results from HMC simulationsαDS scan of the chiral condensate for a volume of 64


Staggered quarks



0.13120.13140.13160.13180.13200.13220.13240.13260.13280.13300.13320.1334

0 20 40 60 80 100

〈ψψ〉 µ

=0

〈ε〉/10−6

HMCDS

Linear fit

Grey band represents results from HMC simulationsLangevin step size extrapolation of the chiral condensate for a volume of 84


Staggered quarks



0.58100.58110.58120.58130.58140.58150.58160.58170.58180.58190.58200.5821

0 20 40 60 80 100 120 140

Pla

quet

te

〈ǫ〉/10−6

HMCDS

Linear fit

Grey band represents results from HMC simulationsLangevin step size extrapolation of the plaquette for a volume of 124


Staggered quarks



Plaquette ψψVolume HMC Langevin HMC Langevin

64 0.58246(8) 0.582452(4) 0.1203(3) 0.1204(2)84 0.58219(4) 0.582196(1) 0.1316(3) 0.1319(2)104 0.58200(5) 0.58201(4) 0.1372(3) 0.1370(6)124 0.58196(6) 0.58195(2) 0.1414(4) 0.1409(3)

Expectation values for the plaquette and chiral condensate for full QCDLangevin results have been obtained after extrapolation to zero step size


Staggered quarks


Preliminary results at µ > 0 (not extrapolated to zero step size)

−2

0

2

4

6

8

10

12

14

0 0.5 1 1.5 2

〈n〉/T

3

µ/T

Nτ = 2Nτ = 4Nτ = 6Nτ = 8

−2

0

2

4

6

8

10

12

0 0.5 1 1.5 2

∆p/T

4

µ/T

Nτ = 2Nτ = 4Nτ = 6Nτ = 8

VeryPreli

minary

VeryPreli

minary

Vertical lines indicate position of critical chemical potential for each temperatureLeft: Density as a function of chemical potentialRight: Pressure as a function of chemical potential


Conclusion

Summary and Outlook

SummaryDynamic Stabilisation

improves convergence of complex Langevin simulationsallows for long runs

HDQCD results with DS verified against reweighting across thedeconfinement transition

Very good agreement with HMC for full QCD at µ = 0

OutlookDetermine the QCD phase diagram, with particular attention phaseboundaries and characteristics


Documents

Felipe Attanasio - INDICO-FNAL (Indico) · 2018-07-23 · Complex Langevin Equation CLE: Complexiﬁcation I Complexiﬁcation [Aarts, Stamatescu, hep-lat/0807.1597] Allow gauge links