NOVEL LATTICE SIMULATIONS FOR TRANSPORT COEFFICIENTS IN GAUGE - - PowerPoint PPT Presentation

NOVEL LATTICE SIMULATIONS FOR TRANSPORT COEFFICIENTS IN GAUGE THEORIES

Felix Ziegler

in collaboration with Jan M. Pawlowski and Alexander Rothkopf Heidelberg University Pawlowski, Rothkopf, arXiv:1610.09531 [hep-lat] Pawlowski, Rothkopf, Ziegler, in preparation Seminar – Theory of Hadronic Matter under Extreme Conditions Bogoliubov Laboratory of Theoretical Physics Dubna (Russia) - February 14, 2018

ISOQUANT SFB1225

Overview

Introduction

• Physics motivation of real-time dynamics from lattice QCD
• Challenges in the reconstruction of spectral functions

Novel simulation approach for thermal fields on the lattice with non-compact Euclidean time

• Setup and scalar fields
• Gauge fields
• Convergence to the Matsubara results
• Energy-momentum tensor correlation functions
• Spectral reconstructions

Summary and outlook

1

INTRODUCTION

Physics motivation

Thermal physics of hot strongly interacting matter produced in heavy ion collisions

• Transport phenomena
• In-medium modification of heavy bound states

Transport coefficients are real-time quantities related to the energy-momentum tensor (EMT) correlation function Example: shear viscosity η lim

ω→0

1 20 ρ(ω, 0) ω , ρ(ω, p)

• d4x

(2π)4 e−iωx0+i

p· x [T12(x), T12(0)] .

⇒ need spectral function ρ(ω, p)

3

Lattice QCD at finite temperature

Gauge fields on links Uµ(x) exp(igaµAa

µ(x) Ta)

Dynamical fermions with realistic masses finite extent in imaginary time 1/T β Nτaτ

β Nτ aτ

τ x

O(U) 1

Z

• DU O(U) exp(−SQCD

E

[U])

P(Uk) e−SQCD

E

[Uk] ⇒ O(U) ≈

1 Ncf

Ncf

• k1

O(Uk)

4

Reconstruction of spectral functions and its challenges

Back to real-time EMT-correlator: ρ(ω, p)

• d4x

(2π)4 e−iωx0+i

p· x [T12(x), T12(0)]

Spectral function connects physical real-time observable with Euclidean time simulation D(τ) ∝

• d3x T12(τ,

x)T12(0, 0)

• dµ cosh[µ(τ − β/2)]

sinh[µ β/2] ρ(µ)

For the reconstruction technique used in the following see Y.Burnier, Alexander Rothkopf, Phys.Rev.Lett. 111 (2013) 182003

5

Reconstruction of spectral functions and its challenges

Two main conceptual problems of standard spectral reconstructions Problem 1: D(τ)

∞

dµ cosh[µ(τ − β/2)] sinh[µ β/2] ρ(µ) Extraction from imaginary time correlator ill-posed exponentially hard inversion problem. → Go to imaginary frequencies and use Källén-Lehman spectral representation D(ωn)

∞

dµ 2 µ ω2

n + µ2 ρ(µ) 6

Reconstruction of spectral functions and its challenges

Two main conceptual problems of standard spectral reconstructions Problem 2: Increasing the number of points along Euclidean time axis does not help!

Most relevant regime for the inverse problem: ω∼T=1/β

Standard lattice simulations only access Matsubara frequencies ωn 2πTn, n ∈ Z .

7

SETUP OF A NOVEL COMPUTATIONAL APPROACH

Thermal field theory on the Schwinger-Keldysh contour

Thermal scalar field theory as a real-time initial value problem with Z Tr ρ(0) e−β H Z

• d[ϕ+

0]d[ϕ− 0 ]ϕ+ 0 |ρ(0)|ϕ− 0 ϕ− 0 |ϕ+ ρ(0)=e-βH initial conditions t

9

Thermal field theory on the Schwinger-Keldysh contour

Thermal scalar field theory as a real-time initial value problem with Z Tr ρ(0) e−β H Z

• d[ϕ+

0]d[ϕ− 0 ]ϕ+ 0 |ρ(0)|ϕ− 0 ϕ− 0 |ϕ+

• [dϕt]ϕ−

0 |eiHt|ϕtϕt|e−iHt|ϕ+ ρ(0)=e-βH φ+(t) φ-(t) quantum dynamics initial conditions t

