QCD Thermodynamics using the complex Langevin equation

Dénes SextyWuppertal University, Jülich JSC

SIGN Workshop 2019 Odense, Denmark 4th of September, 2019

1. Introduction: sign problem in QCD and CLE2. CLE and boundary terms 3. Transition Line 4. Equation of state & improved action

In collaboration withErhard Seiler, Nucu Stamatescu, Manuel Scherzer

S=−14

Tr Fμ νFμ ν+∑1

6ψ̄f (iγ

μ Dμ+mf )ψf

From action to phenomenology

Confinement mechanism?Mass of hadrons?Scattering cross sections?Phase transition to Quark-gluon plasma? Critical point at nonzero density? Equation of state?Compressibility of quark matter? (in neutron stars)Exotic phases: Color superconducting phases? Quarkyonic phase?QCD in magnetic fields? …. and so on

7 parameters

How?Perturbation theoryKinetic theoryEffective models (NJL, Polyakov-NJL, SU(3) spin model, … )Functional methods (FRG, 2PI, Dyson-Schwinger eq.)Lattice (mostly with importance sampling)

S=−14

Tr Fμ νFμ ν+∑1

6ψ̄f (iγ

μ Dμ+mf )ψf

From action to phenomenology

7 parameters

How?

Lattice (not with importance sampling)

shape of deconfinement transition line

pressure, energy density at nonzero chemical potentials

Non-zero chemical potential

Euclidean SU(3) gauge theory with fermions:

For nonzero chemical potential, the fermion determinant is complex

Sign problem Naive Monte-Carlo breaks down

QCD sign problem

Z=∫DUexp(−SE [U ])det (M(U))

Importance sampling is possiblefor det (M (U ))>0

det M (U ,−μ ∗ )=(detM (U ,μ))∗

Hadron masses,EoS at , ...μ=0

In QCD direct simulation only possible at

μTaylor extrapolation, Reweighting, continuation from imaginary , canonical ens. all break down around

μ=0

μq

T≈1−1.5

μB

T≈3−4.5

Around the transition temperature Breakdown at μq≈150−200 MeV μB≈450−600 MeV

Results on

NT=4,N F=4,ma=0.05

Agreement only at μ/T<1

using Imaginary mu, Reweighting, Canonical ensemble

Stochastic process for x:

Gaussian noise

Averages are calculated along the trajectories:

⟨O ⟩=limT→∞

1T∫0

T

O(x (τ))d τ=∫e−S (x)O(x)dx

∫e−S (x)dx

Langevin Equation (aka. stochatic quantisation)

⟨η(τ)⟩=0

Given an action S (x)

⟨η(τ)η(τ ' )⟩=δ(τ−τ ')d xd

=−∂S∂ x

Random walk in configuration space

Numerically, results are extrapolated to Δ τ→0

Stochastic process for x:

d xd

=−∂S∂ x

Gaussian noise

Averages are calculated along the trajectories:

⟨O ⟩=limT→∞

1T∫0

T

O(x (τ))d τ=∫e−S (x)O(x)dx

∫e−S (x)dx

Complex Langevin Equation

⟨η(τ)⟩=0

Given an action S (x)

⟨η(τ)η(τ ' )⟩=δ(τ−τ ')

The field is complexified

real scalar complex scalar

link variables: SU(N) SL(N,C)compact non-compact

det (U )=1, U + ≠ U−1

d xd

=−∂S∂ x

Analytically continued observables

1Z∫ P comp( x )O ( x )dx=

1Z∫ P real ( x , y )O ( x+iy )dx dy

⟨ x2⟩real → ⟨ x2− y2⟩complexified

S [x ]=σ x2+i λ x

Gaussian Example

σ=1+i λ=20

dd τ

(x+i y )=−2σ(x+iy)−iλ+η

CLE

P (x , y )=e−a(x−x0)2−b( y− y0)

2−c (x−x0)( y− y0)

Gaussian distribution around critical point

∂ S (z)∂ z ]

z0

=0

Measure on real axis

Proof of convergence

If there is fast decay

and a holomorphic action

[Aarts, Seiler, Stamatescu (2009) Aarts, James, Seiler, Stamatescu (2011)]

then CLE converges to the correct result

P (x , y )→0 as x , y→∞

S (x)

Loophole: decay not fast enough

∫ dxρ(x )O(x) = ∫ dx dy P(x , y )O(x+iy)

What we want What we get with CLE

Using analyticity and partial integrations boundary terms can be nonzero!

