[PPT] - LATTICE Simulating low dimensional QCD 2016 on Lefschetz thimbles PowerPoint Presentation

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LATTICE

2016

Thimble A thimble is a bell or ring shaped sheath of a hard substance, such as bone, leather, metal or wood, which is worn on the tip or middle of a finger or the thumb to help push a needle while sewing and to protect the finger/thumb from being pricked.

Simulating low dimensional QCD

n Lefschetz thimbles

Christian Schmidt with Felix Ziesché

[source: Textile Research Centre (TRC), Leiden, The Netherlands]

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LATTICE

2016

C. Schmidt, Lattice 2016, Southampton, UK

2

Motivation: The QCD sign problem

The QCD partition function

Z(T, V, m, µ) = Z DU detM[U] | {z } e−SG[U]

complex for µ > 0

302

1. Barbour et al. / Simulations of lattice QCD

/d,a

(a)

0.3

*( 0.0

I

0.5

Re 5,

I

1.0

/.La

(b)

0.6

.< 0.0 0.5 1.0

Re 5,

Fig. 2. The distribution of eigenvalues )t of the Dirac matrix for staggered fermions at fl = 0. Shown are

the eigenvalues obtained from 6 random gauge configurations (fl = 0 quenched) on a 4 3 x g lattice for different values of the chemical potential, t* = 0.3 (a), 0,6 (b), 0.9 (c), and 1,2 (d).

[det M(µ)]∗ = det M(−µ∗)

standard MC techniques not applicable
highly oscillatory integral with

exponentially large cancellations Lattice Dirac spectrum

Barbour et al., 1986 Muroya et al., 2003

T = 0 T > 0 43 × 8 44

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LATTICE

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C. Schmidt, Lattice 2016, Southampton, UK

3

Idea: Deforming the domain of integration

Standard 1d-example: the Airy integral Ai[1] = 1 2π Z ∞

−∞

dx exp ⇢ i ✓x3 3 + x ◆

6
4
2

2 4 6

6
4
2

2 4 6 J1 real domain

1
0.5

0.5 1

6
4
2

2 4 6 Re[exp(-S)] along real domain 0.1 0.2 0.3 0.4 0.5 0.6

6
4
2

2 4 6 Re[exp(-S)] along J1

x → z = x + iy z numerical integration easy! numerical integration hopeless!

see also Witten: 1001.2933, 1009.6032

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LATTICE

2016

C. Schmidt, Lattice 2016, Southampton, UK

4

6
4
2

2 4 6

6
4
2

2 4 6 J1 real domain

x → z = x + iy z

see also Witten: 1001.2933, 1009.6032

Idea: Deforming the domain of integration

Standard 1d-example: the Airy integral Ai[1] = 1 2π Z ∞

−∞

dx exp ⇢ i ✓x3 3 + x ◆

use the real valued function

as a Morse function SR(z) = Re[−i(z3/3 + z)] Theory behind: Picard-Lefschetz theory

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LATTICE

2016

C. Schmidt, Lattice 2016, Southampton, UK

5 x → z = x + iy z

see also Witten: 1001.2933, 1009.6032

Idea: Deforming the domain of integration

Theory behind: Picard-Lefschetz theory

6
4
2

2 4 6

6
4
2

2 4 6 J1 real domain

σ1 σ2

find all separated saddle points ( )

Standard 1d-example: the Airy integral Ai[1] = 1 2π Z ∞

−∞

dx exp ⇢ i ✓x3 3 + x ◆

use the real valued function

as a Morse function SR(z) = Re[−i(z3/3 + z)] σi

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LATTICE

2016

C. Schmidt, Lattice 2016, Southampton, UK

6 x → z = x + iy z

see also Witten: 1001.2933, 1009.6032

Idea: Deforming the domain of integration

σ1 σ2

6
4
2

2 4 6

6
4
2

2 4 6 J1,J2 K1,K2 real domain

associated with each saddle point ( ),

find one stable ( ) and one unstable thimble ( ) as solutions of the steepest descent/ascent flow equation σi Ji Ki dz dt = ⌥rSR(z) J1 J2 K2 K1 Standard 1d-example: the Airy integral Ai[1] = 1 2π Z ∞