10

Thermal field theory on the Schwinger-Keldysh contour

Thermal scalar field theory as a real-time initial value problem with Z Tr ρ(0) e−β H Z

• d[ϕ+

0]d[ϕ− 0 ]ϕ+ 0 |e−βH|ϕ− 0 ϕ− 0 |ϕ+

ϕ−

ϕ+

DϕeiSM[ϕ+]−iSM[ϕ−]

ρ(0)=e-βH φ+(t) φ-(t)

quantum dynamics initial conditions

t 11

Thermal field theory on the Schwinger-Keldysh contour

Thermal scalar field theory as a real-time initial value problem with Z Tr ρ(0) e−β H Z

• ϕE(0)ϕE(β)

DϕEe−SE[ϕE] ϕ−(t0)ϕE(β)

ϕ+(t0)ϕE(0)

DϕeiSM[ϕ+]−iSM[ϕ−]

φ+(t) φ-(t)

quantum dynamics initial conditions

t φE(τ) τ

β=1/T

τ0=t0

So far, we have only rewritten the partition function. The imaginary time is a mathematical tool to sample initial conditions ϕ+(t0) and ϕ−(t0).

12

Path contribution

Thermal equilibrium ⇒ G++ ≡ ϕ+ϕ+ correlator sufficient to compute spectral function ρ, see e.g. Laine and Vuorinen, Basics of Thermal Field Theory, Springer 2016 G++(p0, p)

• dq0

2π ρ(q0, p) q0 − p0 + iε − n(p0)ρ(p0, p)

φ+(t) φ-(t)

quantum dynamics initial conditions

t φE(τ) τ

β=1/T

τ0=t0

Using time translation invariance we can set t0 → −∞. Introducing ε > 0 and e−iHt → e−iHt(1+iε) correlations between any finite t on forward branch and endpoint become exponentially damped.

13

Analytic continuation and general imaginary frequencies

From no on, we focus on the forward ϕ+ path. Idea: Cut open real-time path at t ∞ and rotate path to (additional) non-compact imaginary time axis. Z

• ϕE(0)ϕE(β)

DϕE e−SE[ϕE] ϕ−(t0,

x)ϕE(β) ϕ+(t0, x)ϕE(0)

Dϕ ei SM[ϕ+]−i SM[ϕ−]

φ+(t) φ-(t) quantum dynamics initial conditions t φE(τ) τ β=1/T τ0=t0

14

Analytic continuation and general imaginary frequencies

From no on, we focus on the forward ϕ+ path. Idea: Cut open real-time path at t ∞ and rotate path to (additional) non-compact imaginary time axis. Z

• ϕE(0)ϕE(β)

DϕE e−SE[ϕE] ϕ+(∞)

ϕ+(τ0)ϕE(0)

Dϕ+e−SE[ϕ+] ϕ−(τ0)ϕE(β)

ϕ−(∞)

Dϕ−e−SE[ϕ−] Simulation recipe Sample init. conditions ϕE(¯ τ0) ϕ+(τ0) from e−SE[ϕE] on compact Euclidean time lattice, ¯ τ ∈ [0, β]. Concurrently sample ϕ+(τ) from e−SE[ϕ+] with τ ∈ [0, ∞).

φ+(t) φ-(t) quantum dynamics initial conditions t φE(τ) τ β=1/T τ0=τ0=t0 φ+(τ) τ

15

Simulating scalar fields

SE

• dτ

1 2(∂τϕE)2 + 1 2 m2ϕ2

E

• S0

E

+ λ 4! ϕ4

E

• Sint

E

• ∂t5ϕ+(ωl) −

δS0

E

δϕ+(ωl) − δSint

E

δϕ+(τj) δϕ+(τj) ϕ+(ωl) + η(ωl) Use Stochastic Quantization and sample ϕE and ϕ+ concurrently from Langevin equations Imaginary frequency update in Fourier space → kinetic term diagonal and improved convergence

∂t5ϕE(¯ τk) − δSE δϕE(¯ τk) + η(¯ τk)

Temperature in ϕE via compact temporal path Temperature in ϕ+ via initial condition ϕ+(t0)

