[PPT] - Positivity of center subsets for QCD Mkiang whteigs pitoisve Falk PowerPoint Presentation

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Positivity of center subsets for QCD

Mkiang whteigs pitoisve Falk Bruckmann (Regensburg University)

SIGN 2015, Atomki/Debrecen, Oct. 2015

with Jacques Bloch 1508.03522

Falk Bruckmann Positivity of center subsets for QCD 0 / 21

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Introduction

QCD at high density/with chemical potential µ has a sign problem: path integral weight det / D ≥ 0 ⇒ no importance sampling ‘subsets’: add up configurations until the weight is positive works in: random matrix model with µ

Bloch 11

fugacity expansion

Bloch, FB, Kieburg, Splittorff, Verbaarschot 12

lattice QCD in 0+1 dimensions

Bloch, FB, Wettig 13

no gauge action, solvable

(Bilic, Demeterfi 88, Ravagli, Verbaarschot 07)

here: higher-dimensional lattices analytical & numerical evidence for positivity of subsets

Falk Bruckmann Positivity of center subsets for QCD 1 / 21

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Subset idea: an example

‘partition function’: (from the random matrix model) Z = ∞

−∞

dx f(x) , f(x) = e−x2 1 + x2 − cosh(µ) x

≶ 0

2 1 1 2 x 0.5 0.5 1.0 1.5 2.0

fx

Falk Bruckmann Positivity of center subsets for QCD 2 / 21

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Subset idea: an example

‘partition function’: (from the random matrix model) Z = ∞

−∞

dx f(x) , f(x) = e−x2 1 + x2 − cosh(µ) x

≶ 0

2 1 1 2 x 0.5 0.5 1.0 1.5 2.0

fx

subsets: add values at x and −x = integrate over even part Z = ∞

−∞

dx 1 2

f(x) + f(−x)
e−x2(1 + x2) > 0

Falk Bruckmann Positivity of center subsets for QCD 2 / 21

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Subsets in a random matrix model

partition function for Φ1,2 complex N × N matrices

Osborn ’04

Z(µ) =

dΦ1,2 e−N Tr (Φ1Φ†

1+Φ2Φ† 2) det

m

e µΦ1 − e−µΦ†

2

−e−µΦ†

1 + e µΦ2

m

subsets: measure and Gaussian invariant under Φ1,2 → eiθ Φ1,2

determinant changes (not a kind of gauge trafo!) add up the weight of those rotated matrices but before that: interpret the rotation as an imag. µ Z(µ) =

dΦ1,2 .. det
m

e µeiθΦ1 − e−µe−iθΦ†

2

−e−µe−iθΦ†

1 + e µeiθΦ2

m

= Z(µ+iθ)

∀θ = 1 n

n−1

k=0

Z(µ+ik/n)

Falk Bruckmann Positivity of center subsets for QCD 3 / 21

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fugacity expansion: Z(µ) =

2N

q=−2N

e qµZq Zq: canonical partition functions range of q from det being a polynomial in e ±µ of order 2N put together: Z(µ) =

2N

q=−2N

e qµ 1 n

n−1

k=0

e 2πik/n·q

δq mod n, 0

Zq sum of nth roots (and most of their powers) vanish choose n > 2N: Z(µ) = e0Z0 = Z(µ = 0) “no mu, no cry” actually, the µ-dep. disappears on the level of the integrand ⇒ the integrand is that at µ = 0 and thus is positive

Falk Bruckmann Positivity of center subsets for QCD 4 / 21

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Subset definition in lattice QCD

add up configurations ← multiplication of links with group elements (not gauge trafos!) formally using the invariance of the group measure [Haar] Z =

DU w[U ] =
SU(3)
µ,x

d

Uµ(x)
w[U ]

Falk Bruckmann Positivity of center subsets for QCD 5 / 21

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Subset definition in lattice QCD

add up configurations ← multiplication of links with group elements (not gauge trafos!) formally using the invariance of the group measure [Haar] Z =

