[PPT] - Complex Langevin Dynamics in 1+1D QCD at finite densities SIGN PowerPoint Presentation

SLIDE 1

Complex Langevin Dynamics in 1+1D QCD at finite densities

SIGN workshop Sebastian Schmalzbauer supervisor: Jacques Bloch September 29th, 2015

SLIDE 2

1 1 Motivation

QCD phase diagram
many other systems also plagued by sign problem

⇒ find new methods to cure it

But why low dimensional QCD?
sign problem already present
study viability of the complex Langevin method
can compare with analytical results (0+1D)
r other methods (1+1D)

1 / 22

SLIDE 3

1 1 QCD at finite µ: Partition Function

partition function (1 flavour) after integrating over fermions

Z =

D[U] det D[U]e−SG[U]
staggered Dirac operator at chemical potential µ:

Dk,l = mδk,l+

d−1

ν=0

ηk,ν 2a

Uk,νeaµδν,0δk+ˆ

ν,l − U−1 k+ˆ ν,νe−aµδν,0δk−ˆ ν,l

,

with quark mass m, staggered phase η = ±1, antiperiodic boundary conditions in time direction and U ∈ SL(3, ❈)

chiral symmetry not explicitly broken for m = 0
µ = 0

det D ∈ ❘ µ = 0 det D ∈ ❈ ⇒ importance sampling not possible

2 / 22

SLIDE 4

1 1 Complex Langevin Dynamics

Langevin equation for Gell-Mann representation (λa)

(dUx,µ) U−1

x,µ = −

a

λa (Da,x,µS(U)dt + dwa,x,µ) , with independent Wiener increments dwa,x,µ and group derivative Da,x,µS(U) = ∂αS(eiαλaUx,µ)|α=0

discrete time evolution = SL(3, ❈) rotation

U

′

x,ν = eiλa

a(ǫKa,x,µ+√ǫηa,x,µ)Ux,ν
drift Ka,x,µ different for Euler, Runge-Kutta schemes1
gaussian noise

ηa,x,µ = 0 ηa,x,µηb,y,ν = 2δabδxyδµν

1Chang ’87, Batrouni et al. ’85, Bali et al. ’13

3 / 22

SLIDE 5

1 1 Equivalence to Fokker-Planck Equation

FP: real fields with complex probability
dx O(x)ρ(x; t) =
dxdy O(x + iy)P(x, y; t)

CL: complex fields with real probability expect correct expectation values as long as2

solution of FPE asymptotes to correct probability distribution

lim

t→∞ ρ(x, t)

→ det DeSG

boundary terms in partial integration step vanish
sufficient falloff of probability distribution
singular drifts (det D = 0) suppressed enough

⇒ gauge cooling

2Aarts, James, Seiler, Stamatescu ’10 ’11

Nagata, Nishimura, Shimasaki ’15

4 / 22

SLIDE 6

1 1 0+1D Gauge Cooling

Dirac determinant can be reduced to

det(D) ∝ det

eµ/TP + e−µ/TP−1 + 2 cosh(µc/T)
,

with Polyakov loop P = ΠtUt and effective mass aµc = arsinh(am)

reduce unitarity norm

||U|| =

x,µ

tr

P†P +
P†P

−1− 2

via SL(3, ❈) gauge trafos

P → GPG−1

diagonalizing P = maximal cooling

5 / 22

SLIDE 7

1 1 Equivalence to Eigenvalue Representation

diagonalizing P

= working in eigenvalue representation P =

  

eiφ1 eiφ2 e−iφ1−iφ2

  

φ1, φ2 ∈ ❈

gauge cooling not required
additional term in the action (and hence in the drift)

S = − log J − log det D, arising from the Haar measure J(φ1, φ2) = sin2

φ1 − φ2

2

sin2

2φ1 + φ2

2

sin2

φ1 + 2φ2

2

6 / 22

SLIDE 8

1 1 0+1D: Results

quark density nq chiral condensate Σ

0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3

quark density nq chemical potential µ/T analytic uncooled cooled

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.5 1 1.5 2 2.5 3

chiral condensate Σ chemical potential µ/T analytic uncooled cooled

5 10 15 20 0.5 1 1.5 2 2.5 3

|Δnq|/σ

3σ 5 10 15 20 0.5 1 1.5 2 2.5 3

|ΔΣ|/σ

3σ

uncooled results results have significant deviation3
cooled results look good, but not perfect for all µ
Polyakov loop looks fine for all µ