β=1/T

τ τ

τ0=τ0=t0

t μ

Fourier

𝜒E(τ)

analytic continuation initial ¡conditions quantum dynamics

16

NUMERICAL RESULTS FOR SCALAR FIELD THEORIES

0+1 dimensional real scalar field

Two-point correlation function

0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.2 0.4 0.6 0.8 1

λ=24

GE(τ ‾ )=<φE(τ ‾ )φE(0)> τ ‾ /m

QM SQ Nτ=128 Nτ=64 Nτ=32 Nτ=16

• 0.004
• 0.002

0.002 0.004 0.3 0.6 0.9

τ ‾ /m ΔG/GQM vs. SQ

Figure: QM (an-)harmonic oscillator

• vs. stoch. quantization result on the

compact Euclidean time lattice

0.1 1 10 2 4 6 8 10 12 14 Free x2

λ=0 λ=24

G++(ω)=<φ+(ω)φ+(-ω)> ω/m

NMB

ω =16

Nω=512 Nω=256 Nω=128 Nω=64 Nω=32 Nω=16 NMB

ω =16

Nω=512 Nω=256 Nω=128 Nω=64 Nω=32 Nω=16

Figure: Free and interacting theory

from general frequency simulations

Pawlowski, Rothkopf, arXiv:1610.09531 [hep-lat]

18

0+1 dimensional real scalar field

Two-point correlation function

0.1 1 10 2 4 6 8 10 12 14 Free x2

λ=0 λ=24

G++(ω)=<φ+(ω)φ+(-ω)> ω/m

NMB

ω =16

Nω=512 Nω=256 Nω=128 Nω=64 Nω=32 Nω=16 NMB

ω =16

Nω=512 Nω=256 Nω=128 Nω=64 Nω=32 Nω=16 3.2 3.4 3.6 3.8 4 4.2 4.4 0.2 0.4 0.6 0.8 1

G++(ω) ω/m

• 0.005

0.005 0.01 0.015 0.02 0.025 0.03 0.035 10 20 30 40 50

(Gimag-Gmats)/Gmats ω/m

NFree

ω =16

Nω=16

Convergence properties of the correlator

19

0+1 dimensional real scalar field

Spectral functions

101 102 103

• 0.05

0.05 0.1 ρ(µ) (µ-µfree

0 )/µfree

QM tmaxm=8x105 Nτ =16 NMB

ω =16

Nω=16 Nω=32 Nω=128 Nω=512 10 100 1000

• 0.04 -0.02

0.02 0.04 0.06 0.08 0.1 ρ(µ) (µ-µint

0 )/µint

QM tmaxm=8x105 NMB

ω =16

Nω=16 Nω=32 Nω=128 Nω=512

General imaginary frequencies capture physical properties correctly. Information from standard compact Euclidean simulation insufficient.

20

0+1 dimensional real scalar field

Spectral reconstruction from a standard compact Euclidean time correlator GE(¯ τ) does not improve by simply increasing the number

• f temporal lattice points.

10 100 1000

• 0.04 -0.02

0.02 0.04 0.06 0.08 0.1 ρ(µ) (µ-µint

0 )/µint

QM tmaxm=8x105 Nτ =16 Nτ =32 Nτ =64 Nτ =128

21

3+1 dimensional complex scalar field

1 10

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 D(iω) [Lat] iω [Lat] standard MB Im

• agfrq. sim

PRELIMINARY

Figure: Field correlator

0.0054 0.0056 0.0058 0.006 0.0062 0.0064 0.0066 0.0068 0.5 1 1.5 2 < T12T12> (iω) [Lat] iω [Lat] standard MB Im

• agfrq. sim

PRELIMINARY Figure: EMT correlator

Pawlowski, Rothkopf, arXiv:1710.02672 [hep-lat]

22

GAUGE FIELDS

Simulating gauge fields

Wilson plaquette action SE[U] 2 g2

• x
• µ<ν

Re[1 − Uµν(x)]

• a4

2 g2

• x
• µ,ν

tr[Fµν(x)2] + O(a2) Update algorithm 1) Standard heatbath sweep 2) Therm. init. cond. at ¯ τ τ 0 3) Standard heatbath sweep at τ > 0 U(new)

x,µ

XV†, dP(X) dX exp 1 2 aβTr(X)