DU w[U ] =
SU(3)
µ,x

d

Vµ(x)−1Uµ(x)
w[U ] =
DU w[VU ]

Falk Bruckmann Positivity of center subsets for QCD 5 / 21

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Subset definition in lattice QCD

add up configurations ← multiplication of links with group elements (not gauge trafos!) formally using the invariance of the group measure [Haar] Z =

DU w[U ] =
SU(3)
µ,x

d

Vµ(x)−1Uµ(x)
w[U ] =
DU w[VU ]
cf. group U(1):
U(1)

dU U=eiφ = 1 2π 2π dφ = 1 2π 2π d(−φV + φ) =

U(1)

d(V −1U) (translation invariance on the circle)

Falk Bruckmann Positivity of center subsets for QCD 5 / 21

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Center subsets in lattice QCD

add up configurations ← multiplication of links with center elements center Z3: g = e 2πik/3 13 with k = 0, 1, 2 commutes with all SU(3) group elements center subsets: Z =

DU w[gU ]

g ∈ Z3 ⊗ Z3 ⊗ . . . =

DU

1 #g

g

w[gU ]

≡ σ(U)

?

> 0 if the subset weight σ is positive one can use importance sampling even if the original weight w was not positive

Falk Bruckmann Positivity of center subsets for QCD 6 / 21

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bservables can be measured, too

alternative view: part of the integral = center sum performed deterministically, while remaining part = integral over the coset subject to sampling why center? gauge theories w/o center (SU(N)adj, G2) have no sign problem confinement criterion on Polyakov loop: trP → g trP (approx.) preserved/broken at low/high T (and µ?) technically easy seemingly sufficient below: connected to imag. µ and diagrammatics similarity to clusters in Pott model

Alford, Chandrasekharan, Cox, Wiese 01

Falk Bruckmann Positivity of center subsets for QCD 7 / 21

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Center subsets in 0+1 dimensions

no plaquette ⇒ w = det / D temporal links can be gauged to a single link ← Polyakov loop P

riginal weight, m = 0, one flavor:

w(P) = det

3×3(1 + e µ/TP)(1 + e −µ/TP†)

= e−3µ/T + e−2µ/T 2 tr P + e−µ/T 2 tr P† + (tr P)2 + e 3µ/T + e 2µ/T 2 tr P† + e µ/T 2 tr P + (tr P†)2 + 2 + 2 |tr P|2 > 0 (Re w matters, still > 0) center subsets: 1 3

w(P)+w(e2πi/3P)+w(e−2πi/3P)
= e−3µ/T +e 3µ/T +2+2 |tr P|2 > 0

positive, used for importance sampling

nly baryon chemical potential µB = 3µ enters (see below)

Falk Bruckmann Positivity of center subsets for QCD 8 / 21

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Collective subsets and imag. µ

lattice implementation of µ on one (say last) time slice: e µ/TU0, e−µ/TU† ‘collective subsets’: same center element, only on that time slice: e µ/TU0 → e µ/T · 1 3

2

k=0

e 2πik/3U0 = 1 3

2

k=0

e µ/T+2πik/3 · U0 & analogously for U† det / Dµ/T → 1 3

2

k=0

det / Dµ/T+2πik/3 & plaquette unchanged = adding/averaging 3 complex µ’s = Roberge-Weiss symmetry (on integrals) utilized on integrands

Falk Bruckmann Positivity of center subsets for QCD 9 / 21

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fugacity expansion: det / Dµ/T =

q

e qµ/T / Dq →

q

e qµ/T 1 3

2

k=0

e 2πik/3·q

δq mod 3, 0

/ Dq nonzero triality terms removed from the path integrand; the corresponding integrals = canonical partition functions vanish anyhow expansion in 3µ = µB these collective subsets attenuate the sign problem, but in general do not solve it ⇒ independent center multiplications on all temporal links

Falk Bruckmann Positivity of center subsets for QCD 10 / 21

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Positivity of center subsets: numerical evidence