3analytic results: Bilic, Demeterfi ’88

7 / 22

SLIDE 9

1 1 General Effects of SL(3, ❈) Gauge Trafos

measurement of observables are invariant

O(P) = O(GPG−1)

drift term Ka and gauge cooling step commute

Ka(GPG−1)λa = Ka(P)GλaG−1

noise distribution ηa used in the CL step is not invariant4

ηaλa = ηaGλaG−1 ⇒ SL(3, ❈) gauge trafos and CL step do not commute ⇒ apply gauge cooling: different trajectories

4SU(3) gauge trafos leave the noise distributions invariant

8 / 22

SLIDE 10

1 1 0+1D: Effects of Gauge Cooling

effect on det D distribution: • compressed in imaginary direction

avoids the origin

20 20 40 60 80 100 120

Re(det D)

40 30 20 10 10 20 30 40

Im(det D)

⇒

20 20 40 60 80 100 120

Re(det D)

40 30 20 10 10 20 30 40

effect on observable distribution: broad ⇔ narrow distribution

10-5 10-4 10-3 10-2 10-1 100 101

4
2

2 4

distribution chiral condensate Σ Σ = 0.1010(4) Σ = 0.08315(9) Σ = 0.0831 uncooled cooled

10-5 10-4 10-3 10-2 10-1 100

6
4
2

2 4 6

distribution Polyakov loop tr P tr P = 0.647(1) tr P = 0.6587(8) tr P = 0.6590 uncooled cooled

9 / 22

SLIDE 11

1 1 From 0+1D to dD

lattice volume V = NtNd−1

s

size of Dirac operator increases: 3V × 3V
time to compute D−1: ∝ V 3
general gauge trafos

GUG−1 → GxUx,νG−1

x+ˆ ν

⇒ gauge cooling via diagonalizing no longer works!

can include gauge action SG

⇒ extra drift term

we worked in 1+1D with strong coupling on 4 × 4 lattices

10 / 22

SLIDE 12

1 1 Gauge Cooling in Higher Dimensions

unitarity norm ||U|| =

x,µ tr

U†

x,µUx,µ +

U†

x,µUx,µ

−1− 2

minimize via SL(3, ❈) gauge trafos:

Ux,µ → GxUx,µG−1

x+ˆ µ

using steepest descent

10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2 100 100000 200000 300000 400000 500000

unitarity norm ||U|| Langevin time µ=0 cooled uncooled

⇒ even for µ = 0 we need gauge cooling

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 50000 100000 150000 200000 250000 300000

unitarity norm ||U|| Langevin time µ=0.11 uncooled µ=0.25 uncooled µ=0.25 cooled µ=0.11 cooled

distance from SU(3) depends

n µ

11 / 22

SLIDE 13

1 1 Convergence with Stepsize

interested in continuum solution ǫ → 0

µ = 0.07

1.6 1.8 2 2.2 2.4 10-4 10-3 10-2

chiral condensate Σ stepsize ε subsets: 2.22(1) convergence ∝ ε Euler uncooled Runge-Kutta uncooled Euler cooled Runge-Kutta cooled

µ = 0.25

0.8 1 1.2 1.4 1.6 1.8 2 10-4 10-3 10-2

chiral condensate Σ stepsize ε subsets: 2.02(2) convergence ∝ ε Euler uncooled Runge-Kutta uncooled Euler cooled Runge-Kutta cooled

uncooled simulation is stable, but converges to a wrong limit
for some µ even the cooled simulation does not converge

correctly (benchmark: subset method5)

5J. Bloch, F. Bruckmann, T. Wettig, ’12 ’14

12 / 22

SLIDE 14

1 1 1+1D Results: Chiral Condensate

0.5 1 1.5 2 2.5 0.5 1 1.5 2

chiral condensate Σ chemical potential µ m=0.01 m=0.1 m=0.5 m=1.0 m=2.0 uncooled cooled subsets