• ,

X, V A/a ∈ SU(2), a det(A) , A

• νµ

Ux+ ˆ

µ,νU† x+ ˆ ν,µU† x+ ˆ ν,ν

+U†

x+ ˆ µ− ˆ ν,νU† x− ˆ ν,µUx− ˆ ν,ν τ x

β= 1/T

τ τ τ

0=

τ

0=

t0 t μ

Fourier

φE(τ)

analytic continuation initial conditions quantum dynamics

24

Convergence towards the Matsubara results

200 400 600 800 1000 0.670 0.675 0.680 0.685 0.690 0.695 0.700 0.705 0.710 N_τ <P>

Plaquette expectation value

200 400 600 800 1000

• 5. × 10-4

0.001 0.005 0.010 N_τ (<P>M-<P>GF)/<P>M

Relative deviation Lattice sizes: 83 × 8 (Matsubara) 83 × Nτ (General frequencies) β 2.8

25

Wilson loop (confined phase)

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65

• 4
• 3
• 2
• 1

1 2 3 4 |W(R = 1, ω)| Imaginary frequency ω Matsubara Nτ = 8 Nτ = 16 Nτ = 32 Nτ = 64 Nτ = 128 1x10-6 1x10-5 0.0001 0.001 0.01 0.1 1 2 4 6 8 10 12 14 16 W(R = 1, τ) τ Matsubara Nτ = 8 Nτ = 16 Nτ = 32 Nτ = 64 Nτ = 128

Lattice sizes: 43 × 8 (Matsubara) 43 × Nτ (General frequencies) β 1.8 Ncf 8 × 105 configurations

26

Wilson loop (deconfined phase)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

• 3
• 2
• 1

1 2 3 4 |W(R = 1, ω)| Imaginary frequency ω Matsubara Nτ = 8 Nτ = 16 Nτ = 32 Nτ = 64 1x10-6 1x10-5 0.0001 0.001 0.01 0.1 1 8 16 24 32 40 48 56 64 W(R = 1, τ) τ Matsubara Nτ = 8 Nτ = 16 Nτ = 32 Nτ = 64

Lattice sizes: 43 × 8 (Matsubara) 43 × Nτ (General frequencies) β 3.0 Ncf 1.6 × 106 configurations

27

Energy momentum tensor on the lattice

Continuum formula Tµν(x) Fµσ(x)Fνσ(x) − 1

4 δµ,νFρσ(x)Fρσ(x)

Discretization of the field strength tensor on the lattice (clover) Fµν(x) −i 8a2g (Qµν(x) − Qνµ(x)) Qµν(x) Uµ,ν(x) + Uν,−µ(x) + U−µ,−ν(x) + U−ν,µ(x) For the shear viscosity η measure correlation function of time slices T12(0, 0)T12(τ, 0).

ν µ

n

see e.g. Gattringer, Lang, Quantum Chromodynamics on the Lattice, Springer 2010

28

EMT correlator (confined phase)

9x10-5 1x10-4 1.1x10-4 1.12x10-4 1.13x10-4 1.14x10-4 1.15x10-4 1.16x10-4

• 4
• 3
• 2
• 1

1 2 3 4 <|T12(ω, 0)|2> Imaginary frequency ω Matsubara Nτ = 8 Nτ = 16 Nτ = 32 Nτ = 64 1x10-9 1x10-8 1x10-7 1x10-6 1x10-5 0.0001 0.001 2 4 6 8 10 12 14 16 <T12(τ, 0) T12(0, 0)> τ Matsubara Nτ = 8 Nτ = 16 Nτ = 32 Nτ = 64

Lattice sizes: 43 × 8 (Matsubara) 43 × Nτ (General frequencies) β 1.8 Ncf 8 × 105 configurations

29

EMT correlator (confined phase)

9x10-5 1x10-4 1.1x10-4 1.12x10-4 1.13x10-4 1.14x10-4 1.15x10-4 1.16x10-4

• 4
• 3
• 2
• 1

1 2 3 4 <|T12(ω, 0)|2> Imaginary frequency ω Matsubara Nτ = 8 Nτ = 16 Nτ = 32 Nτ = 64 Nτ = 128 1x10-10 1x10-9 1x10-8 1x10-7 1x10-6 1x10-5 0.0001 0.001 2 4 6 8 10 12 14 16 <T12(τ, 0) T12(0, 0)> τ Matsubara Nτ = 8 Nτ = 16 Nτ = 32 Nτ = 64 Nτ = 128