Usually one notices

Distributions with large fluctuations

Power-law decay in histograms

Gauge cooling

complexified distribution with slow decay convergence to wrong results

Minimize unitarity norm∑iTr (U iU i

+ −1)Distance from SU(N)

Keep the system from trying to explore the complexified gauge degrees of freedom

[Seiler, Sexty, Stamatescu (2012)]

Dynamical steps are interspersed with several gauge cooling steps

Empirical observation: Cooling is effective for

β>βmin

but remember, β→∞in cont. limit

a<amax≈0.1−0.2 fm

Can we do more? Dynamical Stabilization soft cutoff in imaginary directions

[Attanasio, Jäger (2018)]

Boundary terms

∫dxρ(x)O (x)=F (t , t) = F (t ,0)=∫ dx dy P (x , y)O (x+iy)

What we want What we get with CLE

[Scherzer, Seiler, Sexty, Stamatescu (2018)]

[See talk from Nucu Stamatescu]

HDQCD Full QCD

More expensive to collect statisticsNoisy drift termsAt low beta simulation instable

KaF=Tr (M−1 DaM )=ηM−1DaM η

?

Mapping out the phase transition line

Follow the phase transition line starting from μ=0

Using Wilson fermions

[Scherzer, Sexty, Stamatescu in prep.]

Earlier study: Mapping the phase diagram of HDQCD[Aarts, Attanasio, Jäger, Sexty (2016)]

Hopping parameter expansion of the fermion determinantSpatial fermionic hoppings are droppedFull gauge action

Polyakov loop Transition to deconfined state

What is reachable?

How to scan the phase diag?

Temperature given by T=1

aN t

a controlled by β too small β (too high a ) problematic with CLE

→ scan with N t at fixed β

We need:

no polesstable simulations (no runaways)no boundary termsfast simulations

Long runs with CLE

Unitarity norm has a tendency to grow slowly (even with gauge cooling)

Runs are cut if it reaches

Thermalization usually fast – might be problematic close to critical point or at low T

∼0.1

Getting closer to continuum limit

Test with Wilson fermionsIncrease by 0.1 – reduces lattice spacing by 30% change everything else to stay on LCP

β

behavior of Unitarity norm improves

How to detect the phase transition?

Physics: deconfinement transition chiral symmetry breaking

Polyakov loopMeasures free energy of quarks

Chiral condensate

heavy quarks → large explicit breaking

Needs renormalizationNeeds renormalization

3rd order Binder cumulant of the Polyakov loop

Labs=√LLinvL=

1V ∑ Px , y , z L inv=

1V ∑ Px , y , z

−1

B3=⟨(Labs−⟨Labs⟩)

3⟩

⟨(Labs−⟨Labs⟩)2⟩3 /2

Measures left-right asymmetryNo renormalization needed

Zero crossing defines phase transition

High temp:

⟨Labs⟩>0 symmetric distribution

Low temp:

⟨Labs⟩ is small

Labs>0 → asymmetric distribution

Linear fit to find zero crossing

Using β=5.9, κ=0.15, N F=2, N S=12,16

a=0.065(1) fm, mπ≈1.3 GeV

Phase transition line

T c(μB)

T c(μB=0)=1−κ2 (

μB

T C(0) )2

parametrization κ2≈0.0008

Large finite size effects, heavy pion

N s=16N s=12

Looks quadratic for μ/T≫1

How to detect the phase transition?

“shift method” [see also: Endrődi, Fodor, Katz, Szabó (2011)]

Take ϕ(T ,μ) monotonic int T around T c

Determine T c at μ=0 → ϕ(T c ,μ=0)=C

Define T c (μ) as ϕ(T c(μ),μ)=C

e.g. B3, chiral condensate, baryon number susceptibility

Critial point at μ4

Works well for small μ

ϕ(T ,μ)=B3 unsubstracted=⟨Labs

3 ⟩

⟨Labs2 ⟩3 /2

ϕ(T c(μ) ,μ)=C=1.15

T c(μB)

T c(μB=0)=1−κ2 (

μB

T C(0) )2

parametrization

κ2≈0.001

Consistent with the first method

Open questionsPossible for lighter quarks?Finite size scaling?Where is the upper right corner of Columbia plot? Critical point nearby?