13 / 22

SLIDE 15

1 1 1+1D Results: Quark Density

0.5 1 1.5 2 2.5 3 3.5 0.5 1 1.5 2

quark density nq chemical potential µ m=0.1 m=0.5 m=1.0 m=2.0 uncooled cooled subsets

14 / 22

SLIDE 16

1 1 1+1D results: Polyakov Loop

0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.5 1 1.5 2 2.5

Polyakov loop tr P chemical potential µ m=0.1 m=0.5 m=1.0 m=2.0 uncooled cooled subsets

15 / 22

SLIDE 17

1 1 Effect of Gauge Cooling on det D (m=0.1)

µ = 0.07

0.5 0.0 0.5 1.0 1.5

Re(det D)

1e 7 1.0 0.5 0.0 0.5 1.0

Im(det D)

1e 7

⇒

0.5 0.0 0.5 1.0 1.5

Re(det D)

1e 7 1.0 0.5 0.0 0.5 1.0 1e 7

µ = 0.25

1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5

Re(det D)

1e 7 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0

Im(det D)

1e 7

⇒

1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5

Re(det D)

1e 7 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 1e 7

⇒ cooling not enough in some µ ranges for light quarks!

16 / 22

SLIDE 18

1 1 Effect of Gauge cooling on det D (m=2.0)

µ = 1.25

0.5 0.0 0.5 1.0 1.5

Re(det D)

1e17 1.0 0.5 0.0 0.5 1.0

Im(det D)

1e17

⇒

0.5 0.0 0.5 1.0 1.5

Re(det D)

1e17 1.0 0.5 0.0 0.5 1.0 1e17

µ = 1.50

0.5 0.0 0.5 1.0 1.5 2.0 2.5

Re(det D)

1e18 2 1 1 2

Im(det D)

1e17

⇒

0.5 0.0 0.5 1.0 1.5 2.0 2.5

Re(det D)

1e18 2 1 1 2 1e17

⇒ cooling works in all µ ranges for heavy quarks!

17 / 22

SLIDE 19

1 1 1+1D Distribution of Observables (m=0.1)

chiral condensate

10-6 10-5 10-4 10-3 10-2 10-1 100 101

20
15
10
5

5 10 15 20

distribution chiral condensate Σ Σ=2.058(3) Σ=2.232(2) Σ=2.23(1) µ=0.07 uncooled cooled

10-6 10-5 10-4 10-3 10-2 10-1 100

20
15
10
5

5 10 15 20

distribution chiral condensate Σ Σ=1.931(3) Σ=2.177(3) Σ=2.20(1) µ=0.15 uncooled cooled

10-6 10-5 10-4 10-3 10-2 10-1 100

20
15
10
5

5 10 15 20

distribution chiral condensate Σ Σ=1.597(3) Σ=1.801(3) Σ=2.10(1) µ=0.23 uncooled cooled

cooled distribution adjusts to uncooled “skirts” as µ increases

Polyakov loop

10-5 10-4 10-3 10-2 10-1 100 101

4
2

2 4

distribution Polyakov loop tr P tr P=0.100(4) tr P=0.093(4) tr P=0.100(7) µ=0.07 uncooled cooled

10-5 10-4 10-3 10-2 10-1 100 101

4
2

2 4

distribution Polyakov loop tr P tr P=0.107(4) tr P=0.093(4) tr P=0.090(4) µ=0.15 uncooled cooled

10-5 10-4 10-3 10-2 10-1 100

4
2

2 4

distribution Polyakov loop tr P tr P=0.145(4) tr P=0.133(4) tr P=0.134(7) µ=0.23 uncooled cooled

no “skirts” noticeable, but distributions also seem to converge

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SLIDE 20

1 1 Other Cooling Schemes

general approach: minimization of a specific norm
idea6: try other norms
anti-hermitian norm

|| ¯ H|| = tr

D + D†2

= 1 2a2

x,ν

tr

U†

x,νUx,νe2aµδ0,ν + (U† x,νUx,ν)−1e−2aµδ0,ν − 2

note similarity to unitarity norm

||U|| =

x,ν

tr

U†

x,νUx,ν +

U†

x,νUx,ν

−1− 2

⇒ “Fireball” plots look the same

6Nagata et al., lattice2015

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SLIDE 21

1 1 Effect of Antihermitian Cooling on det D

µ = 0.07

0.5 0.0 0.5 1.0 1.5

Re(det D)

1e 7 1.0 0.5 0.0 0.5 1.0

Im(det D)

1e 7

⇒

0.5 0.0 0.5 1.0 1.5

Re(det D)

1e 7 1.0 0.5 0.0 0.5 1.0 1e 7

µ = 0.25

1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5

Re(det D)

1e 7 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0

Im(det D)

1e 7

⇒

1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5

Re(det D)

1e 7 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 1e 7

⇒ same as gauge cooling with unitarity norm!