−∞

dx exp ⇢ i ✓x3 3 + x ◆

find all separated saddle points ( )

σi

use the real valued function

as a Morse function SR(z) = Re[−i(z3/3 + z)] (note: remains

const. along flow)

SI(z) Theory behind: Picard-Lefschetz theory

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LATTICE

2016

C. Schmidt, Lattice 2016, Southampton, UK

7 x → z = x + iy z

see also Witten: 1001.2933, 1009.6032

Idea: Deforming the domain of integration

σ1 σ2

6
4
2

2 4 6

6
4
2

2 4 6 J1,J2 K1,K2 real domain

associated with each saddle point ( ),

find one stable ( ) and one unstable thimble ( ) as solutions of the steepest descent/ascent flow equation σi Ji Ki dz dt = ⌥rSR(z) J1 J2 K2 K1 Standard 1d-example: the Airy integral Ai[1] = 1 2π Z ∞

−∞

dx exp ⇢ i ✓x3 3 + x ◆

decompose original integral into thimbles
find all separated saddle points ( )

σi

use the real valued function

as a Morse function SR(z) = Re[−i(z3/3 + z)] (note: remains

const. along flow)

SI(z) (here: ) n1 = 1, n2 = 0, SI(σ1) = 0 Theory behind: Picard-Lefschetz theory Z

R

dz e−S(z) = X

i

ni e−SI(σi) Z

Ji

dz e−SR(z)

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C. Schmidt, Lattice 2016, Southampton, UK

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Idea: Deforming the domain of integration

Original domain of integration Ux,ν ∈ SU(3) real dim. 4 × V × 8 ˜ Ux,ν ∈ SL(3, C) 4 × V × 8 × 2 real dim. Complexified space New domain(s) of integration: Lefschetz thimble U = exp ( −i X

a

ωaTa ) J0 := n ˜ Ux,ν | U(τ) is solution of the SD equation with U(0) = ˜ Ux,ν U(τ → ∞) = N and

N

here denotes the gauge orbit of the unity configuration real dim. 4 × V × 8 U 4V ˜ U 4V J0 + J1 + · · ·

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LATTICE

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C. Schmidt, Lattice 2016, Southampton, UK

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Open questions:

How many relevant thimbles are there in full QCD? How to sample them?

Langevin on the thimble (Aurora-algorithm)

Cristoforetti et al., PRD 86 (2012) 074506

HMC on the thimble

Fujii et al., JHEP 1310 (2013) 147

Use a map of the thimble (projection-, contraction-algorithm)
A. Mukherjee et al., PRD 88 (2013) 051502; A. Alexandru et. al., PRD 93 (2016) 014504
Sample SD paths on the thimble

Di Renzo et al., PRD 88 (2013) 051502

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LATTICE

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C. Schmidt, Lattice 2016, Southampton, UK

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Open questions:

How many relevant thimbles are there in full QCD? How to sample them? How to combine results from different thimbles?

input a number of physical quantities to determine relative weights

Xi = ⌦ eiφOi ↵

0 + α1

⌦ eiφOi ↵

1 + α2

⌦ eiφOi ↵

2

heiφi0 + α1 heiφi1 + α2 heiφi2 i = 1, 2 αi = nieSI(σi)Zi n0eSI(σ0)Z0 , ,

Di Renzo et al., PRD 88 (2013) 051502

here denotes the residual phase (see ) φ

Cristoforetti et al., PRD 89 (2014) 114505

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LATTICE

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C. Schmidt, Lattice 2016, Southampton, UK
1.5
1.0
0.5

0.0 0.5 1.0 1.5 0.60 0.65 0.70 0.75 Re

Im
T=0.01

T=0.05 T=0.5

1.0
0.5

0.0 0.5 1.0

4
2

2 4 Re ( )far SR

11

Open questions:

How many relevant thimbles are there in full QCD? How to sample them? How to combine results from different thimbles?

input a number of physical quantities to determine relative weights
sample multiple thimbles at once, or one manifold that comes

arbitrary close to multiple thimbles

A. Alexandru et. al., JHEP 1605 (2016) 053

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LATTICE

2016

C. Schmidt, Lattice 2016, Southampton, UK

12

Open questions:

How many relevant thimbles are there in full QCD? How to sample them? How to combine results from different thimbles? How to deal with the gauge orbits?

perform simulations in a fixed gauge
make use of the gauge gauge transformations

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C. Schmidt, Lattice 2016, Southampton, UK

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Systems studied so far:

theory

φ4

Cristoforetti et al., PRD 88 (2013) 051501; Fujii et al., JHEP 1310 (2013) 147

Hubbard model, one-site Hubbard model

A. Mukherjee et al., PRD 88 (2013) 051502

Cristoforetti et al., PRD 89 (2014) 114505

(0+1)dim. Thirring model

Fujii et al., JHEP 1511 (2015) 078; Fujii et al., JHEP 1512 (2015) 125;

Chiral random matrix model

Di Renzo et al., PRD 88 (2013) 051502

. . .

(also applications to QM-systems in real time)

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C. Schmidt, Lattice 2016, Southampton, UK

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Agenda:

QCD in (0+1) dim. with std. staggered quarks

simulations in Polyakov loop diagonal form
simulations with a general Polyakov loop

QCD in (n+1) dim. with std. staggered quarks

simulations at strong coupling
simulations away from strong coupling

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2016

C. Schmidt, Lattice 2016, Southampton, UK

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Agenda:

QCD in (0+1) dim. with std. staggered quarks

simulations in Polyakov loop diagonal form
simulations with a general Polyakov loop

QCD in (n+1) dim. with std. staggered quark

simulations at strong coupling
simulations away from strong coupling

}

this talk :-)

}

not yet :-(

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LATTICE

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C. Schmidt, Lattice 2016, Southampton, UK

16

(0+1) dimensional QCD

partition function in the reduced form diagonalize Polyakov loop P = diag(eiθ1, eiθ2, e−i(θ1+θ2)) J(θ1, θ2) = 8 3π2 sin2 ✓θ1 − θ2 2 ◆ sin2 ✓2θ1 + θ2 2 ◆ sin2 ✓θ1 + 2θ2 2 ◆ Z(Nf ) = Z dθ1dθ2 e−Seff (Nf ,θ1,θ2)

Ammon et al., arXiv:1607.05027

(see e.g. )

Bilic et al. Phys. Lett. B212 (1988) 83

Seff = −(ln J + Tr ln D) Z(Nf ) = Z dP detNf [A + eµ/T P + e−µ/T P −1] | {z } D A = 2 cosh(ˆ µc)13 ˆ µc = arcsinh( ˆ m)

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C. Schmidt, Lattice 2016, Southampton, UK

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(0+1) dimensional QCD

find saddle points: minimize , with

4
2

2 4

4
2

2 4 Re(1) Re(2)

µ/T = 0 m/T = 0.2

0.05 0.1 0.15 0.2 1 2 3 4 5 6 µ/T 0.1 · Re(1) Im(1)

0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01

0.01 1 2 3 4 5 6 µ/T Re(2) Im(2)

thimbles are separated by lines of zero probability (infinite action)
saddle points are -dependent

µ

all thimbles are equivalent (give the same contribution)

||rzSR

eff||

z = (Reθ1, Reθ2, Imθ1, Imθ2)t

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C. Schmidt, Lattice 2016, Southampton, UK

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(0+1) dimensional QCD

find tangent space of the thimble at the saddle points: diagonalize hessian (at the saddle point) ∂zi∂zjSR

eff

eigenvectors with positive eigenvalues span the tangent space

sample the thimble using the contraction algorithm ( )