Pressure of the QCD Plasma at non-zero density

p

T 4 =ln Z

V T 3 Derivatives of the pressure are directly measureable Integrate from T=0

Other strategies:

Measure the Stress-momentum tensor using gradient flow

Shifted boundary conditions

Non-equilibrium quench

First integrate along the temperature axis, then explore μ>0

[Suzuki, Makino (2013-)]

[Giusti, Pepe, Meyer (2011-)]

[Caselle, Nada, Panero (2018)]

Taylor expansion [Allton et. al. (2002-), … ]

Simulating at imaginary to calculate susceptibilities [Bud.-Wupp. Group (2018)]

μ

Pressure of the QCD Plasma at non-zero density

Δ ( pT 4 )=∑n>0,evencn(T ) (

μ

T )n

If we want to stay at μ=0

Δ ( pT 4 )= p

T 4 (μ=μq)−p

T 4 (μ=0)

c4=1

241

N s3N T

∂4 ln Z

∂μ4

c2=12

NT

N s3

∂2 ln Z

∂μ2

∂2 ln Z

∂μ2 =N F

2⟨T 1

2⟩+N F ⟨T 2⟩

∂4 ln Z

∂μ4 =−3 (⟨T 2⟩+⟨T 1

2⟩ )

2+3 ⟨T 2

2⟩+⟨T 4⟩

+⟨T 14⟩+4 ⟨T 3T 1⟩+6 ⟨T 1

2T 2⟩

T 1/N F=Tr (M−1∂μ M )

T i+1=∂μT i

T 2/N F=Tr (M−1∂μ

2 M )−Tr ((M−1∂μM )

2 )T 3/N F=Tr (M−1∂μ

3 M )−3 Tr(M−1 ∂μM M−1 ∂μ2 M )

+2 Tr ((M−1 ∂μ M )3 )T 4 /N F=Tr (M−1

∂μ2 M )−4 Tr (M−1

∂μM M−1∂μ

3 M )

−3 Tr (M−1 ∂μ2 M M−1∂μ

2 M )−6 Tr ((M−1 ∂μ M )4 )+12 Tr ((M−1

∂μM )2M−1

∂μ2 M )

Measuring the coefficients of the Taylor expansion

Δ ( pT 4 )= p

T 4 (μ=μq)−p

T 4 (μ=0)=1

V T 3 ( lnZ (μ)−ln Z (0))

If we can simulate at μ>0

ln Z (μ)−ln Z (0)=∫0

μ

dμ∂ ln Z (μ)

∂μ=∫0

μ

dμΩn(μ)

Using CLE it’s enough to measure the density – much cheaper

Pressure of the QCD Plasma using CLE[Sexty (2019)]

n(μ)=⟨Tr(M−1(μ)∂μM (μ))⟩

Integration performed numericallyJackknife error estimates

Pressure calculated with CLE

Improved actions for lattice QCD

Carrying out continuum extrapolation a→0

Fitting some observable

Simulate at multiple lattice spacings

O(a)=O0+O1a+O2a2+...

Change action such that is eliminatedO1

Gauge improvement

Include larger loops in action

Symanzik action: S=−β ( 53 ∑ ReTr −

112 ∑ Re Tr )

Analyticity must be preserved: 2ReTrU=TrU+TrU + → TrU+TrU−1

Straightforwardly implemented in CLE

Improved gauge actions

Gauge improvement

Include larger loops in action

Symanzik action: S=−β ( 53 ∑ ReTr −

112 ∑ Re Tr )

Analyticity must be preserved: 2ReTrU=TrU+TrU + → TrU+TrU−1

Straightforwardly implemented in CLE

plaquette U=U 0U 1U 2U 3

D0(TrU+TrU−1)=iTr (λaU 0U 1U 2U 3)−iTr (U 3−1U 2

−1U 1−1U 0

−1 λa)

real if U∈SU(3)

Improved fermion actions

Wilson fermions: clover improvement adds a clover term

Staggered fermions: naik or p4 take into account 3-link terms

Changeing the Dirac operator

Fat links

Smear the gauge fields inside the Dirac operator

V μ=(1−α)Uμ+α∑ staplesAPE, HYP

Stout

U 'μ=ProjSU(3)V μ

U 'μ=eiQμUμ Qμ=ρ∑ staplesUμ−1

essentially one step of gradient flow with stepsize ρ

not analytic

( ∼Langevin eq. with no noise )

Stout smearing

U 'μ=eiQμUμ Qμ=ρ∑ staplesUμ−1

Usually multiple steps: U→U (1)→U (2)

→ ...→U (n)

For the Langevin eq. we need drift terms:

∂ Seff

∂U=

∂ Seff

∂U (n)

∂U (n)

∂U (n−1)...