Lattice sizes: 43 × 8 (Matsubara) 43 × Nτ (General frequencies) β 1.8 Ncf 8 × 105 configurations

30

EMT correlator (deconfined phase)

2x10-5 3x10-5 4x10-5 5x10-5 6x10-5 7x10-5 8x10-5 9x10-5 10-4

• 3
• 2
• 1

1 2 3 4 <|T12(ω, 0)|2> Imaginary frequency ω Matsubara Nτ = 8 Nτ = 16 Nτ = 32 Nτ = 64 10-10 10-9 10-8 10-7 10-6 10-5 10-4 2 4 6 8 10 12 14 16 <T12(τ, 0) T12(0, 0)> τ Matsubara Nτ = 8 Nτ = 16 Nτ = 32 Nτ = 64

Lattice sizes: 43 × 8 (Matsubara) 43 × Nτ (General frequencies) β 3.0 Ncf 1.6 × 106 configurations

31

EMT correlator (deconfined phase) and spectral function

4x10-6 6x10-6 8x10-6 1x10-5 1.2x10-5

• 4
• 3
• 2
• 1

1 2 3 4 <|Txy(ω, 0)|2> ω Nτ aτ / (2 π) Matsubara Nτ = 8 General freq. Nτ = 64 1x10-6 2x10-6 3x10-6 4x10-6 5x10-6 6x10-6 7x10-6 8x10-6 9x10-6 2 4 6 8 10 a3ρ(µ) / µ aµ m = 10-1 m = 10-2 m = 10-3 m = 10-4 m = 10-5 m = 10-6

PRELIMINARY

Lattice sizes: 83 × 8 (Matsubara) 83 × 64 (General frequencies) β 2.8 Ncf ≈ 106 configurations

Pawlowski, Rothkopf, Ziegler, work in progress

32

EMT correlator spectral function

1x10-6 2x10-6 3x10-6 4x10-6 5x10-6 6x10-6 7x10-6 8x10-6 9x10-6 2 4 6 8 10 a3ρ(µ) / µ aµ m = 10-1 m = 10-2 m = 10-3 m = 10-4 m = 10-5 m = 10-6

PRELIMINARY

10-7 2x10-7 3x10-7 4x10-7 0.1 0.2 0.3 0.4 0.5 0.6 a3ρ(µ) / µ aµ m = 10-1 m = 10-2 m = 10-3 m = 10-4 m = 10-5 m = 10-6

PRELIMINARY

Lattice sizes: 83 × 8 (Matsubara) 83 × 64 (General frequencies) β 2.8 Ncf ≈ 106 configurations

Pawlowski, Rothkopf, Ziegler, work in progress

33

Summary

Thermal fields as initial-value problem formulated in an additional non-compact Euclidean time promising Numerical implementation provides significantly improved access to real-time spectral quantities Formalism easy to implement for gauge fields

34

Outlook

Near future: extract spectral functions and transport coefficients from the energy-momentum tensor correlator Extension to SU(3) gauge theory and full QCD (work in progress) Formal developments Resolving correlators at small momenta

35

Outlook

Near future: extract spectral functions and transport properties from the energy-momentum tensor correlator Extension to SU(3) gauge theory and full QCD (work in progress) Formal developments Resolving correlators at small momenta

Thank you very much for your attention!

36

EMT correlator - latest results

5x10-6 7.5x10-6 10-5 1.25x10-5

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 <|T12(ω, 0)|2> ω 83x64 323x64

PRELIMINARY

9.5x10-6 10-5 1.05x10-5 1.1x10-5 1.15x10-5

• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 <|T12(ω, 0)|2> ω 83x64 323x64

PRELIMINARY

Matsubara lattice sizes: 83 × 8 and 323 × 8 β 2.8 Ncf ≈ 106 configurations

Pawlowski, Rothkopf, Ziegler, work in progress

37

Fixed boundary conditions

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 50 100 150 200 250 FBC, Nτ = 8 FBC, Nτ = 16 FBC, Nτ = 32 FBC, Nτ = 64 FBC, Nτ = 128

38