1+1 dimensional QCD

ne (unrooted) staggered quark

µ = 0.3, m = 0 (sign problem worst) no plaquette = strong coupling approx. reweighting factors on 100,000 configurations (* means 1,000):

Nt × Nx 2 × 2 4 × 2 6 × 2 8 × 2 10 × 2 phase-quenched 0.8134(3) 0.4361(4) 0.233(2) 0.130(2) 0.071(1) sign-quenched 0.9271(2) 0.6150(5) 0.355(3) 0.203(2) 0.109(2) collective subsets 0.9778(9) 0.777(4) 0.500(6) 0.303(8) 0.178(3) full subsets 1.0 1.0 1.0∗ 1.0∗ 1.0∗ Nt × Nx 2 × 4 4 × 4 6 × 4 2 × 6 2 × 8 pq 0.7934(5) 0.295(1) 0.0961(9) 0.7364(6) 0.6725(7) sq 0.9197(3) 0.442(2) 0.149(1) 0.8917(4) 0.8523(5) collective 0.959(1) 0.557(6) 0.214(8) 0.912(3) 0.867(2) full 1.0∗ 1.0∗ 1.0∗ 1.0∗ 1.0∗

Falk Bruckmann Positivity of center subsets for QCD 11 / 21

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higher dimensions

n 200 random configurations:

full subsets give reweighting factors 1.0 for 23, 22×4 and 24 with gauge action β = 1, 2, 3, 4, 5 on 2×6 reweighting factors 1.0, 1.0, 0.984(7), 0.964(13), 0.972(17) = mild sign problem subsets can always be used to attenuate the sign problem reweighting factor has to increase (Cauchy-Schwarz)

r combined with other methods

Falk Bruckmann Positivity of center subsets for QCD 12 / 21

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A first physics result

quark number density as a function of µ and T (in lattice units)

Nx=2

0.1 0.2 0.3 0.4 0.5

T

0.1 0.2 0.3 0.4 0.5 0.6

µ

1 2 3

n

Nx=6

0.1 0.2 0.3 0.4 0.5

T

0.1 0.2 0.3 0.4 0.5 0.6

µ

1 2 3

n

0.5 1 1.5 2 2.5 3

(Nt = 2, . . . , 12 and Nt = 2, . . . , 8) Silver-Blaze property visible

Falk Bruckmann Positivity of center subsets for QCD 13 / 21

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Positivity of center subsets: analytical proof for Nt = 2

massless staggered determinant at µ = 0 the usual argument: / D is antihermitian and chiral∗ ⇒ eigenvalues in pairs ±i· real ⇒ det / Dµ=0 ≥ 0 chiral∗: anticommutes with η5, η5(xev,od) = ±1 argument using quarks as Grassmannians: det / Dµ=0 =

D ¯

ψDψ exp( ¯ ψ / Dµ=0ψ) and even-odd decomposition

Falk Bruckmann Positivity of center subsets for QCD 14 / 21

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D ¯

ψDψ exp( ¯ ψ / Dµ=0ψ) =

D ¯

ψevDψod · DψevD ¯ ψod ×

x,ν

exp ¯ ψ(x)aηνUν(x)abψ(x + ˆ ν)b

· exp
ψ(x)aηνU∗

ν(x)ab ¯

ψ(x + ˆ ν)b

f the two hoppings, one is even-odd and the other odd-even

separate the hoppings to the integrals accordingly integration variable change ψ ⇋ ¯ ψ: same integral up to U ⇌ U∗ p[Uev, U∗

d] · p[U∗

ev, Uod] = p[Uev, U∗

d] · p[Uev, U∗
d]∗ ≥ 0
since p is a polynomial in the links with real coefficients

actually, p = det / Deven-odd

Falk Bruckmann Positivity of center subsets for QCD 15 / 21

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massless determinant is positive at µ = 0 antiperiodic boundary conditions do not interfere similar argument with mass (mass terms just saturate some Grassmann integrals) violated by real µ’s, on ν = 0 ⇒ need subsets antiperiodic bc.s are actually essential [not discussed here]