∂U (1)

∂U

Replace gauge fields in Dirac matrix det M (U )→det M (U (n))

Calculated by “going backwards”

∂ Seff

∂U with Seff=Sg+ ln det M (U (n))

One iteration:∂U ' i∂U j

=∂ eiQ

∂U j

U i+eiQ δij

local terms + nonlocal terms from staples

Stout smearing and complex Langevin

Adjungate is replaced with inverse for links

Q + is not replaced with Q−1(because its a sum)

Q is no longer hermitian ∂ eiQ

∂UCalculation of becomes trickier

Benchmarking with HMC at μ=0

[Sexty (2019)]

a(β=3.6)=0.12 fm a(β=3.9)=0.064 fm

U 'μ=ei(Qμ−Qμ+)tlUμ Qμ=ρ∑ staplesUμ

−1

For CLE: Qμ=ρ∑ staplesUμ

−1

Qμi=ρUμ∑ staples−1

U 'μ=ei(Qμ−Qμi)tlUμ

small β still a problem

What happens with the configurations?

Real part of gauge fields decay

Unitarity norm slightly rises

Langevin eq with no noise large steps

runaway trajcetory problem if UN≿0.1

∂ Seff

∂U=

∂ Seff

∂U (n)

∂U (n)

∂U (n−1)...

∂U (1)

∂U

∂ Seff

∂U (n)=F0 ,

∂ Seff

∂U (n)

∂U (n)

∂U (n−1)=F1 , ...

∂ Seff

∂U=Fn

What happens with the drift terms?

Average drift term is smaller

Long tail

More prone to runaways

Smaller stepsize needed

Pressure with improved action

is measurable with this action at high T (with O(500) configs.)

C4

naiv action

Pressure with improved actionSymanzik gauge action stout smeared staggered fermions

Good agreementCLE calculation is much cheaper

Direct look at the density

n

T 2μ

=Tmu

∂ ( p /T 4 )∂ (μ/T )

c2 → constant

c4 → linear in μ2

c6 → quadratic in μ2

At small μ : n is small → large relative errors

Before doing integration for the pressure

Thermodynamics of the QCD Plasma using CLE

Energy density accesible via trace anomaly

Δ ( ϵ−3 p

T 4 )=−1

V T 3 a ( ∂β

∂ a )LCP [ 12 μ

2 ( ∂3 ln Z

∂β∂μ2 + ( ∂m∂β )

LCP

∂3 ln Z

∂m∂μ2 )+... ]

LCP=Line of constant physics

a lattice spacing changedm in GeV should stay constant

ma has to be changed

LCP: determined at zero temperature,a(β), ma(β) μ=0

ϵ−3 p

T 4=−

1

V T 3a ( ∂β

∂ a )LCP

[ ∂ ln Z∂β

+ ( ∂m∂β )LCP

∂ ln Z∂m ]

nonzero chemical potential with Taylor expansion

∂ ⟨O ⟩∂β

=−⟨O ∂S∂β ⟩+ ⟨O ⟩ ⟨ ∂ S∂β ⟩

∂ ⟨O ⟩∂m

= ⟨ ∂O∂m ⟩+ ⟨O χ ⟩− ⟨O ⟩ ⟨ χ ⟩

Correlators with gauge action and chiral condensate

Thermodynamics of the QCD Plasma using CLE

Energy density accesible via trace anomaly

ϵ−3 p

T 4=−

1

V T 3a ( ∂β

∂ a )LCP

[ ∂ ln Z∂β

+ ( ∂m∂β )LCP

∂ ln Z∂m ]

Directly accessible at in a CLE simulationμ>0

∂ ln Z∂β

=−Sg

∂ ln Z∂m

=N F

4⟨Tr (M−1

)⟩=VT

χ

Gauge action

Chiral condensate

Further quantities of interest:

χq (μ)=∂

2 ln Z

∂μ2 |

μ=0

+12

μ2 ∂

4 ln Z

∂μ4 |

μ=0

+ ...

Directly accessible in CLE

Need higher derivatives for Taylor exp.

n(μ) , χq(μ)=∂2 ln Z

∂μ2

Comparison of gauge action and chiral condensation terms

Chiral condensation term has smaller fluctuation in both setups

ϵ−3 p

T 4 =−1

V T 3 a ( ∂β

∂a )LCP

[ ∂ ln Z∂β

+( ∂m∂β )LCP

∂ lnZ∂m ]

μ dependence:

Energy density

Dependece significantly milder than for the pressure

Δ ( ϵ−3 p

T 4 )=ϵ(μ)−3 p(μ)

T 4 −ϵ(μ=0)−3 p(μ=0)

T 4

from T=0,μ=0: −0.28 −0.04

Summary

Ongoing effort for full QCD to get physical results Mapping out phase transition line Wilson fermions with heavy pion High chemical potentials possible Large finite size effects

Calculating the pressure & energy dens. High chemical potentials possible Observables with small noise (compared to expansion) Only deconfined phase so far

Using improved actions stout smeared fermions Symanzik gauge action