20 / 22

SLIDE 22

1 1 Other Cooling Schemes

improved version of the anti-hermitian norm

exp(−ξ|| ¯ H|| + const.)

need adaptive cooling algorithm
reproduces usual cooled results for ξ ∈ [−100, −0.1]
the bigger ξ, the broader the distribution

⇒ even worse results than in the uncooled case possible

Polyakov norm (as in 0+1D gauge cooling)

||P|| =

x

tr

P†
xP

x + (P†

xP

x)−1 − 2

only affects links in time direction at t = 0
need adaptive cooling scheme
does not change distribution (results similar to uncooled)

21 / 22

SLIDE 23

1 1 Conclusions and Outlook

no systematic runaways
0+1D:
discrepancies in Gell-Mann representation
improvement using gauge cooling / eigenvalue representation
1+1D:
incorrect results without gauge cooling
gauge cooling: correct results for some m, µ ranges
for light quarks: cooling does not help enough for some µ
gauge cooling necessary but not sufficient for correct results
signal for wrong convergence:
skirts in distribution of some observables
distribution of det D contains origin
include gauge action: improved convergence?
find new norm and study the effect of its minimization on

det D

22 / 22

SLIDE 24

1 1 Benchmark: Subset Method

no analytic results for d > 1

⇒ need to compare with other methods

idea: gather configurations of the ensemble such that the sum
f their complex weights is real and positive7
subset = Z3 rotations and complex conjugation

ΩP =

P, e2πi/3P, e4πi/3P, P∗, e2πi/3P∗, e4πi/3P∗
subset construction in higher dimensions: 3N rotations + c.c.
7J. Bloch, F. Bruckmann, T. Wettig, ’12 ’14

22 / 22

SLIDE 25

1 1 0+1D: Convergence with Stepsize

2.4 2.45 2.5 2.55 2.6 10-3 10-2 10-1

quark density nq stepsize ε analytical Euler Runge-Kutta RK (Bali) 22 / 22

SLIDE 26

1 1 0+1D: Correlation between Observables

skirts when unitarity norm is big skirt-branchcut8correlation

8Mollgaard, Splittorff ’13 ’14

22 / 22

SLIDE 27

1 1 Contribution of Skirts

10-6 10-5 10-4 10-3 10-2 10-1 100 101

20
15
10
5

5 10 15 20

ccurrence

chiral condensate Σ Σ

Σ = 0.172(5)

skirts ~ 0.0471 0.24 (x-1.71)-3.39 0.08 (2.72-x)-2.95 uncooled cooled fit ∝ |x-a|b

⇒“skirts” in distribution not enough to explain discrepancies,

nly a sign for wrong convergence

22 / 22

SLIDE 28

1 1 Optimizing Gauge Cooling

free parameter α
vary α, test adaptive algoritm

1e-5 1e-4 1e-3 0.01 0.1

20
15
10
5

5 10 15 20 Histogram of Re(Σ) Σ=1.931(3) Σ=2.157(3) Σ=2.177(3) Σ=2.179(3) Σ=2.179(3) Σ=2.175(3) Σ=2.179(3) Σ=2.181(3) Σ=2.20(1) (subsets) α=0.0 α=0.01 α=0.1 α=1.0 α=10.0 α=0.1adaptive α=1.0adaptive α=10.0adaptive 1e-5 1e-4 1e-3 0.01 0.1

20
15
10
5

5 10 15 20 Histogram of Re(Σ) Σ=1.461(3) Σ=1.521(3) Σ=1.603(3) Σ=1.640(3) Σ=1.651(3) Σ=1.652(3) Σ=1.648(3) Σ=2.03(1) (subsets) α=0.0 α=0.001 α=0.01 α=0.1 α=1.0 α=10.0 α=adaptive