Falk Bruckmann Positivity of center subsets for QCD 16 / 21

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subsets on Nt = 2: take the two points in the temporal direction together pecularity: U0(x) and U†

0(x + ˆ

0) connect the same sites = contain the same Grassmannians, but with inverse µ-factors subset weight of the form: σ =

D ¯

ψevenDψodd · DψevenD ¯ ψodd ×

x,i

exp ¯ ψ(x)ηiUi(x)ψ(x + ˆ ν)

· exp
ψ(x)ηiU∗

i (x) ¯

ψ(x + ˆ ν)

×
x
α

hα ¯ ψ(1, x), ψ(2, x); U0) · hα

ψ(1,

x), ¯ ψ(2, x); U∗

0)

again: in all terms, one factor is even-odd and the other odd-even ⇒ even-odd structure with Uν ⇌ U∗

ν ⇒ subset weight positive

µ tamed in hα=0:

1 − 2 cosh(µB) ¯

ψ(1, x)3ψ(2, x)3 ·

1 − 2 cosh(µB)ψ(1,

x)3 ¯ ψ(2, x)3

Falk Bruckmann Positivity of center subsets for QCD 17 / 21

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Center subsets and diagrammatics

fermion action and Grassmann character: exp(SF) ∼ exp

e µ ¯

ψ(x)U0(x)ψ(x + ˆ 0)

u
exp
e−µ ψ(x)U∗

0(x) ¯

ψ(x + ˆ 0)

¯

u

=

3

m,n=0

1 m!n!

e µu

m e−µ¯ u n (µ ≡ a µ) σ ∼ 1 + u¯ u + (u¯ u)2 2!2 + (u¯ u)3 3!2

mesonic

+ e 3µu 3 3! + e −3µ¯ u 3 3!

baryonic

+✭✭✭✭✭✭✭

✭

nonzero triality from 16 down to 6 hoppings = ‘building blocks’ per bond: (anti)baryons carry ±3µ = ±µB

Falk Bruckmann Positivity of center subsets for QCD 18 / 21

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same diagrams as if U integrated out completely

Rossi, Wolff 84, Karsch, Mütter 89

because center sums already remove the same nonzero triality hoppings as whole link integrals zero triality terms contain a singlet, e.g. u¯ u ∼ 3 × ¯ 3, u3 ∼ 3 × 3 × 3 some weights simplify: u3 ∼ ( ¯ ψUψ)3 ∼ ¯ ψ3

ǫ...

U.◦U.◦U.◦ ψ3

ǫ◦◦◦

∼ det U = 1 ⇒ (anti)baryons and 3-mesons are link-independent (again as if U integrated out) used in the proof above 1- and 2-mesons remain U-independent (U’s enter traces of closed Wilson loops ⇋ gauge invariance) ⇒ can measure e.g. Polyakov loops

Falk Bruckmann Positivity of center subsets for QCD 19 / 21

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Future plan: hybrid approach

subsets on all temporal links contain 3V summands ← exp. expensive ⇒ combine with diagrammatics in Nt = 2 proof the subset weights consists of many positive terms ⇒ which subset diagrams to add up to achieve positivity configurations: those diagrams constraint: at every site 3 arrows have to go in and out (representing mass terms ¯ ψ(x)ψ(x) as arrows returning to the same site) simulation: worm algorithm

use U-dependence to reweight gauge action back in

Falk Bruckmann Positivity of center subsets for QCD 20 / 21

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Summary and Outlook

adding up configurations obtained by center rotating the temporal links give a positive weight and thus solve the sign problem in 0+1d QCD Nt = 2 × any lattices in strong coupling ← analytical proof various lattices in strong coupling ← numerical evidence always positive? proof? subset diagrams: mesonic and baryonic hoppings in temporal direction (spatial subsets: same hoppings in spatial directions) hybrid approach beyond strong coupling = including the gauge action: numerical evidence: the sign problem might be mild, such that reweighting could be used check volume dependence!

Falk Bruckmann Positivity of center subsets for QCD 21 / 21