A. Alexandru et. al., PRD 93 (2016) 014504

< O >= Z dz O(z)e−Seff (z) = Z d¯ z detJ O(z(¯ z)) e−Seff (z(¯

z))

are elements of the tangent space

¯ z

is defined by flowing along the SA for a fixed time T (note: the SA flow is

numerically stable) z(¯ z) ¯ z

is the Jacobian, which is in practice obtained by transporting a

P (z) parallelepipet along the SA flow: detJ = detP (z(¯ z))/detP (¯ z) Jij = ∂zi/∂¯ zj

has a complex phase (residual phase), sample according to |detJ|e−SR

eff

detJ and take the residual phase into account by reweighting

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LATTICE

2016

C. Schmidt, Lattice 2016, Southampton, UK

19

(0+1) dimensional QCD

find tangent space of the thimble at the saddle points: diagonalize hessian (at the saddle point) ∂zi∂zjSR

eff

eigenvectors with positive eigenvalues span the tangent space

sample the thimble using the contraction algorithm ( )

A. Alexandru et. al., PRD 93 (2016) 014504
1
0.5

0.5 1 1 1.5 2 2.5

uter points

inner points

0.1
0.05

0.05 0.1 1.5 1.52 1.54 1.56 1.58 1.6 1.62 1.64 1.66 1.68 1.7

uter points

inner points

T = 0.7 flow time:

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C. Schmidt, Lattice 2016, Southampton, UK

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(0+1) dimensional QCD

results for the Polyakov loop:

exact results are reproduced
only one relevant thimble found

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.5 1 1.5 2 2.5 3 Re Tr[P] µ/T exact thimble-MC

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LATTICE

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C. Schmidt, Lattice 2016, Southampton, UK

21 sample non-diagonal Polyakov loops fist step: find saddle points (now in 16 dim.)

5

5 10 15 20

25
20
15
10
5

5 10 15 20 25 m=0.0

5
4
3
2
1

1 2 3

25
20
15
10
5

5 10 15 20 25 m=0.1

(0+1) dimensional QCD

µ/T = 0 SR

eff

SR

eff

find 3 thimbles, related to Z(3) symmetry

Re ω8 Re ω8 P = exp ( −i X

a

ωaTa )

at m=0, the thimbles are separated by singular points

SLIDE 22

LATTICE

2016

C. Schmidt, Lattice 2016, Southampton, UK

22 sample non-diagonal Polyakov loops fist step: find saddle points (now in 16 dim.)

(0+1) dimensional QCD

P = exp ( −i X

a

ωaTa )

4
2

2 4 6 8

20
15
10
5

5 10 15 20 mu=0.03

4
2

2 4 6 8

20
15
10
5

5 10 15 20 mu=0.05

4
2

2 4 6 8

20
15
10
5

5 10 15 20 mu=0.08

am = 0.1

4
2

2 4 6 8

20
15
10
5

5 10 15 20 mu=0.1

4
2

2 4 6 8

20
15
10
5

5 10 15 20 mu=0.13

4
2

2 4 6 8

20
15
10
5

5 10 15 20 mu=0.15

Re ω8 Re ω8 Re ω8 Re ω8 Re ω8 Re ω8 SR

eff

SR

eff

SR

eff

SR

eff

SR

eff

SR

eff

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C. Schmidt, Lattice 2016, Southampton, UK

23

find 3 thimbles, related to Z(3) symmetry
at , the thimbles are separated by singular points

sample non-diagonal Polyakov loops fist step: find saddle points (now in 16 dim.) P = exp ( −i X

a

ωaTa )

at , the thimbles are separated by singular points

m = µc m = 0

saddle points are not -dependent

µ

(0+1) dimensional QCD

second step: diagonalize the hessian

implementation is work in progress ...

∂a∂bSR

eff = Tr

⇥ D−1∂a∂bD ⇤ − Tr ⇥ D−1(∂aD)D−1(∂bD) ⇤

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Summary

still many open question in the Lefschetz thimble approach that

need to be clarified before it can be applied to full QCD

(0+1) dimensional QCD is doable (at least in the reduced case)
(n+1) dimensional QCD